Wavelet linear estimation of a density and its derivatives from observations of mixtures under quadrant dependence

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1 ProbStat Forum, Volume 5, April, Pages ISSN ProbStat Forum is a e-joural For details please visit wwwprobstatorgi Wavelet liear estimatio of a desity ad its derivatives from observatios of mixtures uder quadrat depedece Christophe Cheseau Laboratoire de Mathématiques Nicolas Oresme (LMNO), Campus II, Sciece 3, 43 Cae, Frace Abstract The estimatio of a desity ad its derivatives from a fiite mixture uder the pairwise positive quadrat depedece assumptio is cosidered A ew wavelet based liear estimator is costructed We evaluate its asymptotic performace by determiig a upper boud of the mea itegrated squared error We prove that it attais a sharp rate of covergece for a wide class of ukow desities Itroductio The followig mixture desity model is cosidered: we observe radom variables X,, X such that, for ay i {,, }, the desity of X i is the fiite mixture: h i (x) = m w d (i) f d (x), x [, ], d= m is a positive iteger, (w d (i)) (i,d) {,,} {,,m} are kow positive weights such that, for ay i {,, }, m w d (i) =, d= f,, f m are ukow desities For a fixed {,, m}, we aim to estimate f ad, more geerally, its r-th derivative from Pairwise Positive Quadrat Depedet (PPQD) X,, X Let us ow preset a brief survey related to this problem uder various cofiguratios Whe X,, X are idepedet, the estimatio of f has bee cosidered i eg [9], [5] ad [4] The estimatio of its r-th derivative has bee recetly studied by [8] (this is particularly of iterest to detect possible bumps, cocavity or covexity properties of f ) Whe X,, X are idetically distributed ie h = h = = h, Mathematics Subject Classificatio Primary: 6G7; Secodary: 6G Keywords Desity estimatio, mixtures, quadrat depedece, wavelets Received: 3 July ; Accepted 5 March address: christophecheseau@uicaefr (Christophe Cheseau)

2 C Cheseau / ProbStat Forum, Volume 5, April, Pages the estimatio of h for associated X,, X (icludig PPQD) has bee ivestigated i eg [], [4], [] ad [7] The estimatio of h (r) has bee explored by [] However, to the best of our kowledge, the combiatio of these two complex statistical frameworks ie the estimatio of, icludig f, uder PPQD coditios is a ew challege Such a problem occurs i the study of medical, biological ad other types of data The most commo situatio is the followig: for ay i {,, }, X i depeds o a uobserved radom idicator I i takig its values i {,, m} Applyig the Bayes theorem, the desity of X i is h i defied with w d (i) = P(I i = d) ad f d the coditioal desity of X i give {I i = d} Naturally, i some situatios, X,, X are ot idepedet ad this motivates the study of various depedece structures as the PPQD oe Further details ad applicatios o the cocept of PPQD ca be foud i [], [9] ad [] To estimate, several methods are possible as kerel, splies, (see eg [5, 6], [6] ad []) I this study, we focus our attetio o the multiresolutio aalysis techiques ad, more precisely, the wavelet methodology of [4] ad [8] We costruct a liear wavelet estimator ad explore its asymptotic performace by takig the mea itegrated squared error (MISE) ad assumig that belogs to a Besov ball We prove that, uder some specific assumptios, it attais a similar rate of covergece to the oe obtaied i the idepedet case This paper is orgaized as follows Assumptios o the model ad some otatios are itroduced i Sectio Sectio 3 briefly describes the wavelet basis o [, ] ad the Besov balls The liear wavelet estimator ad the results are preseted i Sectio 4 Sectio 5 is devoted to the proofs Assumptios Additioal assumptios o the model are preseted below The itegers r ad refer to those i Assumptio o f,, f m Without loss of geerality, for ay d {,, m}, we assume that the support of f d is [, ] (our study ca be exteded to aother compact support) We suppose that there exists a costat C > such that, for ay d {,, m}, sup d (x) C x [,] () We suppose that, for ay d {,, m} ad v {,, r}, f (v) d (v) () = f () = d () Simple examples are desities of the form f d (x) = c x α ( x) β d (x), x [, ], α > r, β > r, d is a positive fuctio such that d C r ([, ]) ad c = xα ( x) β d (x)dx This icludes some usual distributios as Beta(α, β), Betamixture(α, β), Assumptio o the weights of the mixture We suppose that the matrix Γ = w k (i)w l (i) (k,l) {,,m} satisfies det(γ ) > For the cosidered ad ay i {,, }, we set a (i) = det(γ ) m ( ) k+ γ,k w k(i), (3) k= γ,k deotes the determiat of the mior (, k) of the matrix Γ

3 The a (),, a () satisfy C Cheseau / ProbStat Forum, Volume 5, April, Pages (a (),, a ()) = argmi (u,,u ) m d= U,d U,d = (u,, u ) R ; u i, (4) u i w d (i) = δ,d ad δ,d deotes the Kroecker delta Techical details ca be foud i [9] ad [8] We set z = a (i) (5) For techical reasos, we suppose that (/z ) /(r+) (, ) Assumptios o X,, X We suppose that X,, X are PPQD ie for ay (i, l) {,, } with i l ad ay (x, y) [, ], P(X i > x, X l > y) P(X i > x)p(x l > y) (6) This kid of depedece has bee itroduced by [8] Examples of PPQD variables ca be foud i [] We suppose that, for ay (i, l) {,, }, there exists a costat C > such that sup h i,l (x, y) h i (x)h l (y) C, (7) (x,y) [,] h i,l deotes the desity of (X i, X l ) We suppose that there exists a costat C > such that a (i) a (l) C ov (X i, X l ) Cz, (8) i 3 i C ov (, ) deotes the covariace, a (),, a () are (3) ad z is (5) I particular, if there exist a costat C > ad a sequece of positive real umbers (b ) N satisfyig = b <, for ay (i, l) {,, } with i l, i 3 C ov (X i, X l ) Cb i l, the (8) is satisfied Note that (6), (7) ad (8) iclude the idepedet case

4 3 Wavelets ad Besov balls C Cheseau / ProbStat Forum, Volume 5, April, Pages Throughout the paper, we work with the wavelet basis o [, ] described below Let φ ad ψ be the iitial wavelet fuctios of the Daubechies wavelets dbn with N such that φ ad ψ belog to C r+ Furthermore, metio that φ ad ψ are compactly supported Set φ j,k (x) = j/ φ( j x k), ψ j,k (x) = j/ ψ( j x k) The there exists a positive iteger η satisfyig η N such that, for ay l η, the collectio S = {φ l,k, k {,, l }; ψ j,k ; j N {,, l }, k {,, j }}, with a appropriate treatmet at the boudaries, is a orthoormal basis of L ([, ]) (the set of squareitegrable fuctios o [, ]) For ay iteger l η, ay h L ([, ]) ca be expaded o S as l h(x) = α l,k φ l,k (x) + k= j β j,k ψ j,k (x), x [, ], j=l k= α j,k = h(x)φ j,k (x)dx, β j,k = h(x)ψ j,k (x)dx (3) Details ca be foud i [3] ad [] I this paper we assume that belogs to a subset of Besov space defied below We say that a fuctio h L ([, ]) belogs to B s, (M) if ad oly if there exists a costat M > (depedig o M) such that (3) satisfy sup j η β j,k M js j k= For more details about wavelet basis, see [] ad [] The ext sectio is devoted to our estimator ad its asymptotic performaces i term of MISE 4 Estimator ad results Assumig that j f ˆ (r) (x) = k= ˆα (r) j,k = ( )r B s (M) ad usig S, we defie the liear wavelet estimator ˆ, ˆα (r) j,k φ j,k(x), x [, ], (4) a (i)(φ j,k) (r) (X i ), (4) a (),, a () are (3), j is the iteger satisfyig ( ) /(s+r+) ( ) < j /(s+r+) z z ad z is defied by (5) by

5 C Cheseau / ProbStat Forum, Volume 5, April, Pages The defiitios of ˆα (r) j,k ad j, which take ito accout the PPQD case, are chose to miimize the MISE of f ˆ(r) Note that f ˆ(r) is close to the estimator cosidered by [8, equatio (45)] i the idepedet case Further details o derivatives desity estimatio via wavelet methods ca also be foud i [] ad [7] Theorem 4 below ivestigates the MISE of f ˆ(r) whe B s, (M) Theorem 4 Let X,, X be radom variables as described i Sectio uder the assumptios of Sectio Suppose that B s (M) with s > Let f ˆ (r), be (4) The there exists a costat C > such that ( E ( ˆ ) (x) (x)) dx C ( ) z s/(s+r+) The proof of Theorem 4 uses a momet iequality o (4) ad a suitable decompositio of the MISE The obtaied rate of covergece is the oe related to the idepedet case ie (z /) s/(s+r+) (see [8, Theorem 6 ad Remark 6]) Note that Theorem 4 ca be exteded to Pairwise Negative Quadrat Depedece X,, X (this is due to the Newma iequality [3, Lemma 3] used i the proof of Theorem 4 which still holds i this case) Remark that f ˆ(r) is ot adaptive with respect to s Adaptivity ca perhaps be achieved by usig a o-liear wavelet estimator as the hard thresholdig oe This approach works i the idepedet case (see [4, Theorem 4]) However, the proof of this fact uses techical tools as the Berstei ad the Rosethal iequalities ad it is ot immediately clear how to exted this to the PPQD case 5 Proofs I this sectio, C deotes ay costat that does ot deped o j, k ad Its value may chage from oe term to aother ad may depeds o φ Propositio 5 Let X,, X be radom variables as described i Sectio uder the assumptios of Sectio For ay k {,, j }, let α (r) j,k = (x)φ j,k(x)dx ad ˆα (r) be (4) The there exists a costat C > j j,k such that E(( ˆα (r) j,k α(r) j,k ) ) C rj z Proof of Propositio 5 Proceedig as i [8, equatio (46)], it follows from (4) ad r itegratios by parts with () that E( ˆα (r) j,k ) = ( )r = ( )r a (i)e((φ j,k) (r) (X i )) = ( ) r m a (i) (φ j,k) (r) (x)h i (x)dx d= f d (x)(φ j,k) (r) (x)dx = ( ) r f (x)(φ j,k) (r) (x)dx = a (i)w d (i) (x)φ j,k(x)dx = α (r) j,k

6 C Cheseau / ProbStat Forum, Volume 5, April, Pages E(( ˆα (r) j,k α(r) j,k ) ) = V( ˆα (r) j,k ) = a (i)a (l)c ov ((φ j,k) (r) (X i ), (φ j,k) (r) (X l )) a (i)v((φ j,k) (r) (X i )) + i a (i) a (l) C ov ((φ j,k) (r) (X i ), (φ j,k) (r) (X l )) (5) Let us boud the first term i (5) For ay i {,, }, usig () which implies sup x [,] h i (x) C, the equality (φ j,k) (r) (x) = j / rj φ (r) ( j x k) ad makig the chage of variable y = j x k, we have V((φ j,k) (r) (X i )) E(((φ j,k) (r) (X i )) ) = ((φ j,k) (r) (x)) h i (x)dx C rj (φ (r) (y)) dy C rj a (i)v((φ j,k) (r) (X i )) C rj a (i) = C rj z (5) Let us ow ivestigate the boud of the covariace term i (5) via two differet approaches Boud By a stadard covariace equality ad (7), for ay (i, l) {,, } with i l, we have C ov ((φ j,k) (r) (X i ), (φ j,k) (r) (X l )) = (h i,l (x, y) h i (x)h l (y))(φ j,k) (r) (x)(φ j,k) (r) (y)dxdy h i,l (x, y) h i (x)h l (y) (φ j,k) (r) (x) (φ j,k) (r) (y) dxdy ( C (φ j,k) (x) dx) (r) Moreover, sice (φ j,k) (r) (x) = j / rj φ (r) ( j x k), by the chage of variables y = j x k, we obtai (φ j,k) (r) (x) dx = rj j / φ (r) (y) dy C ov ((φ j,k) (r) (X i ), (φ j,k) (r) (X l )) C rj j (53) Boud Sice X,, X are PPQD, it follows from [3, Lemma 3] that, for ay (i, l) {,, } with i l, C ov ((φ j,k) (r) (X i ), (φ j,k) (r) (X l )) sup (φ j,k) (r+) (x) x [,] C ov (X i, X l )

7 C Cheseau / ProbStat Forum, Volume 5, April, Pages Sice (φ j,k) (r+) (x) = (r+3)j / φ (r+) ( j x k) ad sup x [,] φ (r+) (x) C, we have sup (φ j,k) (r+) (x) x [,] C j (r+3) C ov ((φ j,k) (r) (X i ), (φ j,k) (r) (X l )) C j (r+3) C ov (X i, X l ) (54) Combiig (53) ad (54), for ay (i, l) {,, } with i l, we obtai C ov ((φ j,k) (r) (X i ), (φ j,k) (r) (X l )) C mi( j (r+3) C ov (X i, X l ), rj j ) It follows from (55) ad j < that the secod term i (5) ca be boud as i a (i) a (l) C ov ((φ j,k) (r) (X i ), (φ j,k) (r) (X l )) C(E + F), (56) E = rj j j i a (i) a (l) (55) ad F = i j (r+3) a (i) a (l) C ov (X i, X l ) j Usig xy (/)(x + y ), (x, y) R, we have E C rj j C rj j j Usig (8), it comes F rj i 3 i j i ( a (i) + a (l) ) a (i) = C rj z (57) a (i) a (l) C ov (X i, X l ) C rj z (58) Puttig (5), (5), (56), (57) ad (58) together, we obtai E(( ˆα (r) j,k α(r) j,k ) ) C rj z This eds the proof of Propositio 5 Proof of Theorem 4 We expad the fuctio j (x) = α (r) j,k φ j,k(x) + k= j j=j k= o S as β (r) j,k ψ j,k(x), x [, ],

8 C Cheseau / ProbStat Forum, Volume 5, April, Pages α (r) j,k = We have, for ay x [, ], ˆ (x)φ j,k(x)dx, β (r) j,k = j (x) (x) = k= ( ˆα (r) j,k α(r) j,k )φ j,k(x) (x)ψ j,k (x)dx j j=j k= Sice S is a orthoormal basis of L ([, ]), we have ( ) E ( f ˆ(r) (x) (x)) dx = A + B, A = j k= E(( ˆα (r) j,k α(r) j,k ) ), B = j j=j k= (β (r) j,k ) Usig Propositio 5 ad the defiitio of j, we obtai Sice A C j rj z ( ) C z s/(s+r+) B C B s (M), we have, ( E ( ˆ j=j js C js C ) (x) (x)) dx C The proof of Theorem 4 is complete ( ) z s/(s+r+) ( ) z s/(s+r+) β (r) j,k ψ j,k(x) Refereces [] Cai, ZW, Roussas, GG (997) Efficiet estimatio of a distributio fuctio uder quadrat depedece, Scadiavia Joural of Statistics 4, 4 [] Chaubey, YP, Doosti, H, Prakasa Rao, BLS (6) Wavelet based estimatio of the derivatives of a desity with associated variables, Iteratioal Joural of Pure ad Applied Mathematics 7(), 97-6 [3] Cohe, A, Daubechies, I, Jawerth, B, Vial, P (993) Wavelets o the iterval ad fast wavelet trasforms, Applied ad Computatioal Harmoic Aalysis 4(), 54-8 [4] Dewa, I, Prakasa Rao, BLS (999) A geeral method of desity estimatio for associated radom variables, Joural of Noparametric Statistics, 45-4 [5] Hall, P, Zhou, XH (3) Noparametric estimatio of compoet distributios i a multivariate mixture, The Aals of Statistics 3, -4 [6] Härdle, W, Kerkyacharia, G, Picard, D, Tsybakov, AB (998) Wavelet, Approximatio ad Statistical Applicatios, Lectures Notes i Statistics, New York, 9, Spriger Verlag [7] Hosseiiou, N, Doosti, H, Nirumad, HA () Noparametric estimatio of the derivatives of a desity by the method of wavelet for mixig sequeces, Statistical papers, to appear [8] Lehma, EL (996) Some cocepts of depedece, The Aals of Mathematical Statistics 37, [9] Maiboroda, RE (996) Estimators of compoets of mixtures with varyig cocetratios, Ukrai Mat Zh 48(4) 68-6 [] Mallat, S (9) A wavelet tour of sigal processig, 3 rd editio, Elsevier/Academic Press, Amsterdam

9 C Cheseau / ProbStat Forum, Volume 5, April, Pages [] Masry, E () Multivariate probability desity estimatio for associated processes: Strog cosistecy ad rates, Statistics ad Probability Letters 58, 5-9 [] Meyer, Y (99) Odelettes et Opérateurs, Herma, Paris [3] Newma, CM (98) Normal fluctuatios ad the FKG iequalities, Commu Math Phys 74, 9-8 [4] Pokhyl ko, D (5) Wavelet estimators of a desity costructed from observatios of a mixture, Theor Prob ad Math Statist 7, [5] Prakasa Rao, BLS (983) Noparametric fuctioal estimatio, Academic Press, Orlado [6] Prakasa Rao, BLS (999) Noparametric fuctioal estimatio: a overview, Asymptotics, Noparametrics ad Time Series, Ed Subir Ghosh, 46-59, Marcel Dekker Ic, New York [7] Prakasa Rao, BLS (3) Wavelet liear desity estimatio for associated sequeces, Joural of the Idia Statistical Associatio 4, [8] Prakasa Rao, BLS () Wavelet liear estimatio for derivatives of a desity from observatios of mixtures with varyig mixig proportios, Idia Joural of Pure ad Applied Mathematics 4() 75-9 [9] Prakasa Rao, BLS, Dewa, I () Associated sequeces ad related iferece problems, Hadbook of Statistics, 9, Stochastic Processes: Theory ad Methods, (eds CR Rao ad DN Shabag), , North Hollad, Amsterdam [] Roussas, GG (999) Positive ad egative depedece with some statistical applicatios, Asymptotics, oparametrics ad time series (ed S Ghosh), , Marcel Dekker, New York [] Sacetta, A (9) Strog law of large umbers for pairwise positive quadrat depedet radom variables, Statistical Iferece for Stochastic Processes () [] Tsybakov, AB (4) Itroductio à l estimatio o-paramétrique, Spriger