Variance function estimation in multivariate nonparametric regression with fixed design

Transcription

1 Joural of Multivariate Aalysis Cotets lists available at ScieceDirect Joural of Multivariate Aalysis joural homepage: Variace fuctio estimatio i multivariate oparametric regressio with fixe esig T. Toy Cai a, Michael Levie b,, Lie Wag a a Departmet of Statistics, The Wharto School, Uiversity of Pesylvaia, Philaelphia, PA 904, Uite States b Purue Uiversity, 50 N. Uiversity Street, West Lafayette, IN 47907, Uite States a r t i c l e i f o a b s t r a c t Article history: Receive May 007 Available olie 3 April 008 AMS 000 subject classificatios: primary 6G08 6G0 Keywors: Miimax estimatio Noparametric regressio Variace estimatio Variace fuctio estimatio i multivariate oparametric regressio is cosiere a the miimax rate of covergece is establishe i the ii Gaussia case. Our work uses the approach that geeralizes the oe use i [A. Muk, Bissatz, T. Wager, G. Freitag, O ifferece base variace estimatio i oparametric regressio whe the covariate is high imesioal, J. R. Stat. Soc. B 67 Part ] for the costat variace case. As is the case whe the umber of imesios, a very much cotrary to staar thikig, it is ofte ot esirable to base the estimator of the variace fuctio o the resiuals from a optimal estimator of the mea. Istea it is esirable to use estimators of the mea with miimal bias. Aother importat coclusio is that the first orer ifferece base estimator that achieves miimax rate of covergece i the oe-imesioal case oes ot o the same i the high imesioal case. Istea, the optimal orer of iffereces epes o the umber of imesios. 008 Elsevier Ic. All rights reserve.. Itrouctio We cosier the multivariate oparametric regressio problem y i gx i + V xi z i where y i R, x i S [0, ] R while z i are ii raom variables with zero mea a uit variace a have boue absolute fourth momets: E z i µ 4 <. We use the bol fot to eote ay -imesioal vectors with > except -imesioal iices a regular fot for scalars. The esig is assume to be a fixe equispace -imesioal gri; i other wors, we cosier x i {x i,..., x i } R where i k 0,,..., m for k,...,. Each cooriate is efie as x ik i k m for k,...,. The overall sample size is m. The iex i use i the moel is a -imesioal iex i i,..., i. Both gx a Vx are ukow fuctios supporte o S [0, ] ; we also assume that Vx > 0. The miimax rate of covergece for the estimator ˆV uer ifferet smoothess assumptios o g is the mai subject of iterest. The estimatio accuracy for ˆV is measure both globally by the mea itegrate square error MISE R ˆV, V E R ˆVx Vx x 3 Correspoig author. aress: mlevis@stat.purue.eu M. Levie X/$ see frot matter 008 Elsevier Ic. All rights reserve. oi:0.06/j.jmva

2 T.T. Cai et al. / Joural of Multivariate Aalysis a locally by the mea square error at a poit poitwise risk R ˆVx, Vx E ˆVx Vx. 4 We are particularly itereste i fiig how the ifficulty of estimatig V epes o the smoothess of the mea fuctio g as well as the smoothess of the variace fuctio V itself. Variace fuctio estimatio i heteroskeastic oparametric regressio is importat i may cotexts. Previous work has maily focuse o the uivariate regressio moel. See, for example, [,3,6,7,4]. More recet work iclues [,]. I the multiimesioal setup of, the problem has bee cosiere i [8,5] i the special case of a costat variace fuctio Vx σ. Spokoiy [8] ivestigate the effect of the imesioality o the estimatio accuracy of σ while assumig that the regressio fuctio g is twice cotiuously ifferetiable. A rate optimal proceure is costructe usig resiuals of a local liear fit. Muk et al. [5] use a ifferece base approach to variace estimatio a stuie the effects of both the smoothess of g a imesioality o the optimal rate of covergece for estimatig σ. Muk et al. [5] ote that... Differece estimators are oly applicable whe homogeeous oise is preset, i.e. the error variace oes ot epe o the regressor [5], p. 0. I the preset paper we exte the ifferece base approach of Muk et al. [5] to the case of the o-homogeeous heteroskeastic situatio where the variace V is a fuctio of the regressor x. This paper is also closely coecte to Wag et al. [] where a first orer ifferece base proceure for variace fuctio estimatio was stuie i the oe-imesioal case. The preset paper cosiers variace fuctio estimatio i the multiimesioal case which has some ifferet characteristics from those i the oe-imesioal case. I particular, first orer iffereces are iaequate i the high imesioal case. I fact, as i the costat variace case, it is o loger possible to use ay fixe orer iffereces a achieve asymptotically a miimax rate of covergece for a arbitrary umber of imesios >. The orer of iffereces ees to grow with the umber of imesios. We show that the miimax rate of covergece for estimatig the variace fuctio V uer both the poitwise square error a global itegrate mea square error is { } max 4α, β β+ 5 if g has α erivatives, V has β erivatives a is the umber of imesios; these results are obtaie i the ii Gaussia case. So the miimax rate epes o the smoothess of both V a g. The miimax upper bou is obtaie by usig kerel smoothig of the square iffereces of observatios. The orer of the ifferece scheme use epes o the umber of imesios. The miimum orer ees to be γ /4, the smallest iteger larger tha or equal to /4. With such a choice of the ifferece sequece our estimator is aaptive with respect to the smoothess of the mea fuctio g. The erivatio of the miimax lower bou is base o a momet matchig techique a a two-poit testig argumet. A key step is stuyig a hypothesis testig problem where the alterative hypothesis is a Gaussia locatio mixture with a special momet matchig property. It is also iterestig to ote that, if V is kow to belog to a regular parametric moel, such as the set of positive polyomials of a give orer, the cutoff for the smoothess of g o the estimatio of V is /4. That is, if g has at least /4 erivatives the the miimax rate of covergece for estimatig V is solely etermie by the smoothess of V as if g were kow. O the other ha, if g has less tha /4 erivatives the the miimax rate epes o the relative smoothess of both g a V a, for sufficietly small α, will be completely etermie by it. The larger is, the smoother the mea fuctio g has to be i orer ot to ifluece the miimax rate of covergece for estimatig the variace fuctio V. The paper is orgaize as follows. Sectio presets a upper bou for the miimax risk while Sectio 3 erives a ratesharp lower bou for the miimax risk uer both global a local losses. The lower a upper bous together yiel the miimax rate of covergece. Sectio 4 cotais a etaile iscussio of results obtaie a their implicatios for practical variace estimatio i the oparametric regressio. The proofs are give i Sectio 5.. Upper bou I this sectio we shall costruct a kerel variace estimator base o square iffereces of observatios give i. Note that it is possible to cosier a more geeral esig where ot all m k m, k,..., a x ik is efie as a solutio of the equatio i k m k x i k f k ss for a set of strictly positive esities f k s. We will ahere to a simpler esig throughout this paper. Differece base estimators have a log history for estimatig a costat variace i uivariate oparametric regressio. See, for example, [9,0,6,7,9,]. The multiimesioal case was first cosiere whe the imesioality i [8]. The geeral case of estimatig a costat variace i arbitrary imesio has oly recetly bee ivestigate i [5]. The estimatio of the variace fuctio Vx that epes o the covariate is a more recet topic. For the oe-imesioal case, we ca metio [,3,]. The multiimesioal case, to the best of our kowlege, has ot bee cosiere before. The followig otatio will be use throughout the paper. Defie a multi-iex J {j,..., j } as a sequece of oegative itegers j,..., j. For a fixe positive iteger l, let Jl {J j, j,..., j : J j + j + + j l}. Note that J stas for the sum of all elemets of vector J. For a arbitrary fuctio f, we efie D Jl f l f for all J such x j... x j

3 8 T.T. Cai et al. / Joural of Multivariate Aalysis that J l. For a arbitrary x R we efie x J x j... x j. Also, for ay vector u a real umber v, the set B u + va is the set of all vectors {y R : y u + va for some a A R }. For ay positive iteger α, let α eote the largest iteger that is strictly less tha α, α the smallest iteger that is greater tha α, a α α α. Now we ca state the fuctioal class efiitio that we ee. Defiitio. For ay α > 0 a M > 0, we efie the Lipschitz class Λ α M as the set of all fuctios fx : [0, ] R such that D Jl fx M for l 0,,..., α, a D J α fx D J α fy M x y α. We assume that g Λ α M g a V Λ β M V. We will say for the sake of simplicity that g has α cotiuous erivatives while V has β cotiuous erivatives. I this efiitio, stas for the absolute value a is the usual l orm. I this sectio we costruct a kerel estimator base o iffereces of raw observatios a erive the rate of covergece for the estimator. Special care must be take to efie iffereces i the multivariate case. Whe a there is a set of ifferece coefficiets j, j 0,..., r, such that r j0 j 0, r j0 j we efie the ifferece achore arou the poit y i as r j0 jy i+j. Whe >, there are multiple ways to eumerate observatios lyig arou y i. A example that explais how to o it i the case is give i [5]. For a geeral >, we first select a -imesioal iex set J Z that cotais 0. Next, we efie the set R cosistig of all -imesioal vectors i i,..., i such that R + J {i + j j J, i R} {,..., m}. 6 Agai, a subset of R + J correspoig to a specific i R is eote as i + J. The, the ifferece achore arou the poit y i is efie by D i j y i +j. 7 The cariality of the set J is calle the orer of the ifferece. For a goo example that illustrates this otatio style whe see [5]. Now we ca efie the variace estimator ˆVx. To o this, we use kerel-base weights K h i x that are geerate by either the regular kerel fuctio K or the bouary kerel fuctio K, epeig o the locatio of the poit x i the support set S. The kerel fuctio K : R R has to satisfy the followig set of assumptios: Kx is supporte o T [, ], Kxx 8 T Kxx J x 0 for 0 < J < β a T K xx k <. T Specially esige bouary kerels are eee to cotrol the bouary effects i kerel regressio. I the oe-imesioal case bouary kerels with special properties are relatively easy to escribe. See, for example, [5]. It is, however, more ifficult to efie bouary kerels i the multiimesioal case because ot oly the istace from the bouary of S but also the local shape of the bouary regio plays a role i efiig the bouary kerels whe >. I this paper we use the -imesioal bouary kerels give i [4]. We oly briefly escribe the basic iea here. Recall that we work with a oegative kerel fuctio K : T R with support T [, ] R. For a give poit x S cosier a movig support set S x x+hs x which chages with x a epes o through the bawith h. For example, if, the set S x becomes a iterval [x hx, x + h x] [ hx, h + x h]. Pluggig i x 0 as the left bouary a x as the right bouary brigs us back to the regular support S [0, ]. Usig this varyig support set S x, it is possible to efie the support T x of the bouary kerel that is iepeet of. To o this, first efie the set T x x ht; the subscript agai stresses that this set epes o through the bawith h. This is the set of all poits that form a h-eighborhoo of x. Usig T x a the movig support S x, we have the traspose a rescale support of the bouary kerel as T x h [x {T x S x}] h x {x + hs x} x ht x S T. 9 The subscript has bee omitte sice T x is, iee, iepeet of. Thus, the support of the bouary kerel has bee stabilize. The bouary kerel K with support o T x ca the be efie as a solutio of a certai variatioal problem i much the same way as a regular kerel K. For more etails, see [4]. Usig this otatio, we ca efie the geeral variace estimator as Vx K h i xd i K hx i j y i+j. 0

4 T.T. Cai et al. / Joural of Multivariate Aalysis The kerel weights are efie as h xi x K K hx h i h xi x K h whe x ht S, whe x ht S. It ca also be escribe by the followig algorithm:. Choose a -imesioal iex set J.. Costruct the set R. 3. Defie the estimator Kh i x jy i+j as a local average usig kerel-geerate weights K h i x. I this paper we will use the iex set J selecte to be a sequece of γ poits o the straight lie i the -imesioal space that iclues the origi: J {0, 0,..., 0,,,...,,..., γ, γ,..., γ}. I aitio, we use ormalize biomial coefficiets as the ifferece coefficiets. This is the so-calle polyomial sequece see, e.g., [5] a is efie as / γ γ / k k k γ where k 0,,..., γ. It is clear that γ k0 k 0, γ k0 k, a γ k0 kq k 0 for ay q,,..., γ. Remark. It is also possible to use the local liear regressio estimator istea of the kerel estimator. I this case, the bouary kerel ajustmet is ot ecessary as it is well kow that the local liear regressio ajusts automatically i bouary regios, preservig the asymptotic orer of the bias itact. However, the proof is slightly more techically ivolve whe usig the local liear regressio estimator; i particular, the local liear regressio estimator has to be represete as the kerel estimator where the shape of the fuctio K use to efie the local weights ow epes o the locatio of the esig poits, the umber of observatios a the poit of estimatio x. For etails, see, for example, [3]. Remark. It is possible to efie a more geeral estimator by cosierig averagig over several possible -imesioal iex sets J l, l,..., L, a efiig a set R l for each oe of them accorig to 6. I other wors, we efie L L Vx µ l K h i xd i µ l K hx i j y i+j l l l l l where µ l is a set of weights such that l µ l. The proof of the mai result i the geeral case is completely aalogous to the case L. If some iformatio about the geometry of the surface of Vx is kow, we may be able to choose the collectio of iex sets J l as escribe above i orer to miimize the costat factor i the asymptotic variace of the estimator of Vx. I this paper we limit ourselves to the iscussio of the case L a the efiitio 0 will be use with the set J selecte as i. Like i the mea fuctio estimatio problem, the optimal bawith h ca be easily fou to be h O /β+ for V Λ β M V. For this optimal choice of the bawith, we have the followig theorem. Theorem. Uer the regressio moel with z i beig iepeet raom variables with zero mea, uit variace a uiformly boue fourth momets, we efie the estimator V as i 0 with the bawith h O /β+ a the orer of the ifferece sequece γ /4. The there exists some costat C 0 > 0 epeig oly o α, β, M g, M V a such that for sufficietly large, sup sup E Vx Vx C 0 max{ 4α g Λ α M g,v Λ β M V x S, β β+ } 3 a sup E Vx Vx x C 0 max{ 4α, β β+ }. 4 g Λ α M g,v Λ β M V R Remark 3. The uiform rate of covergece give i 3 yiels immeiately the poitwise rate of covergece for ay fixe poit x S, sup E Vx Vx C 0 max{ 4α, β β+ }. g Λ α M g,v Λ β M V

5 30 T.T. Cai et al. / Joural of Multivariate Aalysis Lower bou Theorem gives the upper bous for the miimax risks of estimatig the variace fuctio Vx uer the multivariate regressio moel. I this sectio we shall show that the upper bous are i fact rate-optimal. We erive lower bous for the miimax risks which are of the same orer as the correspoig upper bous give i Theorem. I the lower bou argumet we shall assume that the errors are ormally istribute, i.e., z i ii N0,. Theorem. Uer the regressio moel with z i ii N0,, if V sup E V V g Λ α M g,v Λ β C max{ 4α M V a for ay fixe x [0, ] if V sup E Vx Vx C max{ 4α g Λ α M g,v Λ β M V where C > 0 is a costat. Combiig Theorems a yiels immeiately the miimax rate of covergece, { } max 4α, β +β,, β +β } 5, β +β } 6 for estimatig V uer both the global a poitwise losses. Theorem is prove i Sectio 5. The proof is base o a momet matchig techique a a two-poit testig argumet. Oe of the mai steps is stuyig a hypothesis testig problem where the alterative hypothesis is a Gaussia locatio mixture with a special momet matchig property. 4. Discussio The first importat observatio that we ca make o the basis of reporte results is that the ukow mea fuctio g oes ot have ay first orer effect o the miimax rate of covergece of the estimator ˆV as log as the fuctio g has at least /4 erivatives. Whe this is true, the miimax rate of covergece for ˆV is β/β+, which is the same as if the mea fuctio g ha bee kow. Therefore the variace estimator ˆV is aaptive over the collectio of the mea fuctios g that belog to Lipschitz classes Λ α M g for all α /4. O the other ha, if the fuctio g has less tha /4 erivatives, the miimax rate of covergece for ˆV is etermie by the relative smoothess of both g a V. Whe 4α/ < β/β +, the roughess of g becomes the omiat factor i etermiig the covergece rate for ˆV. I other wors, whe α < β/β +, the rate of covergece becomes 4α/ a thus is completely etermie by α. To make better sese of this statemet, let us cosier the situatio whe β icreases a ca become arbitrarily large. Clearly, i this case the cutoff β/β+ approaches /4. Thus, whe, ay mea fuctio g with less tha half of a erivative will completely etermie the rate of covergece for ˆV; whe 4, ay mea fuctio with less tha oe erivative will o a so o. As the umber of imesios grows a the fuctio V becomes smoother, the rate of covergece of ˆV becomes more a more epeet o the mea fuctio. I other wors, a ever icreasig set of possible mea fuctios will completely overwhelm the ifluece of the variace fuctio i etermiig the miimax covergece rate. As oppose to may commo variace estimatio methos, our approach oes ot estimate the mea fuctio first. Istea, we estimate the variace as the local average of square iffereces of observatios. Takig a ifferece of a set of observatios is, i a sese, a attempt to average out the ifluece of the mea. It is possible to say the that we use a implicit estimator of the mea fuctio g that is effectively a liear combiatio of all y j, j J, except y 0. Such a estimator is, of course, ot optimal sice its square bias a variace are ot balace. The reaso that it has to be use is because the bias a variace of the mea estimator ĝ have very ifferet iflueces o ˆV. As is the case whe agai, see [], the ifluece of the bias of ĝ is impossible to reuce at the seco stage of variace estimatio. Therefore, at the first stage we use a estimator of g that provies for the maximal reuctio i bias possible uer the assumptio of g Λ α M g, ow to the orer α/. I fact, the variace of the estimator ĝ is high but this is of little cocer; it is icorporate easily ito the variace estimatio proceure. Thus, i practical terms, subtractig optimal estimators of the mea fuctio g first may ot be the most esirable course of actio. Note also that it is ot eough to use here a simple first orer ifferece as has bee oe i the case of by Wag et al. []. The reaso is that this oes ot allow us to reuce the mea-relate bias of the variace estimator ˆV to the fullest extet possible. It is ot eough to cosier oly α < /4 as is the case whe. Istea, whe provig the upper bou result, we have to cosier mea fuctios with α < /4. Thus, higher orer iffereces are eee i orer to reuce the mea-relate bias to the orer of α/ a to esure the miimax rate of covergece.

6 T.T. Cai et al. / Joural of Multivariate Aalysis Proofs 5.. Upper bou: Proof of Theorem We will use M to eote a geeric positive costat throughout this sectio. We shall oly prove 3. Iequality 4 is a irect cosequece of 3. Recall that T [, ] is the support of the kerel K. The followig otatio will be useful: for ay two vectors x x,..., x a y y..., y we efie the ifferetial operator D x,y y k x k y x, k z k where z k is a geeric kth argumet of a -imesioal fuctio while is a graiet operator i R. Usig the otatio that we itrouce earlier, we ca write the ifferece D i as 7 D i j gx i+j + where δ i jgx i+j, Vi Vx j i+j a ɛ i j V / x i+j z i+j δ i + Vi ɛ i 8 / Vx j i+j j V xi+j z i+j has zero mea a uit variace. Thus, D i δ i + V i + V i ɛ i + δ i Vi ɛ i. Without loss of geerality, suppose h /β+. Because the kerel K has a boue support T [, ], we have K hx i h K hx i [,] K uu k 9 where k maxk, k. The first step follows from the fact that the h-eighborhoo of x has h poits while the seco step follows from approximatig the Riema sum by the appropriate itegral. Also, K u Ku whe u T u S a K u K u whe u T u S with costats k a k resultig from oe of these two respective choices. Recall that ˆVx Vx Khx i D i Vx. For all g Λ α M g a V Λ β M V, the mea square error of ˆV at x satisfies E Vx Vx E K hx i D i Vx + o h { E K hx i δ i + K hx i V i Vx + K hx i V i ɛ i + } K hx i δ i Vi ɛ i + o h 5 K hx i δ i + 5 K hx i V i Vx + 5E K hx i V i ɛ i + 0E K hx i δ i Vi ɛ i + o h. Recall that it is eough to cosier oly α < /4. Defie γ /4. Thus efie, γ will be the same as the maximum possible value of α for all α < /4. Defiig 0 u a usig Taylor expasio of gx i+j arou x i, we have for a ifferece sequece of orer γ δ i j gx i+j j gx i α m D xi+j,x i m gx i u α α! D x i+j,x i α gx i + ux i+j x i D xi+j,x i α gx i u m! The first two terms i the above expressio are zero by efiitio of the ifferece sequece j of orer γ. Usig the otatio x k i for the kth cooriate of x i, the explicit represetatio of the operator D xi+j,x i α gives.

7 3 T.T. Cai et al. / Joural of Multivariate Aalysis D xi+j,x i α gx i + ux i+j x i D xi+j,x i α gx i [ α ] x tr i+j xtr i D α gx i + ux i+j x i t t α r [ α x tr i+j xtr i t t α r Now we use the efiitio of Lipschitz space Λ α M g, Jese s a Höler s iequalities to fi that D α gx i + ux i+j x i D α α gx i M g ux i+j x i α x tr i+j xtr i M g x i+j x i α t t α α t t α r r x tr i+j xtr i α α M x i+j x i α x i+j x i α M x i+j x i α ; as a cosequece, we have δ i M α/. Thus, 4 K hx i δ i 4 K h i x M α/ + km 4 4α/ O 4α/. ] D α gx i. I exactly the same way as above, for ay x, y [0, ], Taylor s theorem yiels β Vx Vy D x,y j Vy j j! u β 0 β D x,y β Vy + ux y D x,y β Vyu M x y β u β 0 β u M x y β. 0 So, V i Vx Therefore, we have j Vx i+j Vx β j k j D xi+j,x k Vx + k! K hx i V i Vx K hx i β j k u β j 0 + K hx i β [ Vxi+j Vx ] j 0 u β β D x i+j,x β Vx i+j D xi+j,x β Vx u. D xi+j,x k Vx k! D x i+j,x β Vx i+j D xi+j,x β Vx u. It is fairly straightforwar to fi out that the first term is boue by K h x i β D xi+j,x k Vx j k k! h xi x β K k j x tr h k k! i+j xtr i D k Vx t t k r β M h h k Ku i u k i o h. k To establish the last iequality it is importat to remember the fact that V Λ β M V a therefore D k Vx M V. To hale the prouct k r xtr i+j xtr i the iequality i x i i x, that is true for ay positive umbers i x,..., x, must be use. The equality that follows is base o the fact that kerel K has β vaishig momets. After takig the square the above will become o h ; comparig to the optimal rate of β/β+, it is easy to check that this term is always of smaller orer, o β/β+ β+/β+. Usig 0, we fi that the absolute value of the seco term gives us K h x i u β j 0 β D x i+j,x β Vx i+j D xi+j,x β Vx u Mh β K i h x j O β/β+. From here it follows by takig squares that 5 Kh i x V i Vx is of the orer O β/β+.

8 T.T. Cai et al. / Joural of Multivariate Aalysis a O the other ha, sice V M V, we have ue to 9 5E K hx i δ i Vi ɛ i 5Var K hx i δ i Vi ɛ i 5 5M V α/ β/β+ k K h i x δ i V i 0E K hx i V i ɛ i 0Var K hx i V i ɛ i 0M µ V 4 K hi x 0M µ V 4 h k 0Mµ V 4 β/β+ k. Puttig the four terms together we have, uiformly for all x [0, ], g Λ α M g a V Λ β M V, E Vx Vx C 0 max{ 4α/, β/β+ } for some costat C 0 > 0. This proves Proof of Theorem The proof of this theorem ca be aturally ivie ito two parts. The first step is to show if V sup g Λ α M g,v Λ β M V E Vx Vx C β +β. This part is staar a relatively easy. The proof of the seco step, if V sup E Vx Vx C 4α, g Λ α M g,v Λ β M V is base o a momet matchig techique a a two-poit testig argumet. More specifically, let X,..., X ii P a cosier the followig hypothesis testig problem, betwee a H 0 : P P 0 N0, + θ H : P P Nθ ν, Gν where θ > 0 is a costat a G is a istributio of the mea ν with compact support. The istributio G is chose i such a way that, for some positive iteger q epeig o α, the first q momets of G match exactly with the correspoig momets of the staar ormal istributio. The existece of such a istributio is give i the followig lemma from Karli a Stue [0]. Lemma. For ay fixe positive iteger q, there exist a B < a a symmetric istributio G o [ B, B] such that G a the staar ormal istributio have the same first q momets, i.e. B B x j Gx + x j ϕxx, j,,..., q where ϕ eotes the esity of the staar ormal istributio. We shall oly prove the lower bou for the poitwise square error loss. The same proof with mior moificatios immeiately yiels the lower bou uer itegrate square error. Note that, to prove iequality, we oly ee to focus o the case where α < /4; otherwise β/+β is always greater tha 4α/ for sufficietly large a the follows irectly from. For a give 0 < α < /4, there exists a iteger q such that q + α >. For coveiece we take q to be a o iteger. From Lemma, there is a positive costat B < a a symmetric istributio G o [ B, B] such that G a N0, have the same first q momets. Let r i, i,...,, be iepeet variables with the istributio G. Set θ Mg B m α, g 0 0, V 0 x + θ a V x. Let hx m x for x [ x, ] a 0 otherwise here x m m + + x. Defie the raom fuctio g by g x θ r i hx x i I x [0, ]. i

9 34 T.T. Cai et al. / Joural of Multivariate Aalysis The it is easy to see that g is i Λ α M g for all realizatios of r i. Moreover, g x i θ r i are iepeet a ietically istribute. Now cosier testig the followig hypotheses: H 0 : y i g 0 x i + V0 x iɛ i, i,...,, H : y i g x i + V x iɛ i, i,...,, where ɛ i are iepeet N0, variables which are also iepeet of the r i s. Deote by P 0 a P the joit istributios of y i s uer H 0 a H, respectively. Note that for ay estimator V of V, max{e Vx V 0 x, E Vx V x } 6 ρ4 P 0, P V 0 x V x 6 ρ4 P 0, P M 4 g 6B 4 m 4α 3 where ρp 0, P is the Helliger affiity betwee P 0 a P. See, for example, []. Let p 0 a p be the probability esity fuctios of P 0 a P with respect to the Lebesgue measure µ; the ρp 0, P p 0 p µ. The miimax lower bou follows immeiately from the two-poit bou 3 if we show that for ay, the Helliger affiity ρp 0, P C for some costat C > 0. Note that m 4α 4α/. Note that uer H 0, y i N0, + θ a its esity 0 ca be writte as t 0 t ϕ ϕt vθ ϕvv. + θ + θ Uer H, the esity of y i is t ϕt vθ Gv. It is easy to see that ρp 0, P 0 µ, sice the y i s are iepeet variables. Note that the Helliger affiity is boue below by the total variatio affiity, 0 t tt 0 t t t. Taylor expasio yiels ϕt vθ ϕt k0 vk θ k the costructio of istributio G, v i Gv v i ϕvv for i 0,,..., q. So, 0 t t ϕt vθ Gv ϕt vθ ϕvv ϕt H i t v i θ i i0 Gv ϕt H i t v i θ i i0 ϕvv ϕt H i t v i θ i iq+ Gv ϕt H i t v i θ i iq+ ϕvv ϕt H i t v i θ i Gv + ϕt iq+ H k t where H k! k t is the correspoig Hermite polyomial. A from iq+ Suppose q + p for some iteger p; it ca be see that ϕt H i t v i θ i iq+ Gv ϕt H i t i! θi vi Gv ϕt H i t i! θi v i Gv ϕt H i t i! a ϕt H i t v i θ i iq+ ϕvv ϕt ϕt H i t i! θi v i ϕvv ϕt H i t i! θi where i!! i i 3 3. So from 4, 0 t t ϕt H i t i! θi Bi + ϕt H i t i H i t v i θ i ϕvv. 4 vi ϕvv H i tθ i i ϕt θi H i t i θi Bi θi

10 T.T. Cai et al. / Joural of Multivariate Aalysis a the 0 t tt ϕt H i t i! θi Bi + ϕt H i t i θi ϕt H i t i! θi Bi t ϕt H i t i For the Hermite polyomial H i, we have t θi [ ] i ϕt H i t t ϕt k ii i k + i!! + t k t k k! [ ] i ϕt k ii i k + i!! + t k t k k! i k ii i k + i!! + t k ϕtt k k! i k ii i k + i!! + k!! k k! i ii i k + i!! + k! i i!!. k For sufficietly large, θ < / a it the follows from the above iequality that ϕt H i t i! θi Bi t θ p θ i Bi i! B i θ i p ϕt H i t t θ p eb θ i Bi i! i i!! t. 5 a ϕt H i t i θi t θ i i θ p ϕt H i t t θ i i i i!! θ p i θ i p θ p p+. θ p i i!! i p θ i p The from 5 0 t tt θp eb θp where c is a costat that oly epes o q. So p+ θ p eb + p cθ q+ ρp 0, P 0 t tt cθ q+ c αq+. Sice αq+, lim c αq+ e c > 0 a the theorem the follows. Ackowlegmets The research of Toy Cai was supporte i part by NSF Grats DMS a DMS

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