Sharp minimaxity and spherical deconvolution for super-smooth error distributions

Transcription

1 Journal of Multivariate Analysis 90 (004) Shar minimaxity and sherical deconvolution for suer-smooth error distributions Peter T. Kim, a,,1 Ja-Yong Koo, b, and Heon Jin Park b a Deartment of Mathematics and Statistics, University of Guelh, Guelh, Ont., Canada NIG W1 b Inha University, Incheon, South Korea Received 17 Setember 001 Abstract The sherical deconvolution roblem was first roosed by Rooij and Ruymgaart (in: G. Roussas (Ed.), Nonarametric Functional Estimation and Related Toics, Kluwer Academic Publishers, Dordrecht, 1991, ) and subsequently solved in Healy et al. (J. Multivariate Anal. 67 (1998) 1). Kim and Koo (J. Multivariate Anal. 80 (00) 1) established minimaxity in the L -rate of convergence. In this aer, we imrove uon the latter and establish shar minimaxity under a suer-smooth condition on the error distribution. r 003 Elsevier Inc. All rights reserved. AMS 000 subject classifications: rimary 6G05; secondary 58G5 Keywords: Hellinger distance; Rotational harmonics; Sobolev saces; Sherical harmonics 1. Introduction The sherical deconvolution roblem can be described as follows. Consider the situation Z ¼ ex; ð1:1þ where e is an SOð3Þ (the grou of 3 3 rotation matrices) random element, and Z; X Corresonding author. addresses: kim@uoguelh.ca (P.T. Kim), jykoo@stat.inha.ac.kr (J.-Y. Koo), hjark@anova. inha.ac.kr (H.J. Park). 1 Research suorted in art by NSERC (Canada) OGP4604. Research suorted in art by KOSEF (Korea) through Statistical Research Center for Comlex Systems at Seoul National University X/$ - see front matter r 003 Elsevier Inc. All rights reserved. doi: /j.jmva

2 P.T. Kim et al. / Journal of Multivariate Analysis 90 (004) are S (two-dimensional unit shere) random elements, with e and X assumed indeendent. Let f Z ; f e ; f X denote the densities of Z; e; X; resectively. Then, f Z ¼ f e f X ; ð1:þ where denotes convolution and is defined below. Observations are made on Z and the objective is to recover f X : This roblem was first roosed by Rooij and Ruymgaart [8] and subsequently solved in [4]. Kim and Koo [5] established minimaxity in the L -rate of convergence and in this aer, we imrove uon the latter and establish shar minimaxity, i.e., constant and rate, under a suer-smooth condition on the error distribution. A summary of the aer is as follows. In Section, we briefly go over the necessary Fourier tools for the two-dimensional unit shere and the three-dimensional rotation matrices, as well as the connections between the two. In Section 3, we outline the deconvolution roblem on the -shere. In addition, we define suer-smooth densities on the sace of three-dimensional rotation matrices. This is done in the Fourier domain using the oerator norm. Following this we state the main results. All roofs are collected in Section 4.. Fourier reliminaries For comleteness, we will briefly outline the necessary Fourier analysis. Further details can be found in [4,5]. The well-known Euler angle decomosition says, any gasoð3þ can almost surely be uniquely reresented by three angles ðf; y; cþ; known collectively as the Euler angles, where fa½0; Þ; ya½0; Þ; ca½0; Þ: Consider the function, D c q 1 q ðf; y; cþ ¼e iq1f dq c 1 q ðcos yþe iqc ; where, dq c 1 q are related to the Jacobi olynomials for cq 1 ; q c; c ¼ 0; 1; y: Collectively, the functions f ffiffiffiffiffiffiffiffiffiffiffiffiffi c þ 1 : cq 1 ; q c; c ¼ 0; 1; yg D c q 1 q is a comlete orthonormal basis for L ðsoð3þþ (square integrable functions on SOð3Þ) with resect to the robability Haar measure and are otherwise known as the rotational harmonics. In addition, if we define a ðc þ 1Þðcþ1Þ matrix by D c ðgþ ¼ðD c q 1 q ðgþþ; where cq 1 ; q c; cx0 and gasoð3þ; these constitute the collection of inequivalent irreducible reresentations of SOð3Þ: Let f AL ðsoð3þþ: We define the rotational Fourier transform on SOð3Þ by q c 1 q ¼ Z SOð3Þ f ðgþd c q 1 q ðgþ dg; cq 1 ; q c; c ¼ 0; 1; y and dg is the robability Haar measure on SOð3Þ: Again, taken collectively, we can take the latter as the matrix entries of the

3 386 P.T. Kim et al. / Journal of Multivariate Analysis 90 (004) ðc þ 1Þðcþ1Þ matrix c ¼ðˆ fq c 1 q Þ; cq 1 ; q c; c ¼ 0; 1; y : The rotational inversion can be obtained by f ðgþ ¼ X0 q 1 ;q ¼ c ðc þ 1Þf ˆ q c 1 q D c q q 1 ðg 1 Þ; for gasoð3þ: Sherical Fourier analysis also has similar results. Any oint on S reresented by o ¼ðcos f sin y; sin f sin y; cos yþ t ; can be where ya½0; Þ; fa½0; Þ and suerscrit t denotes transose. Let sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Yq c ðoþ ¼Y q c ðc þ 1Þðc qþ! ðy; fþ ¼ð 1Þq P c q 4ðc þ qþ! ðcos yþeiqf ; where ya½0; Þ; fa½0; Þ; cqc; c ¼ 0; 1; y and P c q are the Legendre functions. In this situation fy c q : cqc; c ¼ 0; 1; yg form a comlete orthonormal basis over L ðs Þ (square integrable functions on S ) with resect to the sherical uniform measure and are otherwise known as the sherical harmonics. We can similarly think of the latter as the vector entries to the c þ 1 vector Y c ðoþ ¼ðY c q ðoþþ; cx0: Let f AL ðs Þ: We define the sherical Fourier transform on S by q c ZS ¼ f ðoþyq c ðoþ do; where do is the sherical uniform measure on S and overbar denotes comlex conjugation. Again we can think of the latter as the vector entries of the ðc þ 1Þ vector c ¼ðˆ fq c Þ; cqc; c ¼ 0; 1; y : The sherical inversion can be obtained by for oas : f ðoþ ¼ X0 q¼ c q c Y q c ðoþ;

4 P.T. Kim et al. / Journal of Multivariate Analysis 90 (004) Let f AL ðsoð3þþ and hal ðs Þ: Define the convolution Z f hðoþ ¼ f ðuþhðu 1 oþ du; SOð3Þ ð:1þ for oas : We have the following convolution roerty for f AL ðsoð3þþ and hal ðs Þ ð f d hþ c ¼ f ˆc ĥ c ; ð:þ for all c ¼ 0; 1; y; see Lemma.1 in [4]. 3. Shar minimaxity Consider the sherical deconvolution roblem as secified in (1.1) and (1.). Through (.1) and (.), the Fourier transforms f ˆ Z c; f ˆ e c and f ˆ ; satisfy ¼ f ˆc e f ˆc 1 Z ; where for ease of notation, we write f ˆc ¼ðˆ f c e 1 e Þ 1 ; cx0: The above is describing the non-euclidean analogue of observations Z made u of the true measurement X; corruted by noise e: Our interest is in the recovery of the unknown f X : It is assumed that f e is known and that f ˆc e exists for a range of c s. 1 Since f X is unknown, f Z is also unknown, hence f ˆ Z c is unknown, however, we assume that a random samle Z 1 ; y; Z n is available. Define n;c X;q ¼ 1 X n n j¼1 s¼ c c e 1 ;qs Y s c ðz jþ; which is an unbiased estimator of f ˆ X;q c ; where jqjc; c ¼ 0; 1; y: This leads to the following nonarametricdeconvolution density estimator of f X on S : fx n ðoþ ¼Xm c¼0 q¼ c ˆ f n; c X;q Y c q ðoþ; ð3:1þ where oas and m ¼ mðnþ-n; as n-n: Estimation will take lace over the Sobolev class of functions H s ðs Þ with Sobolev norm, jjhjj H ¼jĥ0 s 0 j þ X fcðc þ 1Þg s jĥc q j ; ð3:þ cx1 q¼ c where s41: The Sobolev class of functions is characterized by the comletion of C N ðs Þ; the infinitely continuously differentiable functions on S ; with resect to the Sobolev norm (3.), where s41 refers to the order of differentiability. Further details are given in ½4; 11Š:

5 388 P.T. Kim et al. / Journal of Multivariate Analysis 90 (004) For some fixed constant Q40; let H s ðs ; QÞ ¼fhAH s ðs Þ : jjhjj H s 1 þ Q g: Let g n be any estimator of a robability density function f AH s ðs ; QÞ: Consider the maximal risk Rðg n ; H s ðs ; QÞÞ ¼ su Ejjg n f jj : f AH s ðs ;QÞ Define the minimax risk by R min ðn; H s ðs ; QÞÞ ¼ inf Rðg n ; H s ðs ; QÞÞ; g n where the infimum is taken over all estimators g n of f : In deconvolution density estimation the degree to which we can recover the density f X is best characterized in terms of the quality of smoothness of f e : Indeed, following Fan [] for the Euclidean case, we will aroriately define the smoothness of f e sectrally. We will say that the distribution of e is suer-smooth if the rotational Fourier transform of f e satisfies jj f ˆc e 1jj o d 1 0 c b 0 exðc b =gþ and jj f ˆ e c jj o d 1c b 1 exð c b =gþ as c-n; where jj jj o denotes the usual oerator norm, d 0 ; d 1 ; b; g40 and b 0 ; b 1 AR: We have the following shar minimax result which imroves uon the rate minimax result obtained in [5]. Theorem 3.1. Suose f e is suer-smooth. If f X AH s ðs ; QÞ for some s41 and j m ¼ g ln n g k 1=b; lnðln nþz ð3:3þ where Z4ðs b 0 þ 1Þ=b; then Rð fx n ; H sðs ; QÞÞ ¼ R min ðn; H s ðs ; QÞÞð1 þ oð1þþ ¼ Q g s=bð1 ln n þ oð1þþ as n-n We remark that in Theorem 3.1, estimator (3.1) with smoothing arameter m chosen according to (3.3) is shar. A similar result for the Euclidean case was obtained by Efromovich [1], although in the latter, the deendence on g is hidden in the fact that he was considering the case where g ¼ : For an examle of a suer-smooth rotational error distribution, consider the Gaussian distribution which can be defined on general Riemannian manifolds by

6 P.T. Kim et al. / Journal of Multivariate Analysis 90 (004) solving the aroriate heat equation. It is known that D c q 1 ;q are the eigenfunctions of the Lalace Beltrami oerator on SOð3Þ with eigenvalue cðc þ 1Þ=; for jq 1 j; jq jc; c ¼ 0; 1; y: Thus we can write the distribution as f e ¼ X0 q¼ c exð tcðc þ 1Þ=Þðc þ 1ÞD c qq ; for t40; where t reresents the diffusion time and so as in the Euclidean case, can be regarded as the variance. Consequently, e;qj c ¼ exð tcðc þ 1Þ=Þd qj; so that it is an examle of a suer-smooth distribution with d 0 ¼ d 1 ¼ 1; b 0 ¼ b 1 ¼ 0; g ¼ =t and b ¼ : Corollary 3.1. Suose f e is Gaussian. If f X AH s ðs ; QÞ for some s41 and m ¼ 1 t ln n 1 1= lnðln nþz ; t where Z4s þ 1; then Rð fx n ; H sðs ; QÞÞ ¼ Q 1 s t ln n ð1 þ oð1þþ as n-n: 4. Proof We will rove Theorem 3.1 by first finding an uer bound for the suer-smooth case. Following this we will establish the lower bound which will reveal that the uer and lower bounds match so that the resulting bound is minimax. The aroach of Healy et al. [4] and Kim and Koo [5] will be used for calculating the uer bound, while the aroach of Korostelev and Tsybakov [6] and Goldenshluger [3] will be used to find the lower bound Uer bound for suer-smooth case By arguing as in [5], we first note that there exists some constant so that jhjcðs ; QÞoN for hah s ðs ; QÞ; s41: Thus Ejj fx n f X jj CðS ; QÞ n 1 ex mb m b0þ þ Q m s : 4d 0 g ð4:1þ

7 390 P.T. Kim et al. / Journal of Multivariate Analysis 90 (004) Now choose m according to (3.3). We note that n 1 ex mb ¼ exf lnðln nþ Z g¼ðln nþ Z g ð4:þ and m b0þ ¼ g ln n g ðb0 1Þ=b lnðln nþz : ð4:3þ Substituting (4.) and (4.3) into (4.1), we have that Ejj fx n f X jj Q g s=bð1 ln n þ oð1þþ ð4:4þ as n-n: This comletes the uer bound calculation. 4.. Lower bound for suer-smooth case To show that the uer bound is minimax, we calculate the lower bound and show that this is the same as the uer bound. First, consider f j ðoþ ¼ð4Þ 1= þ jqm s Y m 1 0 ðoþ; where jaf 1; þ1g and oas : We note that jj f j jj H s 1 þ Q m s fðm 1Þmg s 1 þ Q ; hence f j AH s ðs ; QÞ; s41; for jaf 1; þ1g: We also have, s n jjf 1 f 1 jj ¼ 4Q m s : ð4:5þ Now define g j ðoþ ¼f e f j ðoþ; where jaf 1; þ1g and oas : Then the chi-squared distance can be calculated w ðg 1 ; g þ1 Þ¼ 1 Z jg 1 ðoþ g 1ðoÞj S g 1 ðoþþg 1 ðoþ do ffiffiffiffiffi 4 Q d1 m b m sþb 1 ex : ð4:6þ g Let P n g j denote the joint density corresonding to a random samle Z 1 ; y; Z n from g j ; for jaf 1; þ1g: We have the following calculation for any estimator g n of f X : su E fx jjg n f X jjx su E fx jjg n f X jj f X AH s ðs ;QÞ f X Aff 1 ; f 1 g

8 X s n 4 su f X Aff 1 ; f 1 g P fx ARTICLE IN PRESS P.T. Kim et al. / Journal of Multivariate Analysis 90 (004) n jjg n f X jjx s o n X s n 4 exf H ðp n g 1 ; P n g 1 Þg X Q m s exf H ðp n g 1 ; P n g 1 Þg; ð4:7þ where Hð; Þ is the Hellinger distance. The third inequality follows from Proosition.3.8 of Korostelev and Tsybakov [6]. The fourth inequality follows from substituting in (4.5). Now by choosing j m 1 ¼ g ln n þ g k 1=b; lnðln nþx where x4ðb 1 sþ=b we have that, H ðp n g 1 ; P n g 1 Þ nh ðg 1 ; g 1 Þ nw ðg 1 ; g 1 Þ ffiffiffiffiffi ( ) 4 Q d1 ðm 1Þ ðs b 1Þ ðm 1Þb n ex : g The second inequality comes from LeCam [7] while the third inequality uses (4.6). Thus exf H ðp n g 1 ; P n g 1 Þg ¼ 1 oð1þ as n-n: Observe that m s ¼ðm 1Þ s 1 1 s m ¼ g s=bð1 ln n oð1þþ: Consequently, uon substitution into (4.7), we have inf g n su f X AH s ðs ;QÞ E fx jjg n f X jjxq g s=bð1 ln n oð1þþ ð4:8þ as n-n: Finally, because the leading terms in (4.4) and (4.8) are the same, the roof of Theorem 3.1 is now comlete. Acknowledgments Parts of this research was comleted while P.T.K. was visiting the Deartment of Statistics, Inha University, and while J.-Y.K. was visiting the Deartment of Mathematics and Statistics, University of Guelh. They would like to take this oortunity to thank the two institutions for their hositality during these visits.

9 39 P.T. Kim et al. / Journal of Multivariate Analysis 90 (004) References [1] S. Efromovich, Density estimation for the case of suersmooth measurement error, J. Amer. Statist. Assoc. 9 (1997) [] J. Fan, On the otimal rates of convergence for nonarametric deconvolution roblems, Ann. Statist. 19 (1991) [3] A. Goldenshluger, Density deconvolution in the circular structural model, J. Multivariate Anal. 81 (00) [4] D.M. Healy, H. Hendriks, P.T. Kim, Sherical deconvolution, J. Multivariate Anal. 67 (1998) 1. [5] P.T. Kim, J.Y. Koo, Otimal sherical deconvolution, J. Multivariate Anal. 80 (00) 1 4. [6] A.P. Korostelev, A.B. Tsybakov, Minimax Theory of Image Reconstruction, Sringer, New York, [7] L. LeCam, Asymtotic Methods in Statistical Decision Theory, Sringer, New York, [8] A.C.M. van Rooij, F.H. Ruymgaart, Regularized deconvolution on the circle and the shere, in: G. Roussas (Ed.), Nonarametric Functional Estimation and Related Toics, Kluwer Academic Publishers, Amsterdam, 1991,