Indian Statistical Institute

Transcription

Indian Statistical Institute
Completeness, Similar Regions, and Unbiased Estimation: Part I
Author(s): E. L. Lehmann and Henry Scheffé
Source: Sankhyā: The Indian Journal of Statistics (1933-1960), Vol. 10, No. 4 (Nov., 1950), pp.
305-340
Published by: Springer on behalf of the Indian Statistical Institute
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SANKHY?
THE
INDIAN JOURNAL OF
STATISTICS
Edited By : P. C. MAHALANOBIS
10
Vol.
4
1950
Part
COMPLETENESS, SIMILAR REGIONS, AND UNBIASED
I
ESTIMATION-PART
By
E. L. LEHMANN
of California
University
AND
HENRY SCHEFF?
Columbia
1.
University
Introduction
is the study of two classical problems
aim of this paper
of mathematical
of
of
similar
and
estimation.
The reason
unbiased
the
statistics,
regions
problems
is that both are concerned with a family of
these two problems
for studying
together
on this family
same
the
measures
condition
insures a very simple
and that essentially
The
solution
of both.
of similar region and unbiased
concepts
an early stage in the development
of statistical
theory,
seems
rather difficult
On the other hand it
ly fruitful.
of statistics.
systematic
development
The
defined
regions were
of testing
with the problem
composite
H that a random
variable
hypothesis
to
is
distributed
x,
according
points
=
is
If the hypothesis
?jpx
{P*\?eo>}.
the
condition
cal region A must
satisfy
Similar
Pe
Neyman
and Pearson
In a number
replaced
of important
by Neyman
estimate
and
were
both have
to justify
and Pearson
proved extreme
either in a completely
(1933) in connection
one wishes
to test the
hypotheses.
Suppose
on values
in a "sample
X,
space" of
taking
a family
some probability
of
distribution
Pex
to be tested at level of significance
a, the criti
for all 6 in o>.
(A) < a
...
(1.1)
of similarity
this by the much
condition
stronger
= oc
...
for all 6 in <o.
(A)
Pf
cases
at
introduced
the problem
reduces
by this device
305
to that
(1.2)
of testing
Vol.
SANKHY?: THE INDIAN JOURNAL OF STATISTICS
10]
a simple
and furthermore
hypothesis,
is either uniformly most powerful
which
1942: Lehmann,
all
(among
alternative
situations
cases the most
on
those
test
powerful
the specific
very
satisfying
(1.1)) frequently
strongly
depends
at which
In these
the power
is maximized
and
Stein,
1948).
(Lehmann
therefore
the simplifying
of
value
the restriction
(1.2) is considerable.
Somewhat
a
again
valued
the similar regions one
among
most powerful
unbiased
(Seheff?,
or uniformly
in the same
hand,
the other
On
1947).
exists
there
4
[Part
family
function
remarks
?
3?x
analogous
of measures
A reasonable
g(6).
the risk for some given weight
hold
{PBX
estimate
Given
for the problem
of point estimation.
some
real
to
is
it
estimate
desired
\Ge?},
say, which
function;
seem
T would
to be one which
...
E9[T-g(6)f
the expected
E9 denotes
will depend on the value
one clearly chooses T = g(60).
where
mate
value
(1.3)
this esti
Unfortunately,
P9*.
is d0
If
value
is
this
minimized.
(1.3)
this difficulty would be to replace
avoiding
calculated
of 6 for which
One way
minimizes
minimizes
of
with
(1.3) by
...
sup ?9[T-g(e)?.
e
in restricting
consists
approach
condition
of unbiasedness
appealing
the class of estimates
Another
EQ(T)
=
(1.4)
by the
...
for all 0in?.
g(6)
rather
intuitively
(1.5)
one minimizing
estimates
there frequently
exists
among the unbiased
now
cannot
becomes
the variance
in 6.
of T?uniformly
(This clearly
(1.3)?which
of unbiasedness
if we omit the condition
unless g(6) is constant).
happen
It appears
that
In order
completeness
be complete
to obtain
of a family
the
results
of measures
of
to above, we
the notion
introduce
alluded
=--is eaid to
fflx
<*>}.The family 0X
{Mex\6
if
=
(x) 0 for all 6 in ?
d
f f(x) Mex
=
implies f(x)
be boundedly
0 except
on a set N with
complete
be a random variable
if this
Mex(N)
implication
distribution
=
holds
...
(1.6)
0 for all 6 in <*>.The
for all bounded
family
functions
is said to
/. Now
3PX, let T be a sufficient
in the family
P9X
for
of T.
and
the distribution
denote by
(not necessarily
3?x,
real-valued)
P?
function
then for any real valued estimable
shall show that if 3j?l is complete,
We
g(6)
an
that
and
smallest
this
estimate
is
with uniformly
unbiased
there exists
variance,
let X
with
statistic
of g(6) which
estimate
the only unbiased
a theorem
of
immediate
consequence
We
shall show also
estimation.
unbiased
is a function
of T
of Rao
(1945)
that whenever
ffi
similar regions A have a very simple structure
(roughly
of X falling in A is independent
of.), which
probability
This
only.
and Blackwell
is boundedly
that the
speaking,
was
first described
?06
?cnilt
is an
(1947) on
all
complete,
conditional
by Neyman
I
COMPLETENESS, SIMILAR REGIONS AND UNBIASED ESTIMATION?PART
cases is
in these
the original
hypothesis
(1937), and that as a result
composite
a
for
the
to
this method
reduced
The first to employ
finding
simple one.
essentially
was
similar
P.
of
L.
Hsu
(1941).
regions
totality
of a sufficient
The applicability
of the above remarks hinges on the existence
in general
there
Now
is complete
(or boundedly
complete).
are many
has the
and the question
different
sufficient
arises, as to which
statistics,
as to
the
condition.
best chance of satisfying
the completeness
question
Actually,
statistic
which
is the appropriate
arises also in other statistical
sufficient
problems.
statistic
T
Speaking
one makes
intuitively,
of a statistical
problem
plexity
condition
such
that ffi
use of sufficient
in order
statistics
information
losing
one is led to seek
without
the com
to reduce
of value.
The
latter
statistic
that
sufficient
being guaranteed
by sufficiency,
as
as
to
definition
the
far
and
hence
reduces
the statistical
possible,
problem
A sufficient
null set qualifications):
later with appropriate
(to be stated more precisely
statistics.
sufficient
statistic T is said to be minimal
of all other
if T is a function
which
a
a minimal
statistic
sufficient
this definition we prove that whenever
exists,
Using
is
if
it
can
sufficient
of completeness
statistic
equivalent
only
satisfy the condition
the existence
the minimal
statistic
sufficient
in a certain sense. We also establish
the
to
of
distribu
and
the sample space is Euclidean
a
(defined in section 6),
density
possesses
probablity
sample
generalized
and wTe give a method
of constructing
it, which we show to be valid in this case. We
statistic
sufficient
is
of the minimal
remark
that the result of our construction
the minimal
tion
sufficient
statistic
when
of the
statistic
of sufficient
to the definition
by Koopman
adopted
essentially
equivalent
(1936) in a more
setting.
special
are also found in the case
estimation
of unbiased
Some results for the problem
the minimal
where
which
those
characterizes
minimum
formly
end of section
variance
same
-function
is not
statistic
sufficient
functions
estimable
and these
A
complete:
estimates.
formal
unbiased
possessing
Finally
statistics
is obtained
theory
of uni
estimates
a justification
when testing
is given
(at the
to sufficient
(A
hypotheses.
4) for the restriction
and
Rao
was
case
in
of
estimation
the
per
Blackwell).
By
given by
justification
point
test and any sufficient
we
that
show
decision
randomised
any
functions,
given
mitting
and having
statistic
statistic
there exists a test based only on the sufficient
identically
the
po<wer
In this paper
ter generality
than
selves to Euclidean
already
pected
would
in sequential
to arise in connection
test.
and for statistical
analysis
with
stochastic
processes.
arise
even
came
Many
sets.
with exceptional
in the nature of the problems
were
limited to Euclidean
if considerations
in the paper
are inherent
difficulties
to
on minimum
our
attention
variance
while
estimates
this
were
that
problems
null
are associated
It
results
given
in grea
is developed
of section 6) the theory
(with the exception
The reason for not limiting our
in statistical
is customary
papers.
that have
both for problems
is that these are insufficient
spaces
arisen
countered
of these
as the
paper was
obtained
may
of the difficulties
We
treated,
believe
and
be ex
en
that most
that
they
spaces.
in proof that some of our
in papers
eaiLer
by Rao
(1947, 1949) and in an abstract by Seth (1949).
307
Vol. 10 ]
SANKHYA: THE INDIAN JOURNAL OF STATISTICS
[Paet 4
The material
outlined
above forms part I of the present
In part II the gene
study.
ral theory
is applied to a number
the
of more
main
As
special problems.
application
some theorems
on similar regions
tests
1942), and
1941), type Bx
(Scheff?,
(Neyman,
most
are
tests
and
one-sided
extended.
powerful
(Lehmann,
uniformly
simplified
1947)
These
results
are obtained
and Scheff?.
The
are solutions
which
introduced
equations
by solving the differential
by Neyman
densities
theory of part I is then applied to the families of probability
of these equations,
and also to some more general families of probabi
in a previous
results were
summarized
publication
were
some
of
and
these
obtained
and have
1947),
(Lehmann
independently
are also made to some non
since then by Ghosh
been published
Applications
(1948).
a very simple proof is
and
For
of
estimation
example,
testing.
parametric
problems
similar regions in the non-parametric
theorem
case,
(1943) concerning
given of Scheff?'s
lity
densities.
of these
Some
and Scheff?,
this
result
Halmos
is generalized,
(1946)
concerning
and
point
2.
the
is given
solution
of a problem
formulated
by
estimation.
Terminology
and Notation
a considerable
of terminology
seems
and notation
Unfortunately
complexity
to
we
minimize
this
the
conventions:
Several
unavoidable;
adopt
spaces,
following
to be denoted
will
to
have
be
here
is
the
Wx
whole
Wx,
Wt,
etc.,
considered;
by
space
of points x, W* is the space of t, etc.
In each space there will be a fixed
countably
additive
etc.
Here a family Jfx is said to be count
family of sets, Jfx in Wx, Jf* in W\
if
additive
it
contains
and
with
ably
Wx,
and
any set A in jfx its complement
Wx?A,
with
or
finite
number
of
sets
in
union.
also
their
any countable
(i.e.,
denumerable)
Jfx
The sets in Jfx will be called measurable
(jfx). We shall need to define measurable
func
tions only for the case of real-valued
A function/(#)
functions:
defined on Wx is said to
be measurable
is in Jfx. A non-negative
set
(Jfx) if for every real c the set [x\f(x)<c]
function Mx defined for all A in Jfx is said to be a measure
on Jfx ii it is
addi
countably
=
sets
in
Mx
tive, that is, if for any disjoint
A
Jfx,
AX,A2,...
(U^. ) ^MÂ^).
proba
= 1.
on Jfx is a measure Mx on Jfx for which
measures
bility measure
MX(WX)
Probability
will usually
be denoted
etc.
by Px, P\
A
of measures
Mx on Jfx will
be denoted by fflx.
It is convenient
to
the members
of the family by a subscript
6 that takes on values
in an abstract
=
=
if the measures
are
space o),0*
Similarly, we may write $x
{Mex\6eo>}.
{Pgx\fleu}
=
measures.
A
set
A
in
for
which Mex(A)
0 will be called a null set
Jfx
probability
family
index
A set will be called a null set for the family
jf?lx if it is a null
for the measure MQX.
measure
a
set for every
in the family.
If
statement
about
the points of Wx is true for
we shall say it is valid almost everywhere
all x in Wx?N,
if N is a null set for
(MQX)
we shall say it is valid almost everywhere
(fflx ) ifN is a null set for i?lx, and we shall
Mgx,
or
this by writing
abbreviate
An arbitrary
(a.e. Mf?x)
(a.e.0Lx) after the statement.
function
sets
t(x) from Wx
in Wt and a family
in Wt whose
pre-inuâs
to a space W*
jf?l' of measures
("complete
a countably
additive
Jp of
generates
family
on Wl
: the family Jf* consists
of all those sets
counter
are
while
the measure
in
Jf*,
images")
-308
I
=
to Mfx on Jfx is defined for B in Jf* by M9\B)
-flf/ on jf* corresponding
Mx(t'\B)),
=
seen
where
is
with
It
that
this defi
of
JE?.
is
tie
t-l(B)
pre-image
easily
{a; |. (#)e J3}
nition o? Jf \ a real-valued
if
and
function f(t ) defined on Wt is measurable
(Jffc
only
if the function
is
measurable
(Jfx).
)
f(t(x
are disjoint
and cover Wx will be called a deocm
of Tfx; if all the elements D in 30 are measurable
say that 30 is
(jfx) we will
position
a,measurable
on
a
function
is
defined
and a;0 is
If
Wx,
.(#)
(jfx) decomposition.
=
a;0.
any point of Wx, we shall say the set {x\t(x)
.(a;0)} is a contour of .(#) through
a decomposition
on W* determines
of Wx which may be de
t(x ) defined
Any function
A
30 of sets D which
family
of t(x).
of 201 being the contours
any
Conversely,
given
but
can
a
the
is
not
be
found
function
of
30
ranges
Wx,
(it
unique,
t(x)
decomposition
=
are in 1:1 correspondence
of an}^ two such functions
) such that 30
301 :For example,
value at x is the element
of
function whose
set-valued
t(x )may be taken to be the
we
to
the
above
x.
form according
If for this function
30 containing
description
t(x)
noted
301, the
by
elements
of those sets
precisely
family JP, it turns out that it consists
X
n
I30
of Jf l
additive
of elements D of 30 whose union is in Jfx. This countably
subfamily
xi30
function
3B or an associated
t(x ) will be denoted byJf
generated
by a decomposition
if 30 = 301.
or jfx,t.
30 are associated
We
say that a function
t(x) and decomposition
the countably
additive
associated
and statistics
in probability
only the decomposition
it being immateral which of the different
is of importance,
For many
a function
purposes
ted with
same
the
will
We
:Two
for decompositions
of equivalence
sense if there
equivalent
(J?x) in the strict
two kinds
?B and 30' of Wx are
positions
.ZVfor |lx such that on Wx?N
associa
functions
is chosen.
decomposition
encounter
with
the
decompositions
3B and W
x
exists
coincide;
decom
a null
set
30 and
30'
130
'
there
exists an A' in
set A in Jf
(S?x) in the weak sense if for every
x
130' xl30' XI30
'
'
such that A and A'
there exists an A in Jf
and for every A' in jf '
Jf
differ by a null set for $x, that is, (A? A1) U (A'?A
) is a null set for $x.
are equivalent
is the sample space of a statistical
being considered
on Jfx will then be
it by Wx.
The family of measures
problem we shall always denote
X
variable
that the random
a family
distributions.
3?x of probability
By saying
set
as
for
we
that
to
Px
is distributed
mean,
usual,
any
(the "sample")
according
be used in
will
teim
statistic
The
of X falling in A is Px
A in Jfx the probability
(A).
of x de
function
this paper to mean
of X, that is, if t(x) is an arbitrary
any function
If Jfl and $t are defined as above then
fined on WX,T = t(X )will be called a statisticl.
of
If 30t is the decomposition
of T falling in any set B in Jf* is P^B).
the probability
with
the
to say also that it is associated
with
Wx associated
t(x), it will be convenient
sense
are equivalent
shall say that two statistics
statistic T =t{X).
We
(3?x ) in the strict
one
When
1While
to functions
in this
whose
paper.
?does give
for
a
We
statistic
of the spaces
some
it may
purposes
associated
remark
be
are measurable
decompositions
however
satisfying
this
that
to restrict
convenient
our
construction
the definition
nothing
( jfx),
of the minimal
of
is gained
sufficient
restriction.
399
statistic,
by
such
statistic
for example
a restriction
in Section
6
Vol.
or in the weak
consider
sense
if their
the statistical
associated
decompositions
of these two kinds
implications
for this
Fundamental
on
[Part 4
10]
is the notion
study
are.
of sufficient
the definition
to
reader may want
of equivalence
of statistics.
The
of the
We
statistic.
conditional
(1933, p.41)
given by Kolmogoroff
set
of
A
in
T has the value
the
statistic
that
Jfx
any
given
P_(_4J. )
a
6 and A, Pe(A\t)
is
function
real-valued
of t, measurable
point
. : For
(JP).
base
it
probability
each fixed
and
defined
implicitly by the equation
=
Pf(At\t-\B))
P.(A\t)dP?,
\ B
...
(2.1)
is any set in Jp, and (2.1 ) is regarded as an identity
in B.
For fixed 6 and
is
are
not
if
but
defined
and
two determinations
A, P6(A\t)
uniquely,
ge,A (t)
f9,A (t)
a
are
of Pe(A/t),
is
null
set
for
the set Ne,A where
P0K A statistic T is
unequal
they
where
B
said to be a sufficient
statistic for |?x if there exists a determination
of Pe(A\t)
inde
on 6, measurable
dent of 6, that is, if there exists a function P (A\t) not depending
(Jp)
for each fixed A, and such that for every 6 in <?>,
every A in jp, and any determination
=
. in a null set
for PeK We
note
that two
P0(A\t), P(A\t)
B6,A
except for
P,(A\t)
of
determinations
must be equal except on a null set NA for ?Pfc. By putting
P(A\t)
B = Wl in (2.1), we see that if T is a sufficient
then for all A in Jp.
for $x,
statistic
=
Pg{A)
We
shall have
by Kolmogoroff
of x such that
of the
also
need
(1933, p. 46).
Suppose
\ w P(A\t)dPe\
...
(2.2)
as defined
concept of conditional
expectation
a
measurable
real-valued
(Jp) function
<f>{x)is
_?,(*) =f
*
W
X
... (2.3)
<f>(x)dPQ,
<?=
the expected
value
tion Px,
is finite.
of the statistic
under the probability
distribu
<?>(X), calculated
=
a
not real-valued),
then the
If T
statistic
(in general,
t(X ) is
?
of
T
to
conditional
value
calculated
under
be
denoted
and
<_>,
t,
Px,
expected
given
defined
of t, integrable
(jft, P/),
by l?e(<_?l.)> is a point function
implicitly
by
J
<}>(x)dPf=
J
EQ(s>\t)dP?,
...
(2.4)
is in Jf*, and (2.4 ) is regarded as an identity
in JE?. For fixed 6, two determina
are equal except on a null set N0 of P\
the definitions
tions of
By comparing
EQ($\t )
=
seen
of a set A in
is
function
and
is
that if <?> <j>A(x) the characteristic
(2.1 )
(2.4) it
Jfx, then
where
B
P0(A\t)
Returning
to the general
case of
=
(a.e.P/).
Ee(Â\t)
<_?
with
finite
E$
( _>),we
...
remark
(2.5)
that
E9(&\t)
can also
be calculated from the conditional probability PXA1 )if tnis is known for a11 sets ^ in
)Jfx, as was
proved
by Kolmogoroff
(1933, p. 48).
It then
follows
from
310
the above
defi
is a sufficient
I
is finite
for |5X, and if
EQ($ )
a measureable
of 6,
of t, independent
for all 6 in w, there exists
(jP) point function
in ?,
6
for
that
we
has
shall denote by E(?\t)
the property
and which
which
every
=
on a null set
E(<b\l)
NQ forPj.
EQ(<b\l) except
of sufficient
nition
that
if T
for the following
from the calculus of condi
three formulae
if
: Suppose T is a sufficient
is finite
statistic
for 3?x. (i) Then
EQ(Q )
probabilities
from
follows
...
[ E(*\l)dPf.
E,(*)=
This
statistic
use
shall have
We
tional
statistic
= Wfc.
(2.4) by taking B
(2.6)
is a real-valued
(?) If f(t)
measurable
and JE^) are finite for all 0 in ?>,then
...
(a.e.*t).
E{f(Tm)=f(t)E($\t)
(JP)
function of ?,and ifE3(f(T)$)
p.
is proved by Kolmogoroff
easily from
(Hi) It follows
formula
This
105).
(2.7)
(1933, p. 50), differently
by Blackwell
(2.4) that if cx <^(#) ^ c2 then
cx< E(*\t) < c2
(a.e. $<).
...
(1947,
(2.8)
and
of conditional
that the values
remark finally
expecta
probabilities
on
statistic.
with
the
associated
the
a
statistic,
decomposition
depend only
tions, given
=
=
a function
T!
and
T
Of two statistics
t(X)
t'(X) we shall say T is
of T
=
a function
t(x)
\?r(t'(x))(&.e. S?x).
t-=\?r(t') on Wtf to Wl suchthat
(a.e. i?*) if there exists
with
the functions
associated
and
?t'
In terms of the decompositions
t(x) and
3Bt
on
the
that
a
Wx?N
Af
such
set
for
null
means
exists
there
S?x
decomposition
t'(x) this
in an
element
of
is
is
contained
that
30t'
the
subdivides
3Bt,
every
decomposition
JBt/
statis
statistic T for $ix will be called a minimal
of 3Bt. A sufficient
element
sufficient
We
tic for 3?x if, for any
conditions
Sufficient
finding
it, are given
other
for the existence
in section
3.
Given
in Wx,
consider
for 3PX,T is a function
sufficient
of minimal
statistic,
statistic
sufficient
a Family
of
=
jf?lx
integrals
{M*\de<?}
of the form
f(x
in general
of 6, which
(a.e. |?x).
a method
of
of measures
of Measures
on the additive
family
...
f J(x)dM.*,
x
J
where
of T'
and
6.
Completeness
a family
T'
jf* of sets
(3.1)
W
(Jfx ). The value
(if any ) of this integral will
is used;
of the family $[x
(3.1) is a function
on Wx to a func
from a function
as a transform
defined
f(x )
the function
this transformation
that is every
of w. Under
and measurable
) is real-valued
on which measure Mx
depend
we may regard
tion of 6 defined on a part
zero on Wx goes into the function
where
zero on a>.
is everywhere
Completeness
on Wx is the only
function
zero
into the zero
means
function
the
that
going
roughly
of the transform.
The exact
on w ; it is a unicity
definition
is
function
property
the following
: The
family ?lx
f
that
of measures
f(x)dMx
is complete
= 0
if
forall?inco
...
811
(3,2)
Vol. 10 ]
implies f(x
of unbiased
= 0
(a.e. $tx). This definition
)
estimation
; for the problem
of bounded
perty
condition
(3.2) and
is appropriate
of completeness
of similar regions we require
family ffix is called boundedly
The
completeness:
the condition
[Part 4
that f(x)
is bounded
jointly
for the problem
the weaker
pro
complete2 if the
that f(x) = 0 (a.e.
imply
4*tx).
note
We
that
of a family
example
is a slight modification
if jfttx is complete
of measures
which
of an example
and Savage
Mosteller,
it is boundedly
is boundedly
constructed
The
complete.
complete
for a different
simple
following
without
being
complete
by Girshick,
purpose
(1946):
*> is the
interval 0<#<1,
open
Suppose Wx is the real line,
=
x
6 to the
to the points
the measure
and
0, 1, 2..., measure
(l?6)26x
PQx assigns
= ?
zero
measure
to
the
of
set
and
this
of
The
1,
complement
pointa?
points.
condition
(3.2) then becomes
Example
3.1:
f(-l)d+lf(j)(l-d)2V=0.
We
is the coefficient
of 6l in the Taylor
for j = 0, 1, 2..., f(j)
Since
for
for the function ?/(?1)6>(1?O)'2.
|0|<1,
see that
origin
6(l-d)-2
=
series
about
the
?jdi,
ro
it follows that (3.2) is satisfied if and only if
j
/?)-?i/(?1).
=
...
0,1,2,....
=
(3.3)
=
0 for j
is bounded,
satisfies
if f(x)
Hence
?1,
0, 1, 2,..., that is,
f(j)
(3.2) and
=
On
thfe
other
if f(x)
is
and thus 3?x
hand,
complete.
boundedly
0(a.e. $x),
f(x)
= 0
satisfies
0, then it satisfies
(3.2) but not the condition
f(x)
(3.3) and /(?1)_t_-.
is not complete.
(a.e. $x), hence $x
in general completeness
of jf?lxdoes not imply the same for Jflflx,but
of a subset Mi
all
null sets for jRf are null sets for ??tx.
hold provided
It is worth
We
while
shall now
to note
give
that
some
simple
is discussed
completeness
jpxwhose
family
in statistics;
has been used extensively
or bounded
that
this
completeness
does
implication
In each case the
of completeness.
examples
measures
that
will be a family of probability
a
it is of interest that in number of these examp
for a transform
to the problem
of unicity
case the space Wx
literature.
In every
on
be taken as
Wx
and the additive
may
family rjp
reduces
of completeness
les the question
in the mathematical
that has been treated
will
be the real line ?oo <.*:<+oo,
set of
space w of points 6 may be taken as an appropriate
is complete
in which
the family of measures
All the examples
real numbers.
may be
_.
the
subsets
without
certain
modified
proper
completeness.
destroying
by
by replacing
one
con
some
means
of which
in
Part
of
theorems
that
here
II,
We mention
general
by
a
in product
cerns completeness
may be proved for
large number of
spaces, completeness
the class of Borel
2 The
publication
property
The
sets.
here
called
bounded
completeness
was
called
completeness
(1947).
312
by
us
in an
earlier
I
COMPLETENESS,-S?MILAR REGIONS AND UNBIASED ESTIMATION?PART
of measures
families
and
Here
line.
in n-dimensional
Euclidean
use
we
in examples
elsewhere
the
d Px?d[Xx =
to indicate
the measure
that
is absolutely
Px
from these
spaces
"Nikodym
examples
on the real
notation
derivative"
g(x )
with
continuous
respect
to the measure
that
px and
P*(A)= \ g(x)dixx
A
for all A
in Jfx.
If |?x
3.2:
Example
0 and unit variance,
mean
is a family
then
=
dPx?dx
of
normal
-
(27r)-*exp
distributions
probability
t
with
(x-0)2
[-!<*-*) ]
If the
condition
satisfied
(3.2).is
by f(x)
r\ f(x)
J
ior ?oo<6<oo.
exp
exp
and
(?\x2)
(a.e. $x).
?
Thus
Example
0 we
variance
find
r? i
i
-^x2-\-0x
\dx=
...
0
(3.4)
of the function/(#)
transform
Laplace
(3.4) is the bilateral
of this
transform
it follows
that f(x)
theorem
the unicity
= 0
on
measure
and
L
denotes Lebesgue
where
Jfx,
hencef(x)
Now
(?\x2),
exp
we
from
0(a.e. L),
is complete.
$x
For
3.3:
a family
3?x of normal
with
distributions
zero mean
and
have
d
Pex?dx
=
(2*0)-*
exp
(
-
x2
-j?
(-?-)
J
for 0 <
0<
oo.
...
(3.5)
or even boundedly
for every 0 the density
complete because
family is not complete
is an
of x, and hence
(3.2 )will be satisfied by any f(x), which
(3.5) is an even function
of x and such that its product with
odd function
(3.5) is integrable
(L) for all 0>O.
t = x2we find
measures
transformation
the
a
new
of
set
we
to
If
transform
P9l by
ffi
This
fl (2ntd)-1-
dPJIdt^
< 20
10
Condition
(3.2) written
for $t
l
exp(-
for t< 0.
instead
of
3PXgives
[ /(?)r*exp?-?
for 0> 0. Letting r =
transform we find/(i)
-^t)iort>09
-
*
)*'=?
the unicity theorem
(20 )~l,and applying
= 0
is complete.
that
is, ffi
(a.e. ffi),
for the unilateral
313
?
(3-6)
Laplace
3.4:
Example
we
a
For
have
J?x of
family
=
dPx\dx
reasons 3?xmay again be
symmetry
=
x2, the completeness
before, we let.
seen
of the Stieltjes
with
distributions
Cauchy
zero median
for 6>0.
\nd\l+x2\d\-x
For
property
[ Part 4
Vol. 10]
not
of the
to be
resulting
If, as
complete.
from the unicity
boundedly
3P* follows
transform
f fm
? t+6
butions
a
For
3.5:
Example
if 0 is a half
distri
(chi-square
ifa>0,
f2-M~iexp(-te)/r(0)
<?
I 0 if a;< 0,
=
follows
completeness
distributions
gamma
integer),
dP;?dx
0>O,
of
family
the unicity
from
of the Mellin
property
transform
00
0
3.6:
Example
If
is the uniform
Pex
distribution
on the
interval
(0, 6),
fll6if0<x<d,
dPexldx=
i
^0 elsewhere,
the
6>0,
of ^x
completeness
follows
from a theorem
: If
of Lebesgue
9
0J
for all 6 in an
interval,
a function
which
f(x),
then f(x)
is involved
The
3.7:
Example
family
f(x)dx.= 0
=
0 (a.e. _L) on the interval.
in this case, is its indefinite
of
$x
uniform
The
transform
(3.1) of
integral.
on
distributions
the
intervals
i?, 0+1),
[Ii?d<x<0+19
dPe
?oo<0<oo,
is not
boundedly
x?dx=
<{
?^0elsewhere,
: If f(x)
complete
is any periodic
function
1 and
i
[ f(x)dx
o
it
is easily
seen
= 0,
that
314
with
period
I
+ TO
= 0
f"00/(*)d P.'
for
Px
all
0.
so that
of Poisson
distributions,
3j?x is the family
?
x
zero to the
0, 1, 2,..., and measure
e~e0x?x\ to the points
set.
The condition
be
written
(3.2) may
3.8:
Example
the measure
assigns
of this
complement
for 0>O.
and $x
series
of zero
expansi?n
it
is complete.
distributions
of binomial
family
Px
constant
0, 0<#<1,
probability
If $x
trials with
independent
of 0* in the power
coefficient
0 (a.e. $x),
3.9:
Example
to n
is the
Since/(j)
that f(x) =
follows
Suppose
is the
corresponding
the pro
assigns
bability
x =
to the points
0, 1,..., n and
implied by the theorem
of the argument
values
zero probability
elsewhere.
Completeness
of degree n vanishes
that if a polynomial
forw+1
zero.
it is identically
of
3?x is
distinct
distributions
of
the
3.10:
Let
3?x be
Example
family
hypergeometric
in the
of defectives
for fixed lot size N and fixed sample
size n where 0 is the number
==
the
in the sample,
is the probability
of a; defectives
0, 1,..., N; ifPx
lot, 0
Pex assigns
discrete
probabilit?s
(!) (?=9 / (?)
to
the
x =
points
In
0, l,...,n.
=
M
Condition
then
(3.2)
(3.8).
ables
find
Hence
successively
/(0)
|5X is complete.
0ifr<0orr>m.
=
= ?,... m
f<4) (??ZI)
0, /(l)
=
0,
..., f(n) =
0 by putting
0 =
0, 1, ..., w in
... be a sequence of
vari
random
Let Xx, X2,
independent
1 and 0 with probability
p and l?p
capable of taking on the values
on these variables
where
the
a sequential
scheme
Consider
sampling
Example
each being
respectively.
on whether
decision
already
understand
becomes
I
and we
(3.7) we
been
taken
3.11:
or not
ismade
to take
according
an A^+P1
observation,
N
to the value of 21X{.
when
Let
N
observations
have
n be the total number
i=l
315
of
n is arandorn
in one experiment?
taken
observations
for
of bounded
completeness
problem
rule
and
the values
stopping
0<j?<1,
and
Mosteller
S
Savage
to be
(1946),
x{
1=1
/
(N
such a point
scheme being
was
N \
__.
#i? N?
n
?
and
let X ?
2 XL.
the
solved
Wolfowitz
I as
(1946),
coordinates
if it has
to a fixed
|JX corresponding
family
as follows
in a series
of
and
of papers
(1947).
Savage
a point
in
a
plane,
by
For
and
the sequen
under
probability
so that,
tial
considered
but is not a stopping
for this procedure
point
is observed,
in
this point
another
will be taken.
observation
As was shown
when
is
and sufficient
the above papers, a necessary
condition
for bounded
completeness
N
to the same value of
that, given any pair of accessible
points P1,P2
corresponding
sum
N
is
the
of
the
of
all
the
coordinates
points
(for any point P,
P),
lying on the line
define
segment
accessible.
connecting
The
minimality
Px
accessible
variable
i=l
The
Girshick,
[Part 4
Vol. l? ]
and P2
and
positive
valued
integer
having
of completeness
in this section
is
concept
developed
of a sufficient
at
end
of
introduced
the
secton
statistic,
is a sufficient
is a minimal
sufficient
Theorem
If T
3.1:
complete, and if U
sense.
(;PX) in the weak
The
proof
statistic
for
3?x such
statistic
for
3?x, then T
can be based
of this theorem
are
coordinates
also
to that
related
of
2, by
is boundedly
and U are equivalent
that ffi
on the following
=
=
are two sufficient
statistics
t(X) and T'
If T
tf(X)
for
as
and
A
x are equal
in Jp,
and
considered
?Px,
if for every
functions
P(A\t'),
of
P(A\t)
in the weak sense.
(a.e. ?Px), then T and T' are equivalent
the lemma
To prove
let jfxlt = t~x(ix),
and define Jpi^ analogously.
We
a set A' in Jpi*' which
shall prove that to any set A in Jpfc there corresponds
differs
Lemma
3.1:
by a null
from A
set for 3?x. Let <j>A(x)be the
=
(?>A(x)depends on x through t(x), say (?>A(x)
then
P(A\l)
and
=
characteristic
g(t(x)).
By
of the
function
(2.5)and
E(g(T)\t) =g(t)
(a.e.$0,
Ux)
(a.e.W)
set
_4;
(2.7),
thus
P(A\t(x))
=
But by hypothesis
P(A\t(x))
P(A\tf(x))
(a.e. $x). ...
as the set in Jfxt' where the right member
of (3.9) equals 1. Now
ol (3.9) equals
for $x such that on Wx?N
member
the right
and the lemma is established.
of A and A' in Wx?N
coincide,
Define
A'
a null
set N
the parts
=
316
(3.9)
there
exists
hence
(?>A(%>}\
Theorem
for which
3.1 will
an
if we show that if there were
thus be proved
Ax in Jfx
were not equal
then
could
not be
ffi
(a.e. $x),
P(Axl{u(x))y
the real-valued
Consider
measurable
(jfx) function
v(x) defined by
and
P(AJt(x))
boundedly
I
complete.
v(x) =P(Ax\t(x)-P(Ax[u(x)).
We
note
that
and that
(a.e. $*),
|v(a:)|<l
the set in Jfx where
If V = v(X), it follows from (2.2) that
set for $\
v(x) ^
0
E.(V)
for all ? in ?>. Since
for $x
is a minimal
U
V is also a function
and hence
so that the result
sufficient
of T
We
(a.e. $').
is a measurable
of v(x) does
not
invalidate
and
(3.10),
is not
(3.10)
of T
redefine
0 is not a null set for 3?f. This redefinition
boundedly
4.
...
thus
fwt/<i)dzy
for all ? in o), and hence ffi
=0
a null
(a.e. 3?x)
v(x) on a null set
of t, say f(t ), with
the properties
can thus
function
(jfl)
that |/(01< 1, and the set in jp wheref(t) ^
it is a function
statistic
is not
Similar
= o
complete.
Regions
in jfx is said to be a similar region of size oc for the family |?x of probabi
on Jfx if Pfl%4) = a for all 0 in w. Neyman
(1937) noted that if T is a
lity measures
for J?x and if the set A has the property
statistic
sufficient
A set A
=
...
P(A\t)*(a.e. $'),
then A
is a similar
region
of size a for |5X; this follows
in jfx has the Neyman
did not
(4.1 ). Neyman
set A
structure with
(4.1)
from
(2.2).
sufficient
respect to the
under what conditions,
that a
whall'say
statistic T if it satis
We
statis
given a sufficient
investigate
:
T
to
structure
this
is of
all
for
have
this
T
for
similar
with
tic
^x
respect
|?x,
regions
since there one wants
in the Neyman-Pearson
tests,
theory of optimum
importance
one
therefore
needs to know the
and
of all similar
to choose
the "best"
regions,
fies
totality
Theorem
of
such
regions.
A
partial
answer
is given
by
the
following
corollary
to
4.1.
is boundedly
a sufficient
and if ffi
statistic for $x,
4.1:
Corollary
// T is
then a set A in Jfx is a similar region for |?x if and only if it has theNeyman
complete,
structure with respect to T.
is that sets A
arises
of similar regions
problem
tests
of a hypo
as possible
critical
in Jfx are being considered
regions of statistical
is rejected by
: If the sample X falls in the critical region A the hypothesis
thesis
a
similar region
If A is
is accepted.
the hypothesis
if it falls in WX~A,
the test, while
in $x.
distribution
for
all
is
constant
the
of rejection
for |5X,
probability
probability
x
such that
of points
to employ a third category
it is convenient
For many
purposes
or
one
does
not
is observed
when one of these points
accept,
always
reject
always
The
situation
in which
the
317
[Part 4
SANKHY? :THE INDIAN JOURNAL OF STATISTICS
10 ]
Vol.
but rejects according
to a chance method
(say, with the help of a table of random mum
the probability
is a predetermined
of rejection
number
bers), for which
<?>(x),0<$[x)
whole
< 1. One may then extend
to
of the function
the
the definition
sample
<?>(x
)
= 0 on the
= 1 on the
and
set
set,
space Wx, by setting
acceptance
<p(x)
<?>(x)
rejection
which
and is thus led to the notion of critical function
and
may
Stein,
1948),
(Lehmann
as a special case of the randomized
of Wald
be regarded
functions
decision
(1947):
x
0
1.
which
A critical function
for
of
is
measurable
function
<
<f>(x)^
(Jp )
<?>(x) any
one rejects the hypothesis
with proba
Its use in testing hypotheses
is that when X=x
a
of
the
random
to
random
pro
process
(?>(x) according
independent
bility
statistically
cess governing X.
calculated
under
value
of rejection
is then the expected
defined
Critical
regions
by (2.3).
PQX, namely,
JS?(?_>)as
are characteristic
to the special case of critical functions
which
correspond
of sets in Wx,
The
that
probability
take on only
is, which
0 and
the values
is constant
and equal to a for all 6 in <owe shall
Eq(<P)
critical function
of size a for 3?x. Clearly,
similar regions
are characteristic
case of similar
critical
functions
which
special
is its characteristic
If A is a set in Jp and <f>A
a
if
similar
critical function
if and only
0A is
<?>(x)is a
to the
correspond
of sets :
functions
with
that
is a similar region for 3?x
func
shall
We
say that a critical
to a sufficient
statistic T for |?x if
a
if
critical function
<f>is the charac
structure with
<j>(x)has the Neyman
respect
= a
(a.e. ?J?*). It is obvious from (2.5) that
E(<P\t)
of a set A in Jp then $ has the Neyman
teristic function
structure
say
then A
function,
for $x.
tion
and only if A has the Neyman
follows from
functions
1.
If
similar
of <_>
=<?>{X)
seen to
are
structure
to T.
respect
with
to T if
respect
Therefore
Corollary
4.1
4.1:
statistic
3?x,a necessary
andsufficient
If T is a sufficient
for
all
structure
similar
with respect
critical
to
have
the
for
Neyman
functions
for $*
i? that $fc be boundedly
complete.
Theorem
condition
to T
shall
We
of
size
oc for
for all ?
prove
3?x.
in ?,
Then
first
from
all
6
/(.)
=
E&\b)-*.
...
Since
O<0(.c)<l,
(a.e. $*) by
(2.8).
0<l?(*)Ji)^l
determination
of E ($|.) in (4.3) and then redefining/(.)
assume /(.)
to be bounded.
is boundedly
Since ffi
(a.e. ê),
that
Next
complete
is, </>(x)has the Neyman
we
there
shall
would
(4.2)
Jv/(t)aP0?
in w, where
prove
exist
function
= *
= 0 ...
for
critical
(2.6),
\Ê(*\t)dP?
so that
<j>(x) is a similar
Suppose
sufficiency.
necessity
similar
structure
by
critical
with
showing
functions
(4.3)
a particular
by taking
a
on
null set for $)fc, we may
= 0
(4.2) implies /(.)
complete,
Hence
respect
that
to T.
if ffi
were
(of every
318
not
boundedly
not
size a, 0<a<l)
having
there
the Neyman
structure
with
exists a bounded measurable
to T.
respect
function
(jP)
is not
If$fc
f(t)
such
I
complete
boundedly
that
?JWu
for all 0 in w, but
A, ={t|/(i)^0}
Suppose |f(t)\ <M, and define
=
flW cf(t)+z,
is not a null set for$'.
where
0<c<Jf-1min
Then
{a, 1?a}.
O<0(i)<l,
...
{J W tg(<)dPe'=a
(4.4)
for all 0 in ?, and
...
[?(âonif.
as critical
take
Now
?($!*)
=
function
0(a)
=
Then
g(t(x)).
E(g(T)\i), and hence by (2.7)
E($\t)
We
(4.5)
=
(a.e. $*).
fl<*)
...
(4.6)
is a similar critical
function of size a for $*,
(2.6), (4.6) and (4.4) that <j>(x)
have the Neyman
structure wsth respect to T.
(4.6) and (4.5) that it does not
see from
and from
be appropriate
there exist
for which
It may
butions
with
respect
to a sufficient
a simple example
of a family $x
do
have
which
not
the Neyman
similar regions
statistic T.
to give here
of distri
structure
be a random
4.1:
Let X = (XVX2,...,
sample of size n>l
Xn)
Example
The
on the
intervals
distribution
space Wx
(0? J, 0+J).
from the uniform
sample
as
Borel
sets
the
additive
in Wx,
the
as a Euclidean
family jp
w-space,
may be taken
?
of the sample is the uniform
P
bo<#< + oo. The distribution
and toas the real line
Qx
?
=
i
Let tx(x) = min
cube
on the 7i-dimensionsl
l,2,...,n.
distribution
\x- 0\<%,
= max
%i>h(x)
i
dimensional
probability
&..
The
Euclidean
t=
transformation
space
W\
and
t(x)
the
=
(^(cr), t2(?)) maps
statistic
T =
?(X)
=
Tfx into
the
i
two
(?\,
has
the
Ta)
density
fc(?a-?i)n"a
if ?-|<î<<2<^+i,
[0 otherwise.
=
is known
6.4) that T is a sufficient
(it also follows from Example
dpi'
dtxdt2. It
is not boundedly
5.3 that W
for $x; it will be shown in Example
complete.
statistic
with
to T can be based on
structure
the
respect
not
Similar
Neyman
having
regions
a continuous
distribution
of 0, there
independent
the range R =t T^T
r Since B has
where
319
Vol.
10]
exists
for any a(0<a<l)
The set
R<ra.
[Part 4
THE INDIAN JOURNAL OF STATISTICS
SANKHY?:
a constant
r? such
=
Aa
all
for
that
6
the
probability
is a
that
{x\t2(x)-~tx(x)<ra)}
is thus a similar
Since the characteristic
of the set Aa can be expres
function
region.
sed as a function
takes only the values
of., it follows from (2.5) and (2.7) that P(Aa\t)
0 and 1 (a.e. i?1-), that is, Aa does not have the Neyman
structure with respect to T.
we
of similar
illustration
take n =
1, an even simpler
If, in this example
a
can
to
not
the
structure with respect
sufficient
statistic
possessing
regions
Neyman
measure
a and
be given. On the interval
(0,1) in Wx take any Borel set _4 xof Lebesgue
1
so
A
is
and
coincides
function
then define
that its characteristic
periodic with period
is a sufficient
statistic for $x and P(A\x) = 0
that of A x in (0,1 ). Then X = X
with
or 1 (a.e. $x).
conclude
We
statistics:
section
this
? be a set
Let
the
by noting
in the
?-space
following
containing
"justification"
of sufficient
and
<i>,let 3?x={Pex\deQ},
given any critical function
sup
is sufficient
for iPx. Then
T =t(X)
pose
<?>(x)
on
x
there exists a critical function
for testing H0:de(?,
<?>x(x)
depending
only through
as <?>(x). The proof is similar to that of Rao
the same power function
t(x), and having
Let ijf(t) = __?(<_?].)
and let <Px(x)= ijr(t(x));. \?r is
and Blackwrell's
theorem
(Section 5):
of 6 in Q since T is sufficient for |?x. The power
of rejecting
independent
(probability
the
statistic
) is
H0) of <?>(x
=
E9(^)
and
by
(2.6)
this
equals
Eg(&).
be remarked
It should
E9(f(T)) =Ee(E(*\T)),
that
an even
stronger
was given
of sufficient
justification
by Halmos
statis
and Savage
of statistical
inference
for all problems
(1949)
means
a
use
of
the
statistic
and
out
sufficient
of
that
random
variables
by
pointed
a statistic
to construct
it is possible
the same distri
with known distribution
having
as
Their
that
however
there exists
the original
bution
presupposes
argument
sample.
tics
who
measure
is a probability
that
probability
P(A\t)
of this supposition
has been established
(1948, p.399)
by Doob
is a Euclidean
The
for
statistics
sufficient
space.
justification
of the
a determination
(a.e. $*) ; the validity
in the case that Wx
given by us above
out any restriction
conditional
for the problem
on the sample
5.
of testing
has
the
advantage
of being
valid
with
space.
Ujsbiased
Estimation
will denote
the family
of all distributions
to
={PQx\deo)}
a
a
of statistical
is restricted
attention
inference.
which
priori in
problem
particular
as
statistics
solutions
of
of
real-valued
unbiased
esti
In considering
problems
possible
to those with finite second moments.
we shall restrict ourselves
Let V be the
mation
=
statistics
V
for
is
which
class of all real-valued
v(X )
v(x ) measurable
(Jfx) and E9(V2)
?.
in
w.
is
V
all
For
also
6
for
in
in
finite
0
all
for
We shall say
finite
%J,EQ( V)
is
any
In
this
section
$x
920
COMPLETENESS, SIMILAR REGIONS AND UNBIASED ESTIMATION-PART
a real-valued
that
function
is estimable
g(0)
=
g(0) for all 0 in <o. An unbiased
variance will be called a minimum
is a minimum
V
variance
=
g(0)
'Eg(V)
(?)
estimate
variance
a V
exists
if there
in V
I
that
such
Eq(V)
is of uniformly
of g(0) which
minimum
of g(0), that is, a statistic V in
estimate
estimate of (7(0 ) if
for all 0 in <*,and
(n) Var? (F)<Var^( V ) for.all 0 in o>and all F' in V statisfying (i).
the notion
applying
the
following
easily
By
obtains
to a result
of completeness
of Rao
and Blackwell,
one
that $*
there exists a sufficient
statistic T for 3?x such
If
is complete
variance
has a minimum
then every estimable function
estimate, and a statis
ic V in V is a minimum
variance
estimate
of its expected value if and only if it is a
function
of T (a.e. 3?x).
5.1:
Theorem
The
is a con
of Rao
this theorem
(1945) and Blackwell
(1947) of which
is a sufficient
statistic for |lx and V in V an unbiased
=
<*>
if i/r is defined by i/r(f) = E(V\t),
\?r(T) is also an unbiased
result
states
sequence
of g(0), then
that
if T
estimate
estimate
of g(0) with
Var^ (^)< Var^ (V), equality holding for all 0 if and only if F is a function
of ?7 (a.e. $x).
To prove
then there exists
function;
g(0) is an estimable
Let * be defined as above.
To see that * is a mini
5.1 suppose
Theorem
=
a V in V for which
g(0).
Eg(V)
of g(0 ), suppose! Vr in V is any other
estimate
==
and define
Then
f'(t)
E(V"\t).
by the Rao-Blackwell
unbiased
estimate
of g(0) and
for all 0.
Var?(^')<Var?(F')
mum
variance
of g(0),
estimate
=
theorem
?'
i}f'(T) is an
=
Since
0,
E0(*'-*)
unbiased
J w
for
all 0
in <o, and
hence
it follows
from
the
Thus
(a.e. $fc).
Var?(*)=
Var?(^')<Var?(Fr),
estimate
of g(0).
last part of the theorem
The
in the Rao-Blackwell
theorem.
equality
The
application
of Theorem
5.1
is obtained
is illustrated
5.1:
Example
3?x is the
Sappo3e
family
of size n>l
from a normal population
with
It
is well-known
-
(27r02)-i*
of
distributions
t2{x)
=
S xjn,
i=l
=
S
condition
for
of
a random
0X and variance
02. With
=
...
Wx, djax
dxn, and
dxx dx2
S
(*i-0i)a
that with
tx(x)
the
mean
- ~
exp
from
by
sample
x =
in the Euclidean
(xvx2,...,
xn) a point
space
0 = (0V 02) a point
a>=
in the half plane
{0|02>O},
dPo'ld/f
=
of $l
that
\?r(t)
i?rf(t)
is a minimum
variance
completeness
and so *
?i-^?x)
,
321
...
(5.1)
Vol. 10 ]
[Part 4
T =
=
statistic
t(X) is a sufficient
sufficient
and that
statistic),
(TVT2)
as a minimal
6.2 it will
(in Example
and _T2 are independently
for $x
be derived
Tx
distributed
with joint probability density
! J--f->r
''
^
if t2 >
o,
(2?2)
\?(nMV*Th^\
0 if ?2<0.
is complete.
II of this study it will be proved that $[
From
the com
of the sample mean
at once
it follows
functions
that the following
are
estimates
of the indicated
minimum
variance
func
variance T2?n
In Part
of $'
pleteness
sample
-\ancl
tions of 0-and
-""-of.
8t:
cn
^"l+VnA/â
from
mined
and
x; _Y(n-l)
=
of the population,
(lower)
normal
respectively,
where
2^(i(n-l))r(in);
100 p percent
point
so
tables
that
probability
?f tne
y/0t,
.-yT,..02and
where
ap is deter
aP
(2?)-?
f-00
xp(-?Z2)dZ
= ?>.
5.1 that any real
are all special cases of the statement
from Theorem
following
a
is
minimum
variance
variance
of its
estimate
valued
of (TVT2) with finite
function
value.
expected
These
Theorem
5.1
: can be
extended
easily
to the case
of simultaneous
estimation
Cramer (1946), we express the concen
of 6 if, following
of several real valued-functions
of concentra
in terms of its ellipsoid
its mean,
about
estimate
tration of an unbiased
=
of 6, and let g(d)
functions
tion. Let gL(6),...,
(gx(6), ...,gm(6)).
gm(6) be m real-valued
=
an
of g(6) if V{ is an unbiased
estimate
is
unbiased
shall say that V
We
...Vm)
(Vv
of g(6)
V will be said to be an estimate
The statistic
of g{(6) for i = l,...m.
estimate
for
of g(d), if
estimate
if it is an unbiased
concentration
with maximum
E9(V2)?x>
=
estimate
concentration
the
if
unbiased
other
for
in
and
U,
and all 6
<o,
i
any
l,...,m
of U.
T for |?x such that 3Pfe
statistic
sufficient
=
...,
then every estimable function
gm(0)) for which gx(6), ...,
g(6)
(gx(6),
concentration.
has an estimate with maximum
linearly independent,
The
: If there
5.2
Theorem
is complete,
gm(d), 1 are
in that
is contained
of F
elliposid
proof
of this theorem
follows
m
quadratic
forms:
* Pointed
out
Let
II
a
exists
easily
from
the
following
be
two
positive
result3
concerning
definite
quadratic
m
a.-y^y.,
to us by Professor
E. W.
___
bi,yiy-:
Barankin.
322
m
m
and
forms,
2
let
I
2
a'^y^.,
h ri
be
b^yy^
i,M
m
the
inverse
corresponding
forms.
Then
m
S
i, j=l
S
?ij-Mj <
i, j-1
fttf-Mj for all real y's
implies
m
m
2
*, j=i
let F =
Now
let ?7 =
and
Then
for any
m
of s
i, j=i
b3 any unbiased
(Vv...,Vm)
(Uv...,Um).
&ij#i2/j for all real y's.
2
"" >
ftij/A2/j
estimate
of g(0),
m
S ^Fj
i=i
real yv...ym,
=
let ^
is an unbiased
m
2/i??i(0)3 and by Theorem
the variances
Comparing
estimate
m
is a minimum
5.1, S yiUi
i-l
i<L i=l
two
of these
^F^?7),
estimates
m
we
estimate
variance
of S y^S?).
find
m
are the covariances
and
Further
pf (Ui, U. ) and (Fi5Fj ) respectively.
Au.j
A.^
so
are
1 are linearly
and
since gi(0),...,grm(0),
Fl5...,Fm,l
Uv...,Um,l,
independent
and hence the above quadratic
definite.
forms are positive
It follows that
where
more
m
m
S
i,j-i
which
the theorem
proves
since
the
=
A^^y.
and
m+2
shall now
answers
which
minimum
any
shall
develop
about
questions
which
but
estimates,
statistic
T,
three
the
may
defined
class
ra+2,
are defined
respectively.
a single real-valued
estimates
variance
of estimating
of minimum
the problem
a formal theory
to the class V of statistics
sufficient
=
Xv'hj^.
once more
certain
variance
In addition
for
consider
of 0, and we
function
S
i,j=l
*,j=l
We
of U and F
of concentration
m
2
equations
i,j-i
the ellipsoids
m
by
S Av^
Vj2/i?/j >
of estimable
functions
possessing
in specific problems.
to apply
of this section we define,
at the beginning
be difficult
classes
?T,
^Tft,
and
V?1.
Here
T will
in
to be
be expected
theory may
all
class
of
the
be
for 3?x. Let VT
simplest
We define $T? as the subclass of sta
are funct'ons
in V which
of T (a.e. $x).
statistics
=
0 for all 0 in o>. If $fc is complete,
VT? consists
tistics F in VT for which
E$(V)
= 0
In statistical
F = v(X) for which v(x)
(a.e. $x), and conversely.
only of statistics
are
estimates
of T which
of those functions
consists
say the VT?
language we may
subclass
be
the
to
defined
is
of zero (unbiased,
with finite variance).
VTX
Finally,
that FF? is in ?T? for every F? in VT?.
the condition
F satisfy
of ?T whose members
general
not
be real-valued,
if T is a minimal
and
the application
statistic
sufficient
of the
323
Vol.
[Part 4
10]
=
= 0 for
0, or Cov9(VV?)
every
all real constants.4
contains
An
is that
condition
equivalent
Eg(VVQ)
class
Vo
in
The
every
VT?.
#/
always
5.3:
Theorem
A
statistic
0 in *>and
10 is a minimum
in
variance
estimate of its
a
it
is
The
class
and
member
all
estimable
Jg-A
S
expected
of
func
if
only if
of
tions g(6) possessing
minimum
variance estimates is thus obtained by applying
the operator
V
value
E9
to themembers of V^.
strict
If we identify all V in V which are equivalent ($x) in the
then the correspondence
sense,
between
is 1:1.
<? and Vt1
=
To prove the theorem
variance
suppose first that V
v(X) in V is a minimum
=
=
estimate
of its expected
value.
Define
Then
i?r(t) E(V\t).
v(x)
(a.e. $x)
f(t(x))
Rao
and
is
Blackwell's
and
be
thus
V
in
Let
Vo
element
of #T?
theorem,
by
any
?T.
=
and let U
V + XVo, where
? is a constant.
estimate
Clearly U is also an unbiased
of 0(0)
=E0(V),
and
=
V&Ye(U)-VM9(V)
A2V&t9(V?)+2?Cov9(VV?).
This
function
of ? cannot be negative
since F is a minimum
esti
variance
quadratic
=
and
it is easily found in either of the cases
mate,
0, that
consequently
)>0 or
Var^F0
F? )= 0 for all 6 in <*and Vo in ?_?.
is
Hence
in
V
VTl.
Cov^F
Next
g(d)
=
is in V?1
V
that
suppose
=
E9([V). Then if w(t)
and
Also
is a minimum
F
The
last
Lemma
V =
V1 with
statement
5.1:
tion
one for
;> 1; hence
p(d)
4 it
for
Pg
should
be defined
of
(a function
<& is the minimum
the
efficiency
is defined
is one
seems
the members
=
Var?(F)+Var?(F?) > Var^(F).
5.3 follows
are
F1
from
minimum
the following
variance
estimates
of
then
g(6)
<_..
all 6 in
-
to us
of
/>(#)
in the
and
?^,1,
then
variance
to the
=
1, and
of Theorem
light
that
there
be
defined
estimate
variance
lower bound
of g($).
5.3
that
lh(d)+mO)+ih(0)P(d).
exist
the
as opposed
(absolute,
V is unbiased
only when
e) may
=
Var^F+JF1)
constants
Ag,Bg
that
of the minimum
relative
0 for all
=
=
let h(6) =
If for some d,
0, then
Var^(F)
Var^F1).
h(6)
= V1.
one for this 6, V' =
If h(d)>0,
let p(6) be the correla
g(0)
of F and F1, form the unbiased
U = ^(V + V1), and note that
estimate
probability
coefficient
probability
=
EQ(W?V)
this
A(0)<Var0(U)
Thus
is in VT,
W-V
estimate.
in Theorem
V and
If
probability
To prove
with
variance
of
estimate
now have
Var^f/) > VarÎF)
Var?(F+F0)
Hence
unbiased
is also unbiased, and
V*rg(W) <
E(V\t), W =w(T)
theorem.
Var?(C7) by Rao and Blackwell's
6 in o), and so Vo =
IF? F is in #T?. We
is another
U
that
is a member
and g($)
as
feel
#
for the variance
that
may
"efficient
if
be
efficiency
less than
of unbiased
estimate"
class
Vare(F)
shculd
>
not
0,
if Var?(7)
be defined
1, as is the case
estimates
estimate
<g of the theorem
1
given
by
; its efficiency
=
in such
the Cramer-Rao
324
reserved
V of g($)
for example
lity.
that the
be
should
of an
"efficiency"
of the
Varfl(^r)/Var?(F)
We
estimate
term
to relative)
such
0,
where
a way
if
that
efficiency
inequa
V1 =
Ae+BeV.
Hence
E9{V*)
=
I
...
A0+B?E9(V);
=
=
(5.2)
=
=
so
1. If
-1,
2g(0) from
(5.2),
Var^F1)
BQ
Ag
thatP^2
'B9*Vnr0(V),
=
=
and with
Hence
and
to
one, F1
0;
2g(0)- V
U=g(6).
equal
VarÊ/)
probabilityPQ
=
we see that
but
From this contradiction
hence from (5.2),
1,
Var6>(C/)>A(0)>O.
BQ
=
= F1 with
to one.
thus
and
F
0,
equal
probability
Ae
PQ
also,
5.3 in the case
of Theorem
shall now give twro examples
of the application
statistic
is
and
not complete.
$5fc
(actually minimal)
=
x in Example
5.2 : We
3.1.
take t(x)
In defining
statistics
Example
= ?
x
of
definition
matters
the
the
at
function
the points
F ?v(X),
1,
v(x)
only
= V is d3fin3?
th3
class
of
fun3fcions
for
which
series
the
0, 1, 2,... VF
power
v(x)
by
We
where
is a sufficient
T
in0
j=0
has
a radius
^>1. From
of functions
of convergence
T0t? is defined
by the class
*(j)
for such
=
v(?l)
series
functions
the
radius
=
of
of convergence
=
(5.3)
is equal
series
above
the
...
1, 2,...);
0,
l(po, if
the above
to
v(x) satisfying
^Txis defined
by the class of functions
Finally
all
vQ
for
and such that v(x)v?(x ) satisfies
condition
(5.3),
(x) satisfying
(5.3 )
all
v?(x)
such
...
~jv(-l)v?(-l)
v?(j) =-jvO(-l).
we
find that V = v(X) is in ^t1
(5.5),
(5.4) and
<? of extimable
two-parameter
(i
v(j)=v(-l)
Then for F in 3?T\
Eg(V)
class
=
=
=
0}+v(-l)P${X ^
v(0)Pe{X
=?(o )(i-ey+v{-1
){2d-e*).
minimum
g{6 )possessing
of quadratic
functions
=
that
only
In this
constants
0}
variance
estimates
is thus
cx+c2(i-0)2,
of g(0 ) is the statistic
estimate
variance
the minimum
=
X
0 and the value cx if
^ 0.
cx+c2 if X
5.3:
if
1,2,...),
functions
family
(5.5)
if and only
and
to prove
is,
(5.4)
...
=
g(0)
Example
that
that
with v(0) arbitrary.
the
(j
0).
Combining
The
v(x ) satisfying
-j*>(-l)
v(j)v0(j)
for
see that
3.1 we
of Example
calculations
the
we
example
have minimum
shall
see that
variance
V(X
) taking
on the value
it is sometimes
estimates,
325
without
possible
completely
Vol.
[Part 4
10]
the
not be very difficult
in the present
this would
(although
of distributions
of a random
let 3PXbe the family
As in Example
4.1,
example).
on the interval
We
from a rectangular
sample of size n>l
6+\).
population
(d?\,
=
4.1 and consider
define T
From (4.7) we see
the class ^T?.
(TVT2) as in Example
co
that a statistic
of the type F is
in VT? if and only if
f(Tv T2) (a.e. $x)
Eg(V2)<
and
determining
for all 6 in o. =
class VT?
(?oo,
may
of a non-trivial
ct2
J
-foo).
we
be constructed;
r0+?
J
e-\
We
B~\
shall
indicate
in passing
remark
0 ...
f(t1J2)(t2-tlr*dt1dt2=
that
how
certain
there
are
(5.6)
also
of
solutions
periodic
(5.6)
solutions
non-periodic
kind.
Let
on the 45? line, constituting
is a right triangle with hypotenuse
the upper
Ag
=
to
at
of a unit
and
sides
with
square Sg centered
parallel
(6,0),
(tvt2)
=
0
Hold
and in
the axes.
define a Borel-measurable
60 fixed for the moment,
Ag
=
function/
t2) such that the condition
(5.6) is satisfied for 6
60, while
=f(tv
so that
left half
0< ? f
We
the
next
extend
the definition
f2(t2?Qn-2dtxdt2<oo.
to the
of /
square
by
Sg
defining
/
in
symmetrically
lower right half,
/(Mi)=/Ma)>
and we
then
extend
the definition
of/
to the
strip
by the periodicity
0<?2?tx<l
condition
(v,/*
Rt1+v,t2+p)=f(tvt2)
With
the aid of a figure
is satisfied
showing
the triangles
estimates
A0
0,?l,
and
?2,...).
A?
it is easy
to see
that
(5.6)
for all 6, while
Ee(V>)
We
=
=
Eeo(V*)<ao.
can now prove that the only
are constants.
Suppose f(Tv
estimable
T2)
isin^Tx
functions
with
so that
minimum
iff(tv
variance
t2) satisfies
(5.6)
and
E9[f(TvT2)]2<\v(de<*),
so does
...
the product
It will suffice to show
f(tvt2)f(tx,t2).
is a constant
(a.e. L), where L denotes
measure;
Lebesgue
be the same for all
Let 0Q be any value of 6, henceforth
Ag.
(5.7)
that
in every
the
constant
held
fixed,
326
\?r(tv t2)
Ag,
must
then
let A+
be the
part
of
that
it is impossible
where
Ago
and
^>0,
that L{A+)
define f(tL, t2) in
A?
let A_
of
the part
be
be
and L{AJ)
both
A0
where
i?r<0.
If
positive.
We
I
shall show
we
they were,
could
from
=
Kkh){h~tiY-2
llL(A+)inA+,
?
I
...
i -\/L(A_)mA_,
I
0 elsewhere
^
in
(5.8)
A0O.
=
Since / satisfies
(5.6) for 0
above.
described
the method
to an / statisfying
0O, it can be extended
or
have
If A +
A_
every
points within
and the condition
is not bounded
in this way
(5.6) for all 0 by
of the
e-distance
(5.7) may not be
are
if
As
and
if L(A+)
and L(A_)
+ and Ae__ are the
positive,
then for sufficiently
small e, L(AS+)
the strip 0<1?2?tx<e,
jjarts of ^4+ and A_ outside
now
If the above definition
modified
and L(Ae _ ) will also be positive.
of/ is
by replac
and
__
satisfies
all
0. But
in
and
A
(5.6)
(5.7)for
(5.8)by Ae+&nd Ae_, then/
ing A +
while
in the rest of
then ^/cannot
(5.6) for 0=0O for in A*+ and Ae__, ^/>0,
satisfy
45? line, the / defined
satisfied.
However,
Ae
, \?rf=
0, and
thus
mz-hY^dt^?.
f ?A
We
now
have
where
(a.e. $?x) is in V^1, then the sets in
T2)
f((Tv
both
where
have
positive
i?r(tv t2) < 0, cannot
f2) > 0, and
constant
for every
But
if F is in V^1 so is V?c
c, and so it
^(?1?
measure.
Lebesgue
follows that for every
Aeo
if F =
that
shown
c the
sets
in
AQ
where
i/r>c
and where
i?r<c
cannot
both
From this it can be shown that for some constant
c, i/f=c(eL.e. L)
positive measure.
=
c
Thus
then
the
V
is
in
if
F
and this proves that
V?1
in
(a.e. $x ).
only estimable
A0O,
are
the
constants.
estimates
variance
with minimum
functions
have
6.
CONSTRUCTION
AND
OF
EXISTENCE
MINIMAL
SUFFICIENT
to impose any restrictions
necessary
in this section we shall assume
However,
So far
it has not
been
p9(x)
integrable
(jp,
STATISTICS
on the
?J9X
family
that the measures
measures.
of probability
is in
with respect to some measure
continuous
in J?x are all absolutely
?ix on Jp which
union
of sets in jp of
that Wx is a countable
of 0 and has the property
dependent
existence
for
the
all 0 in ?>of a
to
is equivalent
finite measure
assuming
?ix. This
function
?ix) such
that
for all A
in jp
Pex(A) =^p0(x)d?ix.
We
shall
refer
to this
situation
by
saying
with respect to ?ix (it is the "Nikodym
pQ(x)
includes of course the two
This situation
a generalized
probability
density
mentioned
in section
derivative''
dP0x/d/ix
3).
cases with fixed sample size
important
known
there
exists
327
Vol. 10 ]
[ Part 4
as the continuous
on a Euclidean
case, where px is Lebesgue measure
space Wx, and the
discrete5
of the sample point X are included
case, where for all 6 the possible positions
in a fixed countable
set {x1} (i = 1,2,3,...)
In the latter case we may
of 6.
independent
take for /?x (A ) the number of points x'1 in A, and for
the probability
that X = x{
pg(xl),
when
the probability
The generalized
distribution
of the sample
is
density pg(x)
Pgx.
is not uniquely
and
two
determined
determinations
for the same
by Pgx
jux; however,
6
are
(a.e.
equal
?jlx
a.e.
and
Pgx).
For most
of the families
there exists
considered
statisticians
S?x ordinarily
by
a generalized
or "most
some
to
and
the
with
if
respect
'"simplest"
density pg(x)
?ix,
for 5PXcan be
natural"
is used, a minimal
sufficient
of
statistic
determination
pg(x)
an operation
to
found by applying
to the family p =
below.
be
described
#
[pg(x ){dea}
on which determination
of the
does depend
the result of this operation
Unfortunately
can
which
This introduces
how
certain measure-theoretic
difficulties,
family p is used.
ever
be
surmounted
and
by
applying
as we
shall
it to some
show
examples
later.
We
of families
the operation
#
begin by defining
of distributions
of some statistical
interest.
these
f(x ) on Wx, and suppose
functions
on values
in
The result
A,i={fg(x)\6eA}.
of the operation
For
of Wx, to be denoted
#
# on f is a ?ecomposition
any point
(().
by
x? in 1FX the element D of & (t ) containing
is defined as the set of all
x?, written D(x?),
on 6, and such
not depending
there exists a function
k(x, x?) ^0,
points x for which
=
we may
all 6 in A. Roughly
say that D(x?)
k(x, x?)f9(x?)?oT
speaking,
tha,tf9(x)
of 6. We note that if
consists of all x for which
the ratio
(a:0) is independent
fg(x)\fg
x' is in D(x?),
then x? is in D(x')\ also, that
Let
f denote
are indexed
a family of real-valued
6 taking
by a subscript
D? =
{x\f9(x)
functions
= 0 for all
6eA}
of # (f). It may be shown (as in the proof
of measurable
number
of a countable
(Jfx ) functions
is an element
consists
of Theorem
then
(6.1)) that if f
a measurable
is
#(*i)
(Jp) decomposition.
consider
Example
binomial
6.1:
population
now
five
examples
=
is
..., Xn)
Suppose X
(XX,X2,
on
with
6, X{
taking
parameter
6 and l?d.
Let /ix assign measure
a
the
random
values
case
In each
the operation
of applying
#.
as
of
Borel
the
taken
a
and
be
Wx is Euclidean
Jfx may
n-space,
family
on
as
measure
be
taken
will
but Example
Jp.
6.1, /.x
Lebsgue
We
sets.
sample
1 and
In all
a
from
0 with
1 to each of the 2n points
probabilities
respective
zero
or 1, and measure
of the points
in the set W+x consisting
(xx, x2,...,xn) with x{=0
take Jp as the family of all subsets of W+x,
In this example we might
to Wx? W+x.
6 In
^
9 _4 rs
present
for
discrete
the general
case, while
can of course be non-denumerable.
\J0A0
each
The
of X
values
$ the possible
case of non-denumerable
constitute
A
treatment.
32&
is not
a countable
included
set
in
the
I
or the family of all subsets of Wx, instead
Interval O<0<1,
For ? we take the open
jÔ
For
D?
this
left member
?(x,
?
a. =
i-l
and
tinct
this
conclusion
x? not
(6.1)
x is in D(x?)
and hence
a:0)^(x0),
*
=jc(XyXo^
i
if and
in D?,
and only
if
only
...
(e.2)
if
...
E ^?,
(6.3)
i-1
if (6.2 )were required
0 and 1. The
between
even
be valid
would
in Wx.
...
if xeW+x,
of 0 if
is independent
of (6.2)
L
i
[0/(1?0)]
The
p0(x)
sets
otherwise.
For
Wx?W+X.
=
such that
p0(x)
specification,
there exists
Jc(x, x?)
for
! j-i
=
family of all Borel
the determination
the usual
fn
77 0^(1-0)1^
=
p(x)
of
and
to hold
only
for two dis
of 0 instead of all values
resulting
decomposition
of (6.1) may thus be des
the application
of the operation
& to the family
{p0(x)}
: The decomposition
cribed as follows
is equivalent
(J?x ) in the strong sense to that
of this. For anyx?in
the statistic
with
function
associated
or
1:1
any
2-Xj
W+* the
values
from
element
D(x?)
consists
of the
n-\-2
elements
D,
contains
The
same
decomposition
of two
{Pex) consisting
points
(s^0/
namely
(6.3). The decomposition
n
sets D where 2 xi=v(v=0,
1,..., n).
i=l
in W + satisfying
D? and the-%4-1
is obtained
if the operation
elements.
or more
#
is applied
to
any
subset
of
6.2:
Let X be a random
with
from a normal
Example
population
sample
mean
5.1. With
the determination
(5.1 )of the proba
02 as in Example
0xand variance
we
of the fraction
is
the
find
D?
the
Since
denominator
bility density p0(x )
empty set.
cannot vanish, D(x?)
of 0. This
is the set where
this ratio is independent
Pq{x)Ip0(x?)
is seer. 13 be the same as the set where
is independent
of 0, namely,
the set where
2î
The
tic
decomposition
(2X., 2 X{2);
i
=
2^?
i
and
2^?
=
2^?.
t
$> is thus that associated
by the operation
is the same as that associated
with the statistic
induced
this
i
with
the
(TX,T2)
statis
defined
in Example
5.1.
It may be verified
the same result if
in this case that one obtains
the operation
to a set of any
of
# is applied
three pf the members
{p0(x )}, say for
?
=
*
are
not
the
three
collinear.
1,2,3, providing
points
(0?5 02i)>
(0V 02)
(0^, 0?i)
329
4
6.3:
Suppose X is a random
Example
with median
-?oo<0<
6, and <t>
?{61
+co}.
bution
[ Part 4
Vol. 10 ]
of n from a Cauchy
sample
the usual
We take for
pg(x )
distri
deter
mination
=
T.-" n
p$(x)
is the set where
Again D? is empty and D(x?)
This
be written
ratio may
n
it
.?1
the ratio
is independent
pg(x)lpg(x0)
of 6.
[(o-xf-iKO-xf+i]
n
[(g-X}-i)(0-Xi+i)]
i=
If this quotient
is independent
^?1.
must have
and denominator
in 6 in the numerator
where
-0)2]-i.
[l+(a-
of 6, the polynomials
of degree
2n
we
conclude
equal roots, from which
of (x\,x2?,...,xnQ).
be a permutation
The
..., xn) must
decomposition
(xvx2,
as that associated
with the set of ??order statistics"
of
induced by ? may be regarded
a
is
of
where
the sample,
rearrangement
namely
Z1îZ2î...^tZn
(Zv Z2,...,Zn),
same
be
for
of
found
subset
The
result
would
any
...,Xn.
Xx, X2,
{p9(x)} corresponding
to 2n+l
values
of d.
distinct
that
6.4: As in Examples
4.1 and
Example
on (d?\, 6+\),
distribution
the uniform
1 if
With
for all i, as 0 otherwise:
|#i?d\ <\
n from
as
see that pg(x)
=
0 otherwise.
pg(x)
i
a
random
of
sample
determine
pg(x)
= max
x{, we
x-v t2(x)
We
i
1 if
0-i<?1(a:)<?2(^)<0+i
?
5.3 let X be
?
oo<?<oo.
= min
tx(x)
...
(6.4)
Therefore
D? = {x\t2(x)-tx(x)^l}.
vanish
for the same
for any x? and x, x is in D(x?) if and only if pg(x) and
pg(x? )
<
if and only
if (6.4)
is
that for x? not in _D?, a; is in D(x?)
set of 6 in >. It follows
w
\
true for all 6 in which
satisfy
Now
x is in D(x?)
if and only if i. (x) = t. (x?), j == 1, 2.
^ is thus equivalent
The decomposition
imposed by the operation
(3PX) in the strong
4.1
same
and
of
5.3. The
sense to that associated with the statistic
Examples
(_T1?T2)
a
to
if the operation
subset of
denumerable
is obtained
ft is applied
decomposition
It follows
{p?x)}
that
for a;0 not
in D?,
to a subset
corresponding
of
_.
everywhere
dense
in c..
and
independent
Zn)
(Z1,Z2,...,Xm)
(Zm+1,...,
means
normal
and
with
from
and
variances
random
samples
populations
6X
dx+S,
of the Behrens-Fisher
The null hypothesis
then
problem
62 and ?3, respectively^
==
?
a
so
is
for
known
that
where
the
the
d
oonstant,
<PXof
S0,
S0
family
specifies
Example
6.5:
Denote
by
330
I
distributions
probability
Euclidean
space
under
satisfying
the null hypothesis
~oo<01<+oo,
02>O,
of a three-dimensional
is the subset
If we determine
03>O.
the
density
as
^0*0
1
it is easy
to show
that
of
The
rrieas?fe-theoretie
a minimal
statistic
sufficient
n
& to this three-parameter
of applying
the operation
the
statistic
with
associated
is the decomposition
the result
functions
density
four real
(Tx, jP2, T3-,TA) with
the two sample variances.
family
i
m
components
of the
consisting
two
sample
?
the operation
in applying
difficulty
sufficient
If a minimal
is the following:
means
and
to construct
statistic
for
the
sense.
If
family
$JX exists then it is unique up to equivalence
($x) in the strong
to
the
with
a
measures
respect
3$x of probability
density
possesses
generalized
pe(x)
for fixed 0 differ
of
two determinations
measure
p0(x)
/?x, then p0(x) is not unique;
?
=
is
if
0.
Now
on a set
is in Jp and for which
which
non-denumerably
/ix(A0)
A0
wmay not be a null set for ?ax\ it need not even be mea
0
in
union
of
the
for
infinite,
A0
of the operation
from the application
The decomposition
surable (jp).
#(p) resulting
determina
on which
densities
# to the family of generalized
depends
p={pg(x)[de<?}
that
it can be shown
6.1)
tion is chosen for p. As long as w is countable
(Theorem
a
sufficient
minimal
with
is
associated
the decomposition
(Jp), and
#(p)is measurable
from
seen that the two decompositions
it is easily
#(p) resulting
sense.
the
in
and
of p are equivalent
strong
determinations
two different
3?x)
(?ix
of p could
<*>
determination
the
of
choice
a
is
not
if
countable,
However,
pathological
of two
the
is not measurable
equivalence
? (p)that
lead to a decomposition
(Jfx), and
This
of p could not be proved.
determinations
two different
from
?(p) resulting
to
the
lead
? need not necessarily
of the operation
that the application
shows
statistic
and
for ^x,
sufficient
minimal
This
statistic.
difficulty
is resolved
as follows:
We
restrict
to families
ourselves
$x
for
sense to be
in a certain
is separable
densities
{pg[x)}of- generalized
There
a
is
Euclidean
space.
this separability
defined below,
always holds when Wx
the operation
in the whole set, and we apply
dense
subset of
is then a countable
{ve{x)}
we noted that in every
subset.
<&to this countable
(Recall that in the above examples
0 to a suitable countable
the operation
case the same result was obtained by applying
below it follows that the result
On the basis of the theorems
subset of the densities).
is a measurable
subset of the densities
of applying
the operation
# to the countable
for the whole
statistic
sufficient
is associated
with a minimal
which
(Jp) decomposition
which
the
family
set J?x of distributions.
Theoiem
jp,
possessing
the operation
?
6.1:
The
Suppose
result
needed
3?x
a generalized
probability
to a particular
is applied
for countable
subsets
is
on
measures
of probability
to ?ix. If
with respect
density function
p0(x)
the
determination
resulting
of the family
{p9(x)}9
is a
countable
set
331
Vol.
10]
[Part 4
a minimal
is measurable
is associated
with
decomposition
(Jp) and
sufficient
the
statistic for 3PX. The decompositions
operation ft to two differ
resulting from applying
are equivalent
in the strong sense.
ent determinations
of the family
(/ix and ^x)
{p9(x)}
The
(1949)
of this theorem
proof
a theorem
generalizing
to be a sufficient
statistic
generalized
with
density
of Neyman
=
_>,(*)
for
a suitable
determination
(Jp) and h(x) is integrable
on
and
Savage
this generalized
pg(x) of
(Jp, p*).
and
{fi(x)},
ft(l)
- (6-5)
.,(*(*))*(*)
6.1 let us denote
prove Theorem
some
determination
(?, and hyf^x)
particular
?=
of Halmos
(1935):
To
.... Write
result6
condition
A necessary
and
for the statistic T=t(X)
sufficient
=
a
a
measures
set
|?x
of probability
for
{Pgx \de<*}possessing
such
that
and
h(x)
respect to ?ix is that there exist functions
gg(t)
6.2:
Theorem
rests on the following
density,
where
gg(t(x))
of the
by 6{ the elements
of the generalized
density^
for the decomposition
generated
by
is measurable
countable
i=
i (x),
the operation
set
1,2,
ft
f.
If Df =
then
ft (?) is measurable
(Jp).
{x\fi(x)=0}9
=
(Jp) since f{(x) is measurable
Di? is measurable
(Jp); hence D?
C\iDi? ^smeasurable
now
a
that
x?
i for which
is
not
smallest
in
exists
there
_D?. Then
(Jp). Suppose
Define Di (x?) as the set where fi(x)^0
and
say I=I(x?).
fi(x?)=?0,
shall
We
show first
that
fi(x)\fi(x)=fi(xO)\f1(x%
Since f{(x) and/i
Wx where fx(x) ^0,
in Wx? Dj?,
the part of
(x) are measurable
(Jp) so is their quotient
and hence the part D{(x?) of Wx?D-?
where
the quotient
has a cons
=
tant value
is measurable
is measurable
If
(]i A(x?)
(Jp).
D(x?)
(Jp). Therefore
for any point x? in Wx, D(x?) denotes
is
of ft(t) containing
the element
x?, it
easily
verified that for x? not in5?, D(x?) = D(x?), and for x? in J9?, D(x?) =D?.
This completes
the proof that ft(t) is measurable
(Jp).
Let
be another
of the generalized
then
determination
densities;
f'=-{/'.(a;)}
is the set where f\
is
also
is
in
Hence N=ViNi
Jfx and /^(.ZVJÔ.
Ni
(x)^f?x),
a null set for
so are ft (t) and ft (V). This
t and i' are identical hence
?ix. On Wx?N,
if N{
proves
the
strong
Next
we
equivalence
(ux and
3PX) of the
two decompositions.
shall prove
contains
if D(x) is the element of ft (i) which
x, and
that,
we
if T?t(X)
is a statistic
could
take
with v ft(t),?in
associated
particular
t(x)=D(x)
?then
6.2.
To this end we
be factored
in the form
(6.5) of Theorem
f{(x) may
choose in each element D of #(f) a single point x?=x(D)If the set of points x? thus
a function
chosen is denoted
x?=i?r(x) from TFX into TF?,
by W?, this choice determines
6We
paper
exactly
are
containing
the
form
to Professors
indebted
this
stated
result
here;
Halmos
and
long before
the
The
publication.
connection
is discussed
Savage
reader
for
will
giving
not find
in the Appendix
us
the
a manuscript
copy of their
6.2 there in
above Theorem
to the
present
333
paper,
I
since x in Wx determines D=D(x) in # (f), and D determines x?=x(D)=X(I){x))=?f(x)>
Take any D in #(f) and hold it fixed. Then for all x inD and x?=x(D)>
where
i=
and
k(x,x?)^=0,
This
1,2,...
may
be written
fi(x)^k(x,xlr(x))fi(x(D)^
For
a in WX~D?
define
define
of
Define
h(x)=0.
now
We
have
/(#).
?(x)=Jfc(a;, f(x))^0,
where
9i(t)=fi(x(D)),
for x in D?
and
D
the elements
of #
-
fi{*)=m*))M*)>
(t) are the
contours
(?.?)
and
{a:|A(x)=0}=D0.
6.2 except
that we do not
form (6.5) of Theorem
(6.6) is of the factored
so as to satisfy
can be determined
the measurability
the factors
know yet whether
this let us denote
in proving
For brevity
conditions
and integrability
of the theorem.
is a union of elements D of # (f); the
by <?t the family of sets in WX each of which
Now
of
members
<gt need
not
GM
be
in jp.
is a sequence
{GJ
show that all Oi are
shall now
K{
=
{x\\{x)
=
=
x\fx(x)=f2(x)
It is clear that
i =1,2,...,
For
...=fi_x(x)=0>
in<?t.
a; in Gk(k =
this with
(6.6), we
(H*-*,)
...nî-in
1,2,...)
A(z)
Combining
get
we
=
have
from
(6.6)
...
gk(Kx))h(x)ô.
for x in 6?k and
?=
<?? (i
=
1,2,...)
-
define
*(*)
=
2-*A(*).
-
(6-9)
and in i((?k) tbfine
ft(0
(6.7)
1, 2,...,
/i(*) = [U^WS^W^Y
In
and we
U iGi = Wx ?D?,
=
so Hx is in <St. Let
0},
= 0 =
} H{ U D?, so Jft is also in <?t. Finally,
0}={x\gtf(x))h(x)
For
/i(*)#0}.
of disjoint
sets in Jfx with
in <gt: Let Hi =
{x\g^x))
oriin?in
is also
define
= 2fi(*)IW*).
333
(6-8)
Vol.
[Part 4
10]
and gt(t)
noting from (6.7) that gk(t)^0 in t(Ck). This defines h(x) in Wx-D?
for
t(Wx-D?)
?=
in D?
1,2,...;
in
define
...
h(x)=0,
(6.10)
in t(D?), g^t) = 0. To see that there is no inconsistency
in our definition
of g^t)
it is important
to note that the sets t(D?) and t(Gk)(k =
are
This
all
1,2,...)
disjoint:
is a consequence
of D? and Gx,02,...
sets in <6t. From
being disjoint
(6.8) we have
and
=
/_(*)
...
g?t(x))h(x).
(6.9) and (6.10) we see that h(x) is measurable
sets 6?k and D?, and since these form a countable
disjoint
in Jfx, h(x ) is measurable
That it is also integrable
(Jp ) in Wx.
(6.11)
From
=
|A(*)|?/**
J TFX
J[
f
with
We
T =
To
denominator
non-vanishing
now
have
it is possible
tf(x) is contained
If T'
the generalized
that
to find a null
in a contour
is sufficient
g^xj)
density,
if T' =
6.2 to be satisfied
set N"
to prove
and hence
it is minimal.
for 3?x,
statistic
t'(X ) is any other sufficient
of
contour
for $?x such that in Wx?N"
every
say/.'(a;),
then by Theorem
such that for i =
(Jfx)-
The
=
1, 2,
a determination
exists
6.2 there
of
...,
gi'(t'(*)W(x),
V ?
determination
{/'i(~)} may
?
=
previous determination?
{f^x)} to the extent that if JV?'
iV/is
W'-N''.
that
(Jp)
(jfx) in TFX.
is measurable
oft(x).
for $x,
is measurable
h'(x)
in Wx?D?,
all the hypotheses
of Theorem
statistic
for iPx; it remains
only
/i'(*)
where
Gk
shown
a sufficient
.(X)is
do this we must
show
then
(Jfx,/?x ) follows
by sets
from
in J_)?while from (6.11) g{ (t(x)) is the quotient of measurable
Since g{(t(x))=0
functions
of Wx
covering
the
= %2-*< oo .
fk(x)dp*
S 2-k f
>
of
each
h(x)dv?=t
kn J\ fk(x)dp
JF-Z)0
4
in
(Jp)
set for ?f. Hence N' = Ui-^Yisanullsetfor/i*
Let N0 = {x\h'(x) = 0}. Then for all ?, /,'(?)
a null
from
3?x, and/?'
the
then
{^{fiix)^^)},
and
=
differ
=/j
on
0 on N0,
J N
i
0
and
so
_V0 is null
WV^/i'i*)
=f,(x)
We
shall now
set
for $x.
LJ^"
= N'
U N0,N"
is a null
set for $x,
and on
and *'(*) ^0.
show
that
in a con
every contour of ?'(a;) is contained
of
and let A(x?) be the part in Fx-r'iV
in Wx?N",
in Wx?N"
tour of t(x). Let x? be any point
the contour of t'(x) containing
x?,
=
A(x?)
{x\xeWx-Nn,
t'(x)
=
t'(x?)}.
334
I
For all x in A (x?)
=
m=fif(x)
9iW))h'(x);
in particular,
fi(x?)
=
9iV(x?W(xQ).
Since h'(x?) ^0,
f{(x)
for
all x
in A(x?),
=
from
containing
x?.
which
-#0.
is contained
in D(x?),
A(x?)
(6.12) that in Wx?N",
This
the
Theorem
of
6.1.
proof
completes
In proving an existence
may be non-denumerably
whose
for minimal
theorem
we
infinite,
shall
sufficient
employ
the
statistics
contour
of t(x)
for families
the following
two
3?x
lemmas
is obvious.
proof
Lemma 6.1: If the statistic T is sufficient for
measures
(6.12)
where
k(x,x?) ^h!(x)\h'(x?)
It follows
...
k(x,xQ)f^)
it is sufficient
for
any
3?x.
of
subfamily
thefamily 3$x of probability
3PX, if it is a minimal
for the family
If T is a sufficient statistic
null
set
for 3?x is a null set
3$xx of 3Px, and if every
sufficient statistic for the subfamily
for 3?x, then T is a minimal
sufficient statistic for J?\
6.2:
Lemma
Our
sarily
of the existence
extension
case
countable
theorem
from
the
countable
to the not
neces
of a function
of separability
space with
=
one.
and
of order
<7?g(x) are two
f(x)
It/
as
(jp, /?x), we define their distance
integrable
the notion
involves
in the mean
pect to convergence
on Wx,
valued functions
*,(/,?)
=
res
real
[ *\f{x-g(xWx.
Jw
each integrable
f=
a family of real-valued
functions,
{f0(x)\0eu>},
a
of
that for every
subset
such
countable
f
fixed
tx
(Jp,/?x), is separable (?ix ) if there exists
in f there is a sequence {g{ \i= 1,2...} of functions
S? (/,?i)->0 as ?->oo.
g{ in tx for which
fg
is dense
subset tx of this definition
to say that the countable
It will be convenient
We
shall
that
say
remark that if Wx is a Euclidean
space and Jp is the family of Borel sets
(?ix) in t. We
to ?ix is separable:
densities with respect
in Wx then any family {p0(x)} of generalized
all
of
functions
of the separability7
is a consequence
This
integrable
(?ix) of the family
a
a
in
of
metric
of
the
and
family
space.
any subfamily
separable
separability
(jp, ?ax),
the operation
# in each of Examples
that the result of applying
We can now conclude
6.1 to 6.5 was
just made
to give
about
Euclidean
obtained
being
composition
of the generalized
i This
(1948,
Chelsea,
may
a minimal
densities,
be
proved
sufficient
statistic:
This
follows
from
(i) the remark
the same de
about
in the examples,
(ii) the remarks
to
is
certain
countable
#
if the operation
applied
theorem.
and (Hi) the following
Wx,
from
the
Approximation
Theorem
on page
4 of Ergodentheorie
N.Y.).
335
subsets
by Hopf
Vol. 10]
6.3:
Theorem
the family
i/
$x
measures
of probability
p=
a generalized density
p0(x) with respect to ux, and if thefamily
[Part 4
on Wx
possesses
of densities
{p0(x) |0e<*>}
is separable (jax), then there exists a minimal
sufficient statistic for* 3?x, and it may
tructed by applying
the operation & to any countable set px dense,
(?ix) in p.
be cons
let tx = {f{(x)}(i = 1,2,...) be a particular
determination
a
of
countable
dense (?ix) in p, and let 3?xx be the countable
px of densities
subfamily
of
to
6.2 we must
3$x corresponding
px. In order to apply Lemma
subfamily
x
is a null
that
if N
set for $xx
show
it is a null set for $x.
Let
be any
P$
a sequence
and let
member
of $x,
let f0(x) =
in tx for which
{grj be
dPgox?djix,
To prove
the theorem
0 as ?->
^/?(/oî) ->
oo. We
have
J?N
for
all
i.
= 0
9i(x)dux
Now
II = L
I| ?*N h^x- I??N gid/i*!I
f?JNMx)dj*= Ij VfN t f^x
I
.< [J N \U-9i\dj?
(f9-gMr?
I
I*J( N
Hence
= 0
f /?(*)i/?x
J N
and N
the
ing
remains
is a null
set for 3$x.
Suppose
T=t(X)
of
dent
sufficient
# to tx. To prove T
operation
to
prove that T is sufficient
only
a real-valued
exists
there
that
is a minimal
and
0,
such
function
statistic
for
is a minimal
$3f, obtained
sufficient
statistic
for
T
P(A\t),
3?\
That
measurable
is sufficient
(Jp) for fixed
by apply
for *$x it
'or
means
3?x
A,
indepen
that
P*{At\t-*{B))
=
...
] t P(A\t)dP?
(6.13)
to 3?xx.
in :Jp, B in Jf*, and 0 in av where &v is the subset of w corresponding
to
show that if (6.13) is valid for all 0 in u>v
for J?x it thus suffices
To prov? T sufficient
of t~\B).
then it is valid for all 0 in c*. Let us write B' as an abbreviation
for all A
We
may
transform
(6.13)
to
(
J AAB'
Where
n(A,
x)
=
P(A\t(x))9
"
?
JB'
tdP^=?n{A,x)dP^
or
^^
? Ane'
We
note
t><w(ii,a;)<l
(a.e. $x*)
by
=
?/?X
^
f b'
(2.5) and
xWx)dPx-
(2.8), and hence
-
(a.e. $*).
336
(6-14)
Let
0?
0?
=
be any 6 in -, write
as
i ->
oo.
in fx for which
(">(?>g^-Ô
From
f
...
fJ b' *?,_>*
gidf=
J aab'
we
let {gr?}be a sequence
g(x), and
i>0o(~)
I
(6.15)
shall prove
=f
gdftx
J
f
J A/IB'
i?>oo and
by letting
are the corresponding
that
showing
the
of
members
limits
...
ngd/f
B'
of the
and
right
(6.16)
of
left members
(6.15)
(6.16).
\gi-g\dn*
,gidp*-J\ AAB' gdfiA^
J[ AilB'
|
[
I J AAB'
<S_-(0i>0)->O.
<
This
completes
[ , \9i-9\fox<*?(gi,9)->0.
J B
of Theorem
the proof
6.3.
APPENDIX
With
torization
of which
only
notational
changes,
on
result
the
sufficient
and
statistics
the
and Savage
stated by Halmos
densities
of generalized
probability
their
a
is
different
is
6.2 above
Theorem
version,
following
slightly
fac
(1949),
additive
1 : Let
<?l be countably
Jp and
families
of sets in the
Corollary
=
t
t(x) be a function from Wx onto Wt such that for
spaces Wx and W{, respectively. Let
be a family
Let
(#)
every set B in <fe\ tr1 (B) is in Jp, and let <?Y ^t'1
Mx={Mgx\de?>}
to
with
the
continuous
on Jp with all
measure
respect
finite measure
of finite
Mgx absolutely
to be a sufficient statis
condition for T=t(X)
and sufficient
on Jp.
Then a necessary
?ix
=
be
in the form
tic8 for ?lx is that for every 6 in <*>,
factorable
fg(x)
fg =gg h,
dMgxjdpx
are
and
and
is measurable
where 0^gg
integrable
(Jfx, px),
h(x)=Q
(f?3^), 0-Â
0^ggh
on every null set for 0lx.
(a.e. /.x)
we have obtained
arise from the restrictions
of the simplifications
take for (gt the family JP of all sets in W* whose pre-images
always
Some
first, we
8 If
Jf?lx
Savage
define
these
definitions
the
only
case
is n?t
statistic,
here
considered
a family
of probability
conditional
except
by
probability,
to remark
us?the
that
definitions
T
measures,
in the
is not
statistic
sufficient
and
case
agree
that 0[x
with
is a
a
random
variable.
for this
case. We
family
of
ours.
337
5
need
probability
that,
are
Halmos
not
in
and
consider
measures?
Vol.
10]
$*> and that, secondly, we
6.2 follows from Corollary
assume
#ftx to be a family of probability
six steps:
of the following
measures.
Theorem
1 by means
that t(x) satisfies
assumption
in <6* are in Jfx is automatically
for
fulfilled
=
take
<gt
Jp as just mentioned.
always
Io.
[ Part 4
SANKHYA: THE INDIAN JOURNAL OF STATISTICS
the
The
condition
an arbitrary
of sets
pre-images
function
t(x) because we
that
out by Halmos
and Savage that in their work the assumption
2?.
It is pointed
on /?x that
that Wx is a countable
?ix(Wx) is finite may be replaced by the assumption
union of sets of finite measure
(/?x).
3?.
The
is non-negative
and integrable
that
assumption
(Jp, /p) may
ggh
of our assumption
is a
be dropped because
of probability
that ?lx
measures,
family
=
so that
and
be assumed
non-negative
ggh
fg may
=
1,
\wx9ehdpx
on
of non-negativeness
since we
be dropped,
gg and h may
as
h
and
then
and
and
redefine
may
\gf?\
gg
|h| without
fg non-negative
or the integrability
the product
of
of h.
the measurability
affecting
gg,
ggh,
to the assump
5?. The assumption
that ge(af) ismeasurable
(<?xlt) is equivalent
2 in the same
ismeasurable
where
tion that it is of the form
(<?*) by Lemma
gg(t)
g9(t(x)),
our choice of <?*= Jp,
is measurable
and Savage.
But with
paper by Halmos
gg(t)
4?.
The
condition
assume
(45*)if and only if g9(t(x)) is measurable
(Jp).
the assumption
h(x)=0
(a.e. ?ix) on every null set for $lx,
proof
condi
a little more effort.
If h satisfies all the remaining
be dropped
may
requires
all the
also satisfies
tions of Corollary
1 we shall show that h can be defined which
=' 0
and furthermore
conditions
(a.e. /?x) on every null set for fflx.
remaining
h(x)
6?.
The
that
=
for which /e>0.
a particular
of
determination
g h denote
dMgx?dfix
Let/e
a countable
subset 0lxx of 0LX,
7 of Halmos
Lemma
there exists
B
and Savage,
=
=
are
null sets for ?fft*.
sets
for
all
null
such that
0XX
say 0li
dx, #2,...},
{Mgx\6
= 0 for all i. Then
Denote
_40 is a null set for fflxx
by _4? the set in Jp where fg, (x)
and hence for $lx.
Define
f
h(x) for x in Wx?
^
0 for a; in _40.
_40,
For all A in Jp,
=
f 9ehdlf= M A A) = Mgx(A-A0)Ja J[ a-ao g9hd,ix
= f
gfi?p= ( 9$df>
J A-Ao J
=
greA
determination
and. hence
as a
bility
A
=
instead
thus use
g h (a.e. /.x). We may
fg
ggh^0
the
condition
and ? obviously
satisfies
of dMx\dpx,
(Jp, ?ix) since h did.
338
of
of
/?
=
ggh
integra
I
It remains
to prove
only
Let N' be the part
(a.e. jix) on N.
Now
if N
that
of N where
0 = M g X(N')=
Let
AT. be
the
of
part
N'
where
= 0.
X *# 0, and thus
If
^.
= Ofor all ?. Hence N'?N"
ggi
0 on N', and
?y=
therefore
/?x(Ar/)
so W-N"
to prove
=
0
=
0.
h(x)
?ix (N')
g0 hd/ix.
so
geh>0,
is contained
On
0.
=
?ix(N")
in A0,
set.
=
/^(JVj)
then
is the empty
for j?lx then
set
null
0; we need
h^
f
JV" = U^,
a
is
and
0,
? =
=
2V'? Ari,
0,
gr^?
on JV'
and
iV",
But
0 on N'?N".
Since N'
contains
unbiased
sequential
=
N'
N",
and
#",
0.
References
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of

Indian Statistical Institute

Transcription

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