On Edgeworth Expansions in Generalized Urn Models

Transcription

1 On Edgeworh Expansions in Generalized Urn Models Sh M Mirahedov Insiue of Maheaics and Inforaion Technologies Uzbeisan (E- ail: shirahedov@yahooco S Rao Jaalaadaa Universiy of California Sana Barbara USA ( rao@psaucsbedu and Ibrahi B Mohaed Universiy of Malaya Malaysia ( iohaed@ueduy Absrac The rando vecor of frequencies in a generalized urn odel can be viewed as condiionally independen rando variables given heir su Such a represenaion is exploied here o derive Edgeworh expansions for a su of funcions of such frequencies which are also called decoposable saisics Applying hese resuls o urn odels such as wih- and wihou- replaceen sapling schees as well as he ulicolor Pólya- Egenberger odel new resuls are obained for he chi-square saisic for he saple su in a wihou replaceen schee and for he so-called Dixon saisic ha is useful in coparing saples Key words and phrases: Edgeworh expansion urn odels sapling wih and wihou replaceen Pólya- Egenberger odel Poisson disribuion binoial disribuion negaive binoial disribuion chi-square saisic saple su Dixon saisic MSC (000: 6G0 60F05 Inroducion Many cobinaorial probles in probabiliy and saisics can be forulaed and indeed beer undersood by using appropriae urn odels which are also nown as rando allocaion schees Such odels naurally arise in saisical echanics clinical rials crypography ec The properies of several ypes of urn odels have been exensively sudied in boh probabiliy and saisics lieraure; see eg he boos by Johnson and Koz (977 Kolchin e al (978 and survey papers by Ivanov e al (985 Koz and Balarishnan (997 One of he ore coon urn odels is he sapling schee wih-replaceen fro a finie populaion which conains objecs labeled hrough ; he probabiliy ha he -h objec will be seleced in each of he sapling seps is equal o p 0 p p If sands for he frequency of he -h objec in a saple of size n (ie afer n independen selecions hen he rando vecor (rvec ( has a ulinoial disribuion wih paraeers (n p p As is well nown one iporan and useful propery of such a ulinoial rvec is ha is disribuion

2 can be represened as he join condiional disribuion of independen rando variables given heir su n where is Poisson ( p for an arbirary posiive real Such a condiional represenaion is indeed a characerisic propery of any urn odels and hus he following definiion includes several coon urn odels as special cases Le ( be a rvec wih independen and non-negaive ineger coponens such ha P{ n } 0 for a given ineger n Also le ( be a rvec whose disribuion is defined by n ( ( ( where ( X here and in wha follows sands for he disribuion of a rvec X oe ha ( iplies ha P{ n } The odel defined in ( is wha we will call a Generalized Urn Model (GUM: when a saple of size n is drawn fro an urn conaining ypes of objecs and rvec saisics represens he nuber of -h ype of objec appearing in he saple; he disribuion of he defines he saple schee hrough ( We are ineresed in he following general class of R ( f ( ( where f ( x f ( x are Borel funcions defined for non-negaive x The funcions f can also be allowed o be rando in which case we will assue ha he rvec ( f ( x f ( x for any collecion of real non-negaive x x does no depend on he rvec A saisic of he ype ( is called a Decoposable Saisic (DS in he lieraure For he case when he ernel funcions f are also rando he saisic ( is called a randoized DS (see for insance Ivanov e al (985 Mirahedov (985 and Mihaylov (993 Alhough he erinology DS is usually reserved for he special case when f are no rando here we will use i for eiher of hese cases The following hree special cases of he GUMs and relaed DSs are os coon in applicaions A Saple schee wih replaceen Le ( Poi( p be a Poisson disribuion wih expecaion p where (0 is arbirary p 0 and p p ; hen he rvec has he ulinoial disribuion M( n p p and we have a saple schee wih replaceen This schee is associaed wih he rando allocaion of n paricles ino cells: he cells are labeled hrough paricles are allocaed ino cells independenly of each oher and he probabiliy of a paricle falling ino h cell is p The classical chi-square lielihoodraio saisic and he epy-cells saisic are exaples of he ype ( enioned above

3 B Saple schee wihou replaceen Suppose ( Bi( is a binoial 3 disribuion wih paraeers 0 and arbirary (0 hen he rvec has he uli-diensional hypergeoeric disribuion: P{ } n where n and 0 This GUM corresponds o a sapling schee wihou replaceen fro a sraified finie populaion of size For insance he saple su and he sandard saple-based Esiae of he Populaion Toal are exaples of DSs of he for ( C Mulicolor Pólya-Egenberger urn odel Le ( B( d be negaive binoial disribuion wih d 0 and arbirary (0 Then P{ } D n n d (3 where D d d is he generalized Pólya-Egenberger disribuion; such a specificaion of he GUM corresponds o he ulicolor Pólya-Egenberger urn odel (see eg Koz and Balarishan (977 Chaper 40 For exaple he nuber of colors ha appear in he saple exacly r ies and he nuber of pairs having he sae color are saisics of he ype ( We noe ha su of funcions of spacings-frequencies under he hypohesis of hoogeneiy of wo saples can be forulaed as a DS in his GUM; see for insance Hols and Rao (98 Jaalaadaa and Schweizer (985 for furher deails and iporan applicaions o esing hypoheses There is exensive lieraure on DSs uch of i relaed o sapling wih and wihou replaceen fro a finie populaion We specifically enion a few: Mirahedov (996 obains a bound for he reainder er in CLT and Craer s ype large deviaion resul for a special class of GUM; Mirahedov (99 and Ivcheno and Mirahedov (99 consider a wo-er expansion wih applicaions o soe special cases of DS in a ulinoial schee under soewha resricive condiions; Babu and Bai (996 obain Edgeworh expansion for ixures of global and local disribuions -- resuls ha can be used when he DS is a linear funcion of frequencies and a GUM is defined by idenically disribued rvs Such resuls are clearly very resricive on he paraeers of he urn odel and on he ernel funcions f The ai of his paper is hreefold: Firs we presen a general approach ha allows one o obain an Edgeworh asypoic expansion o any nuber of ers for he disribuion of a DS in a GUM Second his general approach is used o exend nown resuls for classes of DS in he 3 special cases of GUM jus enioned Third we illusrae hese resuls by obaining general Edgeworh expansions for hree special and ineresing cases of DS viz

4 (i he chi-square saisic in Case A (ii saple-su in a saple schee wihou replaceen ie in Case B and (iii he Dixon spacings-frequencies saisic in Case C The chi-square saisic is considered for he case when he nuber of groups increases along wih he saple size a siuaion ha has been considered by soe auhors including Hols (97 Morris (975 Quine and Robinson ( and Mirahedov (987 We obain here a hree-er asypoic expansion under very general condiions on he paraeers generalizing he resuls in Mirahedov (99 and Ivcheno and Mirahedov (99 The resul in (ii iproves he ain resuls of Mirahedov (983 Bloznelis (000 as well as pars of Theore of Hu e al (007 Asypoic expansion for a DS in he ulicolor Pólya- Egenberger urn odel and for he Dixon saisic as a special case are obained here for he firs ie I should be reared ha alhough we confine our discussion in his paper o he above 3 exaples of GUM and relaed DS for illusraive purposes as well as o eep he lengh of he paper reasonable i should be enioned ha he resuls derived in his paper are generally applicable o any DS in oher specificaions of GUM for insance o he conex of a specified rando foress a rando cyclic subsiuions (cf Kolchin (985 Pavlov and Cherepanova (00 The paper is organized as follows In Secion we presen a syseaic procedure for obaining an asypoic expansion for he characerisic funcion of a DS o ers of any order Our general approach is based on he so-called Barle s ype inegral forula and provides a sipler and ore srealined way of obaining higher order approxiaions han wha previous auhors have used The ain resuls are presened in Secion 3 For he sae of copleeness and o connec o Barle s ype forula we also presen wo heores on asypoic noraliy and Berry-Esseen ype bounds showing how he curren forulaion helps siplify siilar resuls obained in Mirahedov ( Applicaions o he special DSs (i (ii and (iii are given in Secion 4 while he proofs of he ain resuls are posponed o an Appendix I should be enioned ha we are dealing wih riangular arrays where all he paraeers of a GUM vary (including he disribuion of he rvs when boh n and end o infiniy forally hrough a non-decreasing sequence of posiive inegers { n v } { v } as v ; hence i is iporan o express he reainder ers in our asypoic expansions which show heir explici dependence on he n disribuions of he rvs and he ernel funcions f In wha follows c C wih or wihou an index are universal posiive consans which ay depend on he arguen and ay be differen a differen places; all asypoic relaions and liis are considered as n and ( n 4 Barle s ype forula and asypoic expansion of he characerisic funcion of a DS Define

5 A E B Var x ( n A / B 5 Ef ( B cov( f ( g( y f( y Ef ( ( y E Rˆ ( g ( g ( ( Var Varf B ( Under soe ild condiions one can show ha as n and ( n x ER ( x B Eg ( ( E o( B VarR o ( ( Also R ˆ ( R ( x B and Eg ( 0 cov g ( 0 ( Le be a easurable funcion such ha E ( and We have E( ( n E ( because of ( This ogeher wih E( ( e e P{ } E( ( iplies by Fourier inversion i ( ( n i n 0 E ( P{ n} E ( exp{ i ( n} d (3 Se ( exp{ ( ( } E i g i B E i x ( x e ( d (4 B B Then Equaion (3 ogeher wih he inversion forula for he local probabiliy P n gives us he following Barle s ype forula (cf Barle (938: ( x : Ee ˆ ( ( x (0 x i R which provides he crucial forula of ineres Special forulaions of his Forula (5 show up in lieraure; see eg Hols (979 Quine and Robinson (984 Mirahedov ( Also a very special case of (5 is he os coonly used forula of Erdös and Renyi (959 for (5

6 invesigaing he saple su in a wihou-replaceen schee (see eg Babu and Singh (985 Zhao e al (004 and Hu e al (007 Forula (3 is also useful in sudying large deviaion probles (see eg Mirahedov (996 A foral consrucion of he asypoic expansion for ( x defined in (5 proceeds as 6 follows: The inegrand ( is he characerisic funcion (chf of he su of independen wo-diensional rvecs g Because of ( his su has zero expecaion a uni covariance arix and uncorrelaed coponens Fro Bhaacharya and Rao (976 Chaper (his reference will henceforh be referred o as BR i is well nown ha under suiable condiions his ch f ( can be approxiaed by a power-series in / whose coefficiens are polynoials in and conaining he coon facor exp{ ( / } Hence he series can be inegraed wr over he inerval ( As a resul of his inegraion we ge a power series say ( x in / ex we replace (0 x by is series approxiaion which is (0 x Finally we ge he asypoic expansion of ( x by dividing ( x by (0 x The above algorih alhough anageable needs long and coplex calculaions as we show below Assue ha Eg ( s and s E for soe s 3 Le P ( be he well-nown polynoials in and fro he heory of he asypoic expansion of he chf of he su of independen rando vecors (see (73 (76 of BR p5 in our case for he quaniy g g ; he degree of P ( is 3 and he inial degree is ; he coefficiens of P ( only involve he cuulans of he rvs g g of order and less Define polynoials (in of G ( x as x / e G ( x P ( exp i x d 0 (6 ow define Q ( x fro he equaion Then j s 3 v/ j/ v j 0 v 0 j 0 ( G (0 x Q ( x Q ( x j! G (0 x (7 s 3 ji j i i ji! where he suaion is over all ( s 3 -uples ( j j j s 3 wih non-negaive inegers j i such ha j j ( s 3 j s 3 j Le s 3 ( s / v v 0 v 0 W ( x G ( x Q ( x (8

7 oe ha G0 ( x Q0 ( x so ha W (3 ( x For exaple 7 (5 W ( x ( G ( x G (0 x ( G ( (0 (0 ( ( (0 x G x G x G x G x (9 In wha follows we shall use he following addiional noaion ˆ B B gˆ g ( / ˆ ˆ ( E / B ˆ ˆ j j/ j E g ˆ j j j/ E ˆ M ( T inf ( E exp{ i } if T else M ( T (0 T B exp M 03( B 8 s s s 3 T in( ( 3 in( B ( 0 / M (03( B M (03( B / Throughou he paper we assue ha x c alhough he ehod used here allows leing x o increase a a rae of / O((log (see eg Mirahedov (994 In he above lised hree exaples of GUM he paraeer can be chosen such ha x 0 (see also he beginning of Sec 4 Proposiion Le Eg ( s for soe s 3 and 00 There exis s consans c and C such ha if ct hen for j 0 j ( s 8 ( ( s x e W x Ce j 3 Main resuls We use he noaions of Sec The following Theores 3 and 3 follow fro Theore and of Mirahedov (994 and are presened here for he sae of copleeness and o connec o Barle s ype forula (5; also heir applicaion o DS in our exaples of GUM gives weaer condiions for asypoic noraliy and iproved Berry-Esseen ype bound han are nown before Le { A } sand for he indicaor funcion of he se A and 3 ˆ ˆ E 3/ ` ( { } ˆ ˆ E ` ( { } Theore 3 If for arbirary 0 ˆ ˆ ( { } ` L Eg g (i L ( 0 (3

8 (ii ( 0 8 (iii (iv M B ( (4 ( in( ( ( (4 ( B o M B hen he saisic R ( has an asypoic noral disribuion wih expecaion xb and variance given in ( Rear 3 For all he 3 exaples of GUM we consider condiions (ii (iii and (iv being condiions on he paraeers of he urn odel are auoaically saisfied under very general se-up (see Sec 4 so ha all we need is o chec he Lindeberg s condiion (i for ensuring he asypoic noraliy of he DS Le Eg ( s ( s for soe s 3 Define ( ux so ha iu ( s ( s ( ( e d u x W x e (3 ( s The funcion ( ux can be obained by forally subsiuing v d u / ( ( u e Hv ( u / du where ( u e d u for ( i for each in he expression for ( s W ( x (see Lea 7 of BR p 53 where H ( x is he -h order Herie-Chebishev polynoial oe ha (3 ( u x ( u Se Theore 3 Le 0 ( s ( s u (3 sup P{ R ( u x B } ( u x C( ( ( x ( a b { a b} d a b Then here exiss a consan C such ha Theore 33 Le Eg ( s for soe s 3 There exis consans c and C v such ha C ( ct ( s s s Theore 34 Le he saisic R ( be a laice rv wih span h and a se of possible values in If Eg ( s for soe s 3 hen here exis consans c and C such ha uniforly in z d h du sup ( s P{ R ( z} ( uz x C s ( ct / h z z where uz ( z xb / and

9 ( a b { a b} ( x d a b 9 The following general bounds for ( ab are useful in applicaions Wrie ( E exp{ if ( i } ( sup ( a b d a b (33 H ( a b inf H H( E f ( a b (34 where a sands for he disance beween real a and inegers Here and in wha follows for a given rv we define where is an independen copy of Then and ( a b CB ln( b exp d ( a b (35 ( a b CB ln( b exp H ( a b (36 ( a b C B exp H ( a b (37 These inequaliies (35- (37 follow fro he following arguens: Fro forula (3 i follows ha for he ch f ( x in Equaion (5 one can wrie he produc ( insead of ( Since 0 x c (cf (57 below inequaliy (35 follows by using he fac ha ( x / x e On he oher hand by Lea 4 of Muhin (99 we have 4 H ( H This inequaliy ogeher wih (35 iplies he inequaliies (36 and (37 Rear 3 A DS of he special for X arises in any probles in saisics and in discree probabiliy (see eg Sec 4 and Pavlov and Cherepanova (00 This DS is a laice rv wih span equal o wo Also r l l H ( v P( P( l (38 where vrl ( l(( l As in Lea of Pavlov and Cherepanova (00 one can prove ha for all real and such ha / 4 / and any non-negaive ineger and l

10 Fro his i follows ha if ax{ v l v l v l } 0 l 0 j 0 P{ } P{ l} c 0 (39 j l hen H ( a / a / 4 where for each 0 l jl is defined such ha ax{ v v v } v j 0 3 j l 3 j l 3 j l jl l 4 Applicaions In wha follows we will use he noaions of he preceding secions eeping in he ind ha he disribuion of he rv is wha is relevan for he paricular GUM under consideraion oe ha in all our exaples of he GUM he disribuions of he rv s depend on an arbirary paraeer which can be chosen in a suiably convenien anner We will hus choose he paraeer such ha A n in which case x 0 and hence he ers of asypoic expansion ie he funcion ( s ( u0 is considerably siplified For exaple: i is nown ha P0 ( 3 i P ( E( gˆ ˆ 6 3 i P E g E g 4 4 ˆ ˆ ˆ ˆ ( ( ( 3( ( 4 Therefore fro (6 (9 and (3 P ( u / (5 e u ( u0 ( u 30 6 u / 5 3 e u 0u 5u u 3u ˆ u 8 4 ˆ ˆ ˆ (4 where ˆ Eg ˆ i ˆ j ij i j ˆij In wha follows jus in order o eep our calculaions siple we will resric ourselves o such a hree-er asypoic expansion given above 4 Exaple A The rvec ( has he ulinoial disribuion M( n p p p 0 p p and we ae ( Poi( np We assue ha ( n ax p 0 as n We ae n/ i np and p p i i

11 In his classical schee since he bes condiions for asypoic noraliy and he Berry- Esseen ype bound of DS are already given in Mirahedov (996 and 007 we concenrae our aenion on he asypoic expansion resuls Theore 4 Le he saisic R ( be a laice rv wih span h and a se of possible values If 5 Eg ( and n hen uniforly in z (0ln (5 P{ R ( z} ( uz0 h du z d C n n H ct 3/4 5 ( 3 exp where uz ( z / and T is defined as in Sec 3 wih ( n ( in( n / h The paricular DS X : for any ineger is a special case of Theore 4 We shall focus on he os iporan applicaion he chi-square ype saisic X As saed before X is laice wih span equal wo; also in his case g n Hence ( ( ( ( ( n( n 3 ˆ 4 ( 3 ( 4 ( 3 : n n n n ˆ 4 n ˆ ( n( 3 n ˆ 4 n (6 9 8 n ( ˆ n 8 n(64 7 n ( n ( n ( ( ˆ 8n n (9 4 n ( n ˆ ( 6 ( ( ( ˆ 0 0 n 3 n 4 3 Corollary 4 Le c p c for soe posiive c c and all hen uniforly in b { n 0 n( n / } he se of possible values of he r v X one has

12 d P X b u du C 3/ 3/ n (5 { } ( b0 b n exp c ax( where ub ( b / he exac forulae for and he ers of (5 ( ub0 are given above The following Corollary 4 follows fro Corollary 4 by using he Euler- Maclaurin su forula We sae jus a wo-er asypoic expansion o eep he expressions siple Corollary 4 Le c p c for soe posiive c c and all Then u / e u 30 6 ˆ P{ X u } ( u S ( u ˆ C n n exp c ax( (4 where S ( [ ] / x x x is well-nown periodic funcion of period one (see for insance BR p54 and coes up here due o he Euler-Maclaurin su forula We ay rear here ha Corollary 4 already considerably iproves Theore 5 of Ivcheno and Mirahedov (99 which saes he inequaliy (4 wih l exp{ } e l 0 insead of he exponenial er which aes sense under he addiional resricion ha O(ln Rear 4 Applicaion of Theores 33 and 34 o he log-lielihood saisic L ln and o he coun-saisics r { r } give resuls siilar o Theores 4 and 6 respecively of Ivcheno and Mirahedov (99 bu one can obain addiional ers in he expansions hey provide A DS wih ernel funcions f f for all is called a syeric DS ; For exaple he X and r L are all syeric DS I is well-nown (see eg Hols (97 Quine and Robinson (985 Mirahedov (987 ha he chi-square es is asypoically os powerful (AMP wihin he class of syeric ess ie aong ess based on syeric DS for esing he hypohesis of uniforiy agains he sequence of alernaives H given by: n p ( /4 n ; 0 and 0 C C Moreover he chi-square es is he unique AMP es for bounded away fro zero and infiniy; on he oher hand if 0or hen here exis oher AMP syeric ess for exaple he epy cells es when 0 and he log-lielihood es when In view of his Ivcheno and Mirahedov (99 inroduced and sudied he second order asypoic efficiency (SOAE of syeric ess wr he chi-square es Invesigaion of he SOAE is based on he asypoic

13 expansion of he power funcion of such ess In he case ay arise only if 0 hey have shown ha SOAE 3/4 n O( ; for exaple he epy cells es based on he saisic 0 is SOAE for his siuaion; for he case hey could only noe ha when O(ln he SOAE es does no exis because of he resricive choice of O(ln needed in heir asypoic expansions Therefore hey poined ou ha he SOAE proble is open for Corollary 4 does resolve his proble showing ha he chi-square es is sill opial in he sense of SOAE if 3/ n o( ; i is also SOAE wr he log-lielihood es for Mirahedov(99 for furher discussion 3/ n (cf Ivcheno and 4 Exaple B ow we consider he saple schee wihou replaceen fro a sraified populaion of size wih ; he sraa are indexed by ; is he size of he -h srau ; and is he nuber of eleens of he -h srau appearing in a saple of size n In his schee ( Bi( where (0 is arbirary We choose p : n / 3 q p so ha x 0 Se ax We consider he case where he sraa sizes ay increase ogeher wih bu saisfy he following condiion /4 o(( nq (43 Theore 4 If he Lindberg s condiion (3 is saisfied along wih Condiion (43 hen as nq R ( has he asypoic noral disribuion wih expecaion and variance as given in ( Theore 43 For arbirary (0] here exiss a consan C such ha (3 C / nq nq Rear 4 The er / nq can be replaced by ax 6 3 / pq nq nq If ( /(4 ( nq hen he second er on he rhs doinaes he hird one Theore 44 Le 5 Eg ( hen here exis consans c and C such ha 3 (5 C 5 ct 5 nq ( Le now he eleens of -h srau be independen rvs X X We draw a saple of size n wihou replaceen fro he enire populaion Define he indicaor rvs i which equal one if an eleen Xi of he h srau appears in he saple or else i is zero so ha Then ( S X represens he su of eleens of he h srau n i i i

14 which appear in he saple and he su of all he eleens in he saple he saple-su 4 given by ( n n S S is a DS Assue ha he rvs X X have a coon disribuion sae as ha of a rv Y We also assue ha Y Y are independen rvs Then he rv S n is disribuionally equal o a DS wih f (0 0 f ( j X X j : ( S X { } (44 n j j Suppose s EY for soe s 3 Then he expressions in ( have he following for: f ( X { } (45 j j g ( { }( X p( EY j j EY Fro Theores 43 and 44 we iediaely have he following Corollary 43 If (43 is saisfied hen for arbirary p ( E( Y p( E( Y (46 (0] here exiss a consan C such ha sup P{ S u n } ( u C u ( nq n / where ( Eg ( p( p EY (47 Corollary 44 If (43 is saisfied hen here exis posiive consans c and C such ha 3 (5 C 5 ct 5 nq ( where T in nq / 3 ers of he (5 ( u0 in (4 have he following fors: 0 q n ( p ( p 03 q nq ( ( ( p ( p( 3 q p p (

15 q p ( ( p 0 0 p ( 5 3 / ( 3p p p ( p 40 ( 4 4p 3 3( p 6( p where ( i i EY ; also 3 4 3(3 p p ( p 3/ iy ( ct C n ln( exp n sup Ee c T (48 and iy ( ct C n ln( exp nq sup Ee 5 5 c T 5 (49 In he case our Corollary 43 iproves a resul of Mirahedov (985 and a recen resul of Zhao e al (004 for he case when ( / nq (in heir noaions oe ha p 3 3 qp 3 E Y pey q E Y which provides a naural expression showing he exac dependence of he bound on p n / and oens of he eleens of populaion insead of he forula for in Zhao e al (004 This fac is confired by he second er in (5 ( u0 (see (4 and ha 0 Also in his case he hree-er asypoic expansion ie (5 ( u0 coincides wih ha given by Mirahedov (983; furher fro our Corollary 44 follows he ain resul of Bloznelis (000 and i exends Theore of Hu e al (007 giving an addiional er in heir asypoic expansion for he case when p is bounded away fro one; his case is he os ineresing in a saple schee wihou replaceen 43 Exaple C For his case we assue ha ( B( d p wih p n / ( n D j j where Dj d d hen x 0 Puing p / ( p n / D we ge B D ( ( 3 ( ( D D ( D ( 4

16 Theore 45 Le D D o If he Lindberg s condiion (3 is saisfied hen he DS / ( 6 R ( has asypoic noral disribuion wih expecaion and variance as given in ( Theore 46 There exiss a consan C such ha C( E where (3 3 E 3 D 3 D n( D D D D Rear 43 Using he fac ha D D we have E 3 n( D D Theores 45 and 46 do iprove as well as correc Theores 3 and 4 of Mirahedov (996 Theore 47 Le he saisic R ( be a laice rv wih span h and a se of possible values If 5 Eg ( hen uniforly in z (5 P{ R ( z} ( uz0 h du z d 3/4 D D C D ( exp H ct D Dn( h where uz ( z / and T is defined as in Sec 3 wih ( ow consider he following pracical and iporan wo-saple proble: Le X X M and Y Y n be wo saples fro coninuous disribuions F and G respecively defined on he sae A R The classical wo-saple proble is o es he null hypohesis of hoogeneiy H 0 : F G Define he rvs n { Y [ X X ]} i ( ( i where M / is he larges ineger ha does no exceeds M / ineger X( X( M are he order saisics of he firs saple E X X M The r vec ( are called he spacing-frequencies ie frequencies of he second saple falling in beween he spacings creaed by he firs saple A wide class of es saisics for esing H 0 can be expressed in he for (see Hols and Rao (980 and Gao and Jaalaadaa (998 V f ( where f s are real- valued funcions I is easy o chec ha under H 0 he rvec ( saisfies equaion ( wih ( B( p p n /( n M ie he saisic V is DS defined in

17 7 he Pólya- Egenberger urn odel Hence Theores 45 and 46 iediaely lead o he following Corollaries 45 and 46 by puing d and n/ M Corollary 45 If he Lindeberg s condiion (3 is saisfied hen he saisic V has asypoic noral disribuion wih expecaion and variance Corollary 46 There exiss a consan C 0 such ha (3 C 3 n( Consider he following so-called Dixon s saisic: For his saisic we obain: M( ( ( M( ( g ( ( ( ( ( ( ( ( ( 9 76 ( ( (5 3 6 ( 30 3/ 3 3 ( ( Alhough he exac forula for 4 40 is anageable i is quie long and herefore we resric ourselves o is leading er as obaining he following bounds 4 4 ax( ( C ( ( ( / 3 c The Dixon saisic saisfies he condiions of Theore 47 In paricular by evaluaing he oens of he rv g and using Corollaries 45 ( ( ( ( ( 46 and Theore 47 along wih he Rear 3 we obain he following resul We oi he deails Corollary 47 (i If ( hen he Dixon saisic has an asypoic noral disribuion wih ean M( ( and variance M( ( (ii P{ u M ( ( M ( ( } ( u O ( (iii Le and /3 o( M hen we have for any b 0 n( n / M P n b ( ( ( u b / ub e 3 ub ( u 30 6 ( + O ( where ub ( n b M ( ( / M ( (

18 8 Consider he class of syeric ess (ie based on syeric DS for esing he hypohesis of hoogeneiy agains soe of sooh sequence of alernaives which approaches he null a he rae O n /4 (( The asypoic power of syeric ess increase as grows; he Dixon saisic is an exaple of a syeric DS; i is nown o be unique AMP wihin he class of syeric ess for any fixed he sep of spacings see Jaalaadaa and Schweizer (985 Above saed Corollaries allow us o consider he siuaion when ; in his case he AMP es is no unique Coparison of he AMP ess based on heir second order asypoic efficiencies using he asypoic expansion of he power funcion can be done For his purpose above presened asypoic expansion resuls are cenral and such coparisons will be he subjec of anoher invesigaion 5 Appendix Proofs: Proof of Proposiion We need he following 3 Leas o coplee he proof of his proposiion Lea A Se in( There exis consans c 0 and C 0 such ha if / s / s s s s ax c s hen for 0and s 3 v / ( e Pv( v s s 4 C( s s ( e Lea A follows fro Theore 9 of BR because of ( and he fac ha he su of he rvs ( g ˆ ˆ has uni correlaion arix Lea A For any ineger l saisfying 0 l 3v where v 0 s l G 3 ( x c ( l v v l v ( ( x ( v v l Proof Siilar o ha of Lea 95 of BR p7; he only difference being ha in Equaion (9 of BR p7 we use he inequaliy / ( / ( ji / ( r / ji/ r j r o obain i l 3v l Pv ( c( l v ( ( v v l c( l v ( ( (5 3 v l v/( s v/( s s s Lea A follows fro his and (6 Lea A3 Le ax( If 03 3 and 03 3 hen for = 0 and ( exp 0 Lea A3 follows fro Lea A Par ( of Mirahedov (005

19 Pu T ( s in( ( where s 3oe ha T ( s T since / s / s s s / s s 3 Le 9 c T ( s where c 0 is o be chosen sufficienly sall Fro (4 and (6 x s 3 v/ x e G v x v ( : ( ( c s s 3 v / ( v ( v e P d c s 3 e v / Pv ( v d s c s 03 3 ( d 03 3 B ( d 3 4 (5 Applying Lea A (5 and Lea A3 o and 3 respecively we obain afer soe algebraic anipulaion l s s s 6 C( ( e l 3 (53 Se ˆ ( E exp{ igˆ ( i ˆ } and recall ha ( ˆ ( We have ˆ ˆ ˆ ˆ ˆ ig i ig ( (0 E ( e ( e E( e ˆ ˆ ˆ ˆ (0 E g Eg and ˆ i ˆ igˆ ˆ ( (0 Ee ( e ˆ (0 E g ˆ Using hese inequaliies and he fac ha x exp{( x / } and ˆ ( ˆ ( ˆ ( l l ˆ ( we find for = 0 ˆ ( e (54 ( exp in( ˆ ( (0 Choosing c o be sufficienly sall using (54 and ha we ge for c T ( s CB exp M (03( B in( B CB exp M (03( B (55 Fro (5(53 and (54 i follows ha

20 0 x s 3 v/ x e Gv x v ( ( 8 Ce (56 s In paricular (56 iplies Pu x s 3 v/ (0 v (0 s v 0 x e G x C (57 x s 3 ( s v/ x e Gv x v 0 ( ( Then fro (5 (56 and (57 we have ( s ( ( ( x W x ( s ( s ( x (0 x W ( s ( x s 8 ( s ( e s ( s (0 x s (0 x ( x (58 where as usual i oe ha he polynoials Q ( x in (7 are acually he resuls of he j expansion of ( s exp x / (0 x aing ino accoun ha G0 ( x ; also i is obvious ha ( s (0 x c for soe c 0 Using hese facs and (7 (8 Lea A in (58 afer soe algebra we coplee he proof of Proposiion for c T ( s Le now ct ( s ct Then using Lea A i is easy o see ha x 8 ( ( x e Ce s Apply above oulined echnique for he firs er in he rhs of his inequaliy using (4 Lea A wih s=3 Lea A3 and he fac ha c T ( s o coplee he proof of Proposiion ; he deails are oied Proof of Theore 3 In addiion o he noaions of Sec3 define 3 L ( E gˆ { gˆ } 3/ ` Since x c 0 Theore of Mirahedov (996 gives: for arbirary 0 here exiss a consan C 0 such ha C L ( L ( ( ( B ( exp M ( (4 B ( 8 (3

21 ax M ( (4 B ( in( B M ( (4 B ( Since L ( ( and 0 is arbirarily sall Theore 3 follows Proof of Theore 3 Fro Theore of Mirahedov (994 C B exp M ( 4 B ( 8 (3 / / since ( u P( g( g ( u n If M ( (4 B cb for soe c 0 / hen Theore 3 is rue wih C c If M ( (4 B cb hen / B exp M ( (4 B / 8 / / c c since (0] and Theore 3 / follows Proof of Theore 33 By he well-nown Esseen's soohing inequaliy we obain ( s ( s ( x e W ( x 4 d s s ( s ( s ( x e W ( x e W ( x s d ct ct d ct s ( x 4 d s We have ( s ( x e W ( x u ( s ax ( u x e W ( u x u u ( s On he oher hand fro he definiion of W ( x Lea A and he inequaliy /( s 3 s we observe ha ct s ( s e W ( x ct 3 d Ce c s Therefore ( s ( s ct ( x e W ( x d ( s ax ( x e W ( x c ( ct s s

22 Applying Proposiion here coplees he proof of Theore 33 Proof of Theore 34 follows by sandard ehods oulined as for insance in Perov (995 p04-07 which uses he inversion forula and Proposiions ; he deails are oied Proof of Theore 4 The cenral oens of order of he Poi( rv is a polynoial in of order / hence for even j ( / ( j j ( j / c P n ; for odd j one can use he well-nown inequaliy ( l /( s l s 3 l s Siilarly fro he inequaliy (53 of Mirahedov (996 we have M (03 B 3 0 ( n np Theore 4 follows fro hese facs and he inequaliy (37 Proof of Corollary 4 To ge he se of equaliies given before Corollary 4 wrie g np nex o find a higher order cenral ( ( ( ( ( oens of he Poi ( rv using he following recurrence forula of Kenney and Keeping (953: d E v E E d v v v ( ( ( Considering a rv which equals l p wih he probabiliy p and using well-nown inequaliies beween oens one can see l l 3 l l l ; wih equaliy iff p Wrie p ( wih p and pu ( I is easy o observe ha n ( c ( also 40 cn ( / 4 4 Considering separaely he cases when 0 and is bounded away fro zero and infiniy one can show ha c n 4 40 (( ( ( 4 c and T c ( ax( ; he deails oied; here c 0 is a consan and is differen in differen places I is eviden ha he condiion (39 is fulfilled Corollary 4 follows fro Theore 4 and Rear 3 Proof of Theores 4-44 We recall ha in his case is Bi p rv wih p n / To find he cenral oens of he rv we use he following forula: for ineger d E( p pq E( p ( E( p dp (59 We have: B nq 6pq 3 / / / 6 pq 3 pq 7 / 4 nq nq 4nq 3/ nq 3/ B 3 6 pq 3 pq 7

23 ow use he inequaliies 3 Eexp{ i } exp{ 4 pq sin / } sin e u e c c u0 u c (50 o ge M / ( ( ( (4 B 4 ( 6 pq 3 nq 4 e nq e nq since Finally ( 3 / nq Theores 4-44 follow fro Theores 3-33 respecively and he relaions given above s s s s Proof of Corollaries 43 and 44 Use inequaliy( a a n ( a a a 0 s n n o ge (47 Applying (59 we obain he forulas for ij Recall ( E exp{ if ( } i and fac ha is a su of independen Bi ( p rvs Fro (45 we have i ( P( e Ee 0 if ( 0 P( Ee iy Hence iy ( 0 iy P Ee P( P( 0 Ee ( P ( 0 d ( ( p sup Ee ( T s ( T s iy iy p sup Ee since P( 0 ( p Inequaliy (48 follows fro his and (35 On he oher hand ( i ln Ee ( ( iy ( i iy ( i Y ( ( Ee Ee Ee p Ee wih ( Bi( p Hence ( i Y p Ee pq E Y ( ( exp ( cos( Inequaliy (49 follows i iy exp pq Ee Proof of Theore 45 and 46 Recall in his case ( B( d p wih p n / ( n D j j i where D d d and p/ ( p We use ha Ee ( p ( pe j o find he oens of he rv and ha B D ( i d d B 3 ( ( D D 4 i d 4 ( sin Ee

24 Therefore using he inequaliies (5 we ge 4 since 3( e D ( M B e /3 /3 (03( 3 3( 4 3 ( ( DD d ( ( 3 ( ( D D / 3 Therefore Theores and 47 follow fro Theores 3 3 and 34 respecively and he inequaliy (37 by puing and soe siple algebra References Babu GJ and Bai ZD (996 Mixures of global and local Edgeworh expansion and heir applicaions J Mulivariae Anal Babu GJ and Singh E(985 Edgeworh expansions for sapling wihou replaceen fro finie populaions J Mulivariae Anal Barle MS (938 The characerisic funcion of a condiional saisic J London Mah Bhaacharya R and R Ranga Rao (976 oral approxiaion and asypoic Expansions John Wiley & Sons ew-yor 5 Bloznelis M (000 One and wo er Edgeworh expansion for finie populaion saple ean Exac resuls I Lih Mah J 40( Erdos P and Renyi A(959 On he cenral lii heore for saples fro a finie populaion PublMahIns Hungarian Acad Sci Gao R and Jaalaadaa S Rao (998 Sall saple approxiaions for spacings saisics Jour of Sais Planning and Inference Hols L (97 Asypoic noraliy and efficiency for cerain goodness-of-fi ess Bioeria Hols L (979 A unified approach o lii heores for urn odels J Appl Probab 6# Hols L and Rao J S (980 Asypoic heory for failies of wo-saple nonparaeric saisics Sanhya 4 Ser A 9-5 Hu Z Robinson J and WangQ(007 Edgeworh expansion for a saple su fro a finie se of independen rando variables Elecronic J Probab Ivanov VA Ivcheno GI and Medvedev YuI (985 Discree probles of he probabiliy heory (a survey J Sovie Mah3 no Ivcheno GI Mirahedov Sh A (99On lii heores for decoposable saisics and efficiency of he corresponding saisical ess Discree MahAppl

25 5 4 Jaalaadaa SRao and Schweizer RL (985 On ess for he wo-saple proble based on higher order spacing-frequencies Saisical heory and Daa Analysis Edior K Mausia or-holland Johnson L Koz S(977 Urn Models and heir Applicaions Wiley ew Yor 6 Kenney JF and Keeping ES (953 Maheaics of Saisics Par Van osrand Co 7 Kolchin VF Sevas'yanov BA and Chisyaov VP (978 Rando Allocaions VH Winson and Sons Washingon DC 8 Kolchin VF (985 Rando appings Translaions Series in Maheaics and Engg (Ed AV Balarishnan Opiizaion Sofware Inc Y 9 Koz S and Balarishnan (997 Advances in urn odels during he pas wo decades In Advances in Cobinaorial Mehods and Appl o Probab and Sais Birhauser Boson MA 0 Mihaylov A (993 Polynoial and polynoial lie allocaion: Resen developens In: Probab Mehods in Discree Mah Proc TVP/VSP Urech/Moscow VF Kolchin e al (eds Mirahedov Sh A(983An asypoic expansion for a saple su fro a finie populaion Theory Probab Appl v Mirahedov Sh A (985 Esiaes of proxiiy o he noral disribuion in sapling wihou replaceen Theory Probab Appl Mirahedov Sh A(987 Approxiaion of he disribuion of ulidiensional randoized divisible saisics by noral disribuion (ulinoial schee Theory Probab Appl Mirahedov Sh A(99Randoized decoposable saisics in he schee of independen allocaing paricles ino boxes Discree Mah Appl Mirahedov Sh A(994 Lii heores for condiional disribuions Discree Mah Appl Mirahedov Sh A(996 Lii heores on decoposable saisics in a generalized allocaion schees Discree Mah Appl Mirahedov Sh A(005Lower esiaion of he reainder er in he CLT for a su of he funcions of -spacings Sais Probab Leers Mirahedov Sh M * ( 007 Asypoic noraliy associaed wih generalized occupancy proble Sais Probab Leers Morris C(975 Cenral lii heores for ulinoial sus Ann Sais Muhin AB(99 Local lii heores for laice rando variables Theory Probab Appl * Mirahedov Sh M was forerly Mirahedov Sh A

26 6 3 Pavlov YuL and Cherepanova EV (00 Lii disribuion of a nuber of pairs in he generalized allocaion schee Discrenaya Maeaia (Russian 3 Quine MP and Robinson J(984 oral approxiaions o sus of scores based on occupancy nubers Ann Probab Quine MP and Robinson J(985 Efficiencies of chi-square and lielihood raio goodnessof-fi ess Ann Sais Robinson J (978 An asypoic expansion for saples fro a finie populaion Ann Sais Wils SS (963 Maheaical Saisics John Wiley ew-yor 36 Zhao L C Wu C Q and Wang Q(004 Berry-Esseen bound for a saple su fro a finie se of independen rando variables J Theoreical Probab