Generalized Nonparametric Deconvolution with an Application to
Transcription
Generalized Nonparametric Deconvolution with an Application to
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" #$ %& ' ( ) ' ' * ' ) ' +, - ) * -) ) . ' ' / 0) 1) !/01$ ) 234 5678 , #9 , #: ; 7<=568 > ' > ) ! F $ ' ! F $ ' ' ) ( ) ( ' # ! " #$%& > ' ) ' - ' ) ' ' ! + # ! & , # ! $ > ' ) " # * !# $! $ ') ! %- & $ + ) ' ) ! ' ' & (($ & (($ ) * & ((! $ " ) F . + !# + $ - ) ( ) ' h ,-,./ ' - ) ' ' ' ( ) ) ! ' '$ ) . + ' ) 0123 45467 6889:6;2 <5 =: 6>:8= ?3@<A73 8696B3=9<; ><5=9<A4=<:C5 D:9 =23 <C><E<>467 3F3;=5 6C> =23 <CC:E6=<:C5 G3HI HJ K26BA3976<C 6C> L<96C:J MNNNJ L<96C:J . OO . J P3Q3R3 6C> S36C3J . OOO J . OOTU H VW:9 93D393C;35 :C =23 ;7655<;67 >3;:CE:74=<:C 89:A73B J 533 K699:77 6C> L677 GMNXX U J Y26C I GMNN OU J W6C GMNNM U J 6C> K699:77 Z[ \]^ GMNN @ U H _`45R: 6C> a6>:C G. OO . U ;:C5<>39 =23 56B3 53=48 6C> 35=<B6=:9 65 L:9:Q<=b 6C> c69R6=:4 Q<=2 6C 6887<;6=<:C =: I 3C3 3@89355<:CH c:93 93;3C=7dJ e376<I 73J L677 6C> c3<5=39 G. OO X U 26E3 89:8:53> 6 B:><f3> R39C37 35=<B6=:9 =26= B6d 6;2<3E3J 4C>39 5B::=2C355 ;:C><=<:C5J =23 56B3 65dB8=:=<; 839D:9B6C;3 65 <D =23 ><5=9<A4=<:C :D 399:95 Q393 RC:QCH # ) + ) ' ) ) ) ). ' ) . ) # * !# ! $ # * . + ( ) ) ) ) ¤ ( - ) !# +$ $ ! ) / # & &#$ 0. ) , !& ((( $ . ) - ' ( ) ! " #$%& '' ' ) ) ) ! " #$%& ) ) ' ) . +' ' ) % & 1 ' !& (( :$ * ' ' ' ' # * !# ! $ ) ( ' ) ) ') * ) ( ' ) ' ' 12353 3@=3C5<:C5 693 >3E37:83> <C L:9:Q<=b GMNNXJ 8H M .@ M0 U H < 6C> 4:C I 5 35=<B6=:9 265 A33C 453> Ad < Z[ \]^ G OOOU <C =23 ;:C=3@= :D 6 5=94;=4967 64;=<:C B:>37J 6C> <C < G. OO . U <C 6 C:C7<C369 399:95 <C E69<6A735.B:>37H L677 6C> 6: G OO 0 U 89:8:53> 6C 35=<B6=:9 =26= <5 ;7:537d 9376=3> =: < 6C> 4:C I GMNNX U H 376=3> B3=2:>5 26E3 A33C .453> Ad `;23CC6;2 G. OO = 6J . OO = A U <C =23 ;:C=3@= :D C:C7<C369 93 I 9355<:C 6C> C:C8696B3=9<; 93 I 9355<:CJ 93583;=<E37dJ Q23C =23 93 I 9355:95 693 B365493> Q<=2 399:9J 6C> Ad L4 6C> <>>39 G. OOTU <C :9>39 =: >367 Q<=2 B365493B3C= 399:9 Q23C B69 I <C67 <CD:9B6=<:C <5 6E6<76A73H & !, # !! > # #$ 1 ) ' + , * # - - ) . ' - ' ' +' ' - . + !# + $ # * !# ! $ * - ) * ) ) ' /01 * ' ' ' ' - ' . + !# + $ * ). ' ) ) ' ) - ) ) ( 0 & 0 9 ' - 0. ) !& ((( $ 0 : 0 ' ) ' 0 + ) + , 0 $ ) 0 ! ( . ! $ a:=3 =26= L:9:Q<=b 6C> c69R6=:4 6554B3 <C>383C>3C= 52:;R5 Q<=2 <>3C=<;67 5dBB3=9<; ><5=9<A4 =<:C5H 5 =23d 453 K` >6=6J Q2<;2 <5 6 =Q: d369 86C37J 8:55<A<7<=<35 D:9 <>3C=<Dd<C I ;:B873@ 399:9 ;:B8:C3C= B:>375 693 3@=93B37d 7<B<=3>H 9 5 " , ,. , , ! F $ # ., , , , , -, . ! F $ ., - , / , , , , , ,. , . - . , ,. , , - . , , . F ., ,.-, , , , . ., , . ! F $ # ., , , , , -, . ! F $ ,. , , . F ., ,.-, ., , ,! ., , ,., #$%&' ( . ) . . , ,. / ., , . / , , , * 7 + , ) ! $ - ' * 7 + . , + ') ) - * . ( * 7 + / , ' - ' & !# # $( ( &9#$ ( ) ' ' ) - ) 0. ) !& ((( $ * 7 + 0 , * ') ' ( ) ' - ' ) * 1 ') - 2 2 ' ( ! 3 $ ( ) ) ( ) ( - : ) ! + # ! & , # ! $ F " " # # F " & 2 ( !#$ !&$ !9$ !:$ ' ) - ) ) ' ) ' ( . + !# + $ - ' !#$ !:$ ! ( $ ) )) . + !# + $ - ( ' ' ) ) ! , # ! + / & (( :$ ' ' - > : ) h F # ( h h " ! $ h # # ( # ( - # 9 & 9 ! h F h $ ! h h $ # ( # ( ( # # ( # ( ( # ( ( # + ) ' ) `33 L:9:Q<=b 6C> c69R6=:4 GMNN U D:9 6 D:9B67 =936=B3C= :D ;6535 Q2393 =23 >383C>3C= E69<6A73 <5 =23 935<>467 :D 6 89<:9 93 I 9355<:CH 123<9 6C67d5<5 5488:535 5=9<;=7d 3@: I 3C:45 6C> A:4C>3> 93 I 9355:95J 6C> 9::= ;:C5<5=3C= 35=<B6=35 :D =23 93 I 9355<:C ;:3 ;<3C=5H ) ' ' ) ' * 5 ,. . , . ,. . - . , F -, ., ( ') ' ! 0 & (( : $ ( ) ' . ' . ) & # ( & & & F '' * ! $ * ! $ * ! & $ &F & & !+ $ # ' ) ' ) > # ' 9 9 ) ' ! $ ! FF Fh F Fh hh h F h $ ' ! $ ! FF Fh F hh h $ * ! $ ) !+ $ ' !* ! $$ * ! & $ ' & & !$ $ & F $ F * ! $ * ! + ' F F ' + ! " #$%& #$% F $ F $% $ ' # 1 ' ' ' ') - ) 5 / . ) .( 1 ! " #$%& ' ) # ) 9 ' ! # F F '$ ) ) # +! $ # F F ' ) !) , 3 7 8 7 7 7 7 75 567 , F "" -- F h h !! $ ! F h $ ! - F - h $ h # ( # ( # ( # ( ( # ( # 0 9 ( - ( - !! $ !# ( $ 3 7 . 8 7 567 , F " h " " F h F " h " " " F " h " $ - F h !$ ! F h $ ! F h - F - h - $ # h h # ( ( h h (( (( (( # # ( ( # ( # ( h # h ( # ( h ( ( ( # ( ( # h h # ( ( # '- + & ( # ( # ' & ) + ) ) ' ) 0. ) !& ((( + & (( $ # !$ & ! $ ! $ ) ) !$ & # ') & ' - ' - ') +' ' !$ & - ') * & ! $ ! $ ( ) ) ) ) ' > ! & $ # 2 ! $ ! $ !#( $ & a:=3 =26=J D:9 <>3C=<f;6=<:CJ <= 54 ;35 =: 6554B3 =26= =23 53= :D b39:5 :D D6;=:9 ;HDH 5 <5 3A35 I 43 C3 I 7<I <A73J 533 3HI H K69965;: 6C> W7:93C5 G. OOTU H ! !#( $ & - - * ) ' !$ !$ * ' '' # % !$ '' !$ %h !$ '' !$ % * !$ # % !$' ' !$ # %h !$' ) ' 0-) ) ! F $ !##$ !$ & & > !##$ ) F !$ & &F & 3 '' % ( ' ) !$ & & & & F !$ ) ) ' !$ & ' & & & F F !#&$ ' ( !#&$ ) ' !$ $ ( ' ' ) ) ' " !#&$ ! 9$ ) '' ' - ) ' !#&$ F 2 ' !$ !#9$ 2 ' 2 2 F # # 0) !#9$ ) ') ! $ ) ) & * ' ' ) 0 * & % & ' ) ) ( # &2 2 > ) ( ! $ &2 ' & ( & ) ( ) ' !( $ & ( ' . !( $ ( . !#:$ ! $ &2 ' & " !#:$ !! $ ' ! F h $ ! F " h $ " ! F $ " ! h $ ' !$ FFhh !! F h $$ Fhh ! F h $ hh F h #( h # ( ! F " h $ ( ( ! F $ # ( # ! h $ ( ) ! F " h $ ( 2 F ( FFh ! F h $ h ! F h $ !# $ ( Fh ! F $ # F h ( 2 # hh ! F h $ ! h $ 7 5 5 5 , # # ) ' ( hF hF ! $ !#+ $ ! $ Fhh & & ') !( #$ ) ! $ Fhh !( $ F !( $ !# $ $ ( # * !# ! $ 0 !& (( : & (( :$ ) ' !#:$ - ) " & ) F ! #$ 0 F ' h ( ' F ' 35 , > - F !# ( $ ! $ FFh Fhh ! ( $ ! ( $ h ! ( $ ! $ h ! ( $ # * ) ! $ ! $ F !( $ * ! $ * - h " h ! " #$ h ! F " h " $ Fh !# F $! " &$ h ! F h $ ## !#! $ !# $ - " h " #$ ! Fhh " Fh $ " !& " #$ hh ! F h $ ! " &$ 0 F h ( h " h ! " #$ h ! $ Fh !# F $! " &$ h ! h h $ ) '- ' h ( ) # ) h " h ! " #$ h !# $ ! $ Fh !# F $! " &$ h # " & h ! F $ FFh ! ' * ) ) ' ' F . ! F $ - '' ) ) !$ % % !$ % !$ % %h !$ % " %h ! $ ! $ # ! !$$ F #& F F ! $ > ( ) ! $ !& ( $ ! $ &2 ' & " ') & !&#$ ! $ &2 ' & * & ! $ + ) ' ' ' ' !&&$ ! $ &2 ' & ! $ +, ! & h $ ( ) ! $ & ) ' > !#( $ ' !#( $ ' ) ! . # ! #( :$ ( ' ' # ' -) . ( & # 2 ! $ ! $ !&9$ & ! $ ' ) !&&$ !&9$ # # #' > ' ! $ ! $ ( ) ) * h ! $ : ! ! $ # #h #::¤ ! $ & h ! $ # h # # #' R39C37 :D :9>39 <5 6 D4C;=<:C J C:= C3;35569<7d C:CC3 I 6=<E3J 54;2 =26= G <5 <C=3 I 96A73 D:9 677 J G U O D:9 677 MJ 6C> G U O H `33J 3HI HJ H 6: GMNXU 0 U J 8H = O H #9 ( h ' ! 1 ' & (( & $ ) h - ! $ ! $% # * !# ! $ ) ! $ # # #' ) ' 7 7 , , ) ' & ' > ) ' ') ' > ) !1 ' & (($ $ > . & (( !' ' # ' $ ' ') & ' ) !&#$ !&&$ ! $ ( ' ! $ 3 # # ) ) ' ) !&9$ ) # ( ) ) > . ! $ ( ) ' > ) * . ' ' ) ) ) ' ' +' ' ( '' % % ! $ ) ' #: ) % % ( . + # # % % # * !# ! $ F ' ( ) ' ) ) ' > ) ' ! F $ ! #$ % % 47 , , . . - . , , , -, . . - . , , , , , , , , , , .., ,) , ,. . , ., , , , , . ., , .., , ! F $ . . , , , , ,. h ,) , , . ( , , ! <$ . , ! $ ., , 3 ( 3 ! & " !# " $ # # ! % $ ' # ' ' ( ! $ h ( ') h - ( ' ) - & ' # #FF ' ' # * 0 L:9:Q<=b 6C> c69R6=:4 GMNN U >: C:= 6554B3 5488:9= A:4C>3>C355H L:Q3E39J 65 8:<C=3> :4= Ad L4 6C> <>>39 G. OO X U J =23<9 93D393C;3 G8H M = U =: =23:93B . H0 T :D :7769> GMNX = U <5 <C3@6;=J 6C> 5488:9= A:4C>3>C355 <5 <B87<;<=7d C33>3>H 00 23C :C7d =23 3@<5=3C;3 :D =23 f95= B:B3C=5 :D 6C> <5 6554B3>J 6 5=9<;= 48839 A:4C> =: =23 96=3 :D ;:CE39 I 3C;3 <C 3BB6 M <5. 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