Asymptotic theory of statistical inference, by B L S Prakasa Rao

22869_6S0273_0979_1988_15668_6.pdf

254 BOOK REVIEWS

BULLETI

N (Ne w Series ) O F TH

E AMERICAN MATHEMATICAL SOCIETY Volume 18, Number 2, April 1988 ©1988 American Mathematical Society 0273-0979/88 $1.00 + $.25 per page

Asymptotic

theory of statistical inference, b y B . L . S . Prakas a Rao . Joh n Wile y an d Sons , Ne w York , Chichester , Brisbane , Toronto , Singapore , 1987
, xi v + 43
8 pp. , $49.95 . ISB N 0-471-84335- 0

Statistics

, generall y speaking , addresse s th e proble m o f ho w t o deter min e fro m dat a knowledg e o f th e underlyin g mechanism , presume d ran dom , whic h produce s tha t data . Usuall y th e mechanis m i s idealize d a s a probabilit y la w whic h i s assume d t o belon g t o a collectio n o f possibl e laws . I f w e hav e abundan t data , w e expec t tha t w e ca n determin e fairl y accuratel y th e unknow n law , o r som e aspec t o f it , sa y th e mean , // , i n whic h w e ar e interested . Th e asymptoti c theor y o f statistica l inferenc e i s th e stud y o f ho w wel l w e ma y succee d i n thi s pursuit , i n quantitativ e terms . Any functio n o f th e data , whe n th e amoun t o f dat a i s n , i s calle d a "statistic " o r estimato r fi{n) of , e.g. , th e mea n /i . Th e sequenc e {p>{nj} i s sai d t o b e consisten t fo r \i i f p,(n) converge s t o / i a s n goe s t o infinity . Th e sequenc e i s sai d t o b e asymptoticall y norma l (regrettably , languag e i s abuse d thi s way ) i f fi(n) - /i ca n b e normalize d s o tha t th e la w o f th e re sultin g sequenc e converge s t o a norma l distribution . Proof s tha t particu la r estimator s hav e thes e an d othe r nic e propertie s i n variou s version s an d setting s compris e muc h o f th e wor k o f classica l an d moder n asymptoti c statistics . I n purel y mathematica l terms , th e subjec t i s abou t convergenc e o f se quence s o f function s o r measure s i n variou s senses ; i n particula r it s tool s ar e draw n fro m tha t par t o f rea l analysi s an d measur e theor y calle d probabilit y theory . Unti l rathe r recently , som e woul d sa y "classically" , a larg e portio n o f prob abilit y theor y deal t wit h operation s o n sequence s o f independen t rando m variables , an d statistica l model s assume d tha t dat a consiste d o f sequence s o f independen t observations . A s probabilit y theor y bega n t o focu s o n othe r processes - Marko v processe s i n th e 50
s an d 60s
, an d stationar y tim e serie s i n th e 60
s an d 70s
, mathematica l statistic s bega n t o dea l wit h model s wher e observation s wer e assume d t o follo w thes e patterns . Th e las t te n o r fifteen year s hav e produce d a stron g thrus t o f activit y i n sev era l area s associate d wit h stochasti c processes : stochasti c integrals , stochas ti c analysis , stochasti c differentia l equations , wea k an d stron g convergenc e o f stochasti c processes , etc . A clas s o f processe s receivin g a lo t o f attentio n i s th e ver y broa d clas s calle d semimartingales . Durin g th e sam e tim e perio d ther e ha s bee n a burs t o f activity , partl y i n respons e t o computin g powe r an d convenience , i n technique s o f dat a analysis , statistica l softwar e packages , an d adaptiv e statistica l procedures . Th e subjec t o f asymptoti c statistics , buoye d up , perhaps , b y th e prosperit y o f it s neighbors , ha s take n of f energeticall y i n a numbe r o f fres h directions . I n suc h a situatio n i t i s a darin g ste p t o writ e a boo k whos e state d ai m i s t o brin g u p t o dat e th e interfac e betwee n probabilit y theor y an d asymptoti c

BOOK REVIEWS 255

statistics . I t i s indee d t o thro w larg e stone s int o a torren t wit h th e ai m o f providin g som e kin d o f bridge .

Prakas

a Rao' s first an d longes t chapte r collect s a variet y o f probabilit y topics , eac h introduce d wit h severa l line s o f orientation . A numbe r o f point s receiv e specia l emphasis . On e i s recen t wor k o f She u an d Ya o o n a momen t inequalit y fo r embeddin g times . Anothe r i s th e extensiv e developmen t i n th e relate d area s o f absolut e continuit y an d contiguit y o f measures . Th e exposi tio n extend s fro m Kakutani' s well-know n resul t t o recen t wor k o f Liptse r an d

Shiryaye

v an d other s o n contiguit y o f stochasti c proces s measures . Th e topic s i n thi s chapte r ar e indee d primaril y concerne d wit h stochasti c processe s eve n thoug h th e statistica l conten t o f th e boo k is , a s state d i n th e preface , mostl y abou t independen t dat a a s oppose d t o mor e genera l stochasti c proces s data . I t i s implici t her e tha t th e potentia l fo r us e o f thes e topic s i n statistic s i s fa r ahea d o f thei r exploitation . Particularl y o n thi s account , i t woul d hav e bee n convenien t t o hav e som e forwar d indexin g i n th e for m o f additiona l note s i n th e "Remarks " whic h appea r a t th e en d o f eac h section , tellin g u s wher e i n thi s boo k o r elsewher e statistica l applicatio n o r significanc e o f th e probabilit y result s ma y b e found . Th e remainin g chapter s ar e abou t asymptoti c statistics , an d her e Prakas a Ra o ha s don e a larg e jo b i n assemblin g an d selectin g piece s o f wor k fro m a n enormou s literatur e spannin g th e fifteen-year perio d beginnin g abou t 1971
. A s a guidelin e fo r emphasi s h e ha s use d insight s gaine d throug h hi s ow n re searc h o n severa l topics . A s a patter n o f expositio n fo r eac h topic , h e ha s selecte d a particula r autho r o r author s whos e writin g lend s itsel f t o pre sentatio n i n boo k form , whos e method s ar e attractiv e an d innovativ e an d whos e wor k i s importan t fo r tha t topic , an d presente d tha t person' s wor k a s a kin d o f featur e article . I n thi s wa y h e ha s manage d t o cove r a n impres siv e arra y o f topic s withou t gettin g bogge d dow n wit h differen t approache s an d interplays . Th e expens e i s tha t ther e i s littl e integratio n o r amalga matio n o f th e wor k presente d o r o f th e literature . W e have , fo r instance ,

Strasse

r o n globa l an d loca l asymptoti c bound s fo r risk , Sweetin g o n max imu m likelihoo d estimatio n fo r processes , Deshaye s an d Picar d o n a partic ula r change-poin t problem , Khmaladz e o n goodness-of-fi t an d s o on . Eac h o f thes e work s represent s a larg e development . Th e collectio n i s extensiv e an d provide s a usefu l introductio n t o man y topic s an d acces s t o thei r lit erature . Separat e referenc e list s a t th e en d o f eac h chapte r an d larg e au tho r an d subjec t indice s mak e thi s a source-boo k fo r orientatio n an d reference . Ther e ar e occasiona l sign s tha t th e author' s ow n revie w o f th e literatur e ma y no t b e ver y thoroughgoing . Fo r example , followin g Moor e [2 ] h e point s ou t a difficult y abou t chi-squar e test s wher e bin s ar e base d o n a n estimate d parameter . Bu t thi s proble m ha s bee n resolve d b y Dzhaparidz e an d Nikuli n [1] , an d recen t literatur e i n thi s directio n i s no t explored . Eve n a cursor y stud y o f th e boo k wil l yiel d a n impressio n o f th e magnitud e o f th e tas k i t attempt s an d als o o f area s wher e researc h effor t i s especiall y invited . I t i s evident , fo r instance , tha t th e exploitatio n o f result s abou t rat e o f convergenc e an d o f "martingale " method s fo r provin g wea k convergenc e an d fo r constructin g statistic s ha s onl y jus t begun .

256 BOOK REVIEWS

REFERENCE

S 1 . K . O . Dzhaparidz e an d M . S . Nikulin , On a modification of the standard statistics of

Pearson,

Theor y Probab . Appl . 1 9 (1974) , 851-853
. 2 . D . S . Moore , A chi-squared statistic with random cell boundaries, Ann . Math . Statist . 4 2 (1971) , 147-156
.

PRISCILL

A E . GREENWOO D

BULLETI

N (Ne w Series ) O F TH

E AMERICAN MATHEMATICAL SOCIETY Volume 18, Number 2, April 1988 ©1988 American Mathematical Society 0273-0979/88 $1.00 + $.25 per page

Invariant

manifolds, entropy and billiards; smooth maps with singularities, b y Anatol e Kato k an d Jean-Mari e Strelcyn , wit h th e collaboratio n o f F .

Ledrappie

r an d F . Przytycki . Lectur e Note s i n Mathematics , vol . 1222
,

Springer-Verlag

, Berlin , Heidelberg , Ne w York , London , Paris , Tokyo , 1986
, vii i + 28
3 pp. , $23.60 . ISB N 0-387-17190- 8 Man y dynamica l system s arisin g i n physics , meteorology , chemistry , biol ogy , engineerin g an d othe r fields exhibi t chaoti c behavior . Ther e i s n o precis e definitio n o f "chaos " ; however , i n simpl e terms , chaoti c behavio r mean s tha t a typica l orbi t seem s t o wande r aimlessl y i n th e phas e spac e wit h n o iden tifiabl e patter n an d it s futur e i s unpredictabl e althoug h th e syste m itsel f i s deterministi c i n nature . Th e onl y know n caus e o f chao s i s hyperbolicity . Sup pos e w e begi n movin g alon g a hyperboli c orbi t wit h th e spee d prescribe d b y th e syste m an d observin g th e relativ e motio n o f nearb y orbit s tha t star t o n a codimensio n 1 transversa l t o ou r orbit . The n i n a n appropriat e coordinat e syste m th e relativ e motio n u p t o first-order term s wil l b e th e sam e a s i n a neighborhoo d o f a saddl e poin t x = Arc , y = My. Th e eigenvalue s o f A hav e strictl y negativ e rea l parts ; th e eigenvalue s o f M hav e strictl y positiv e rea l parts . Thes e rea l part s ar e calle d Lyapuno v characteristi c exponent s (LCEs ) an d giv e u s th e exponentia l rate s wit h whic h nearb y trajectorie s mov e t o o r awa y fro m ou r orbit . I f al l orbit s ar e hyperbolic , al l LCE s ar e uniforml y sep arate d fro m 0 an d al l estimate s ar e uniform , the n w e hav e a n Anoso v system ; a goo d exampl e i s th e geodesi c flow o n a compac t surfac e o f curvatur e - 1 . D . Anoso v an d Ya . Sina i studie d suc h system s abou t 2 0 year s ago . The y constructe d invarian t familie s (o r foliations ) o f stabl e an d unstabl e manifold s an d use d the m t o prov e ergodicit y (i.e. , chaos ) fo r Anoso v system s preserv in g a n absolutel y continuou s measure . I n th e mid-70 s Ya . Pesi n generalize d th e whol e theor y fo r smoot h nonuniforml y hyperboli c dynamica l systems , i.e. , system s fo r whic h LCE s ar e no t bounde d awa y fro m 0 an d som e ma y actu all y equa l 0 . H e followe d a simila r pat h b y constructin g an d usin g th e stabl e an d unstabl e manifold s an d prove d tha t th e measure-theoreti c entrop y equal s J2j ƒ Vj dm , wher e th e //j;' s ar e positiv e exponent s an d m i s a n absolutel y con tinuou s invarian t measure . Late r D . Ruelle , R . Man e an d other s simplifie d an d generalize d som e o f Pesin' s results . I n thei r boo k A . Kato k an d J . M . Strelcy n generalize th e Pesi n theor y fo r th e cas e o f a dynamica l syste m wit h singularities . A n exampl e o f suc h a sys te m an d on e o f th e mai n motivation s fo r th e boo k i s a billiar d system , i.e. ,