TIKHONOV S REGULARIZATION TO DECONVOLUTION PROBLEM

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1 TIKHONOV S REGULARIZATION TO DECONVOLUTION PROBLEM Dang Duc Trong a, Cao Xuan Phuong b, Truong Trung Tuyen c, and Dinh Ngoc Thanh a a Faculty of Mathematics and Computer Science, Ho Chi Minh City National University, 7 Nguyen Van Cu, District 5, Ho Chi Minh City, Viet Nam. b Faculty of Mathematics and Statistics, Ton Duc Thang University, Nguyen Huu Tho, District 7, Ho Chi Minh City, Viet Nam. c Department of Mathematics, Indiana University, Bloomington, IN 47405, USA. Abstract We are interested in estimating the pdf f of i.i.d. random variables X,..., X n from the model Y j = X j +Z j, where Z j are unobserved error random variables, distributed with the density function g and independent of X j. This problem is known as the deconvolution problem in nonparametric statistics. The most popular method of solving the problem is the kernel one in which, we assume g ft t 0, for all t R, where g ft t is the Fourier transform of g. The more general case in which g ft t may have real zeros has not been considered much. In this paper, we shall consider this case. By estimating Lebesgue measure of the low level sets of g ft and combining with the Tikhonov s regularization method, we give an approximation f n to the density function f and evaluate the rate of convergence of E f n f L R. A lower bound for this uantity is also provided. f F,K g G s0,γ,m,t AMS 000 subject classifications: 6G07, 6F05. Key words: Deconvolution, Tikhonov s regularization, Cartan s theorem, Fourier transform. Introduction In this paper, we are interested in estimating the density function f of the random variables i.i.d X, X,..., X n based on the direct random variables Y, Y,..., Y n from model Y j = X j + Z j, j =,,..., n. Here Z j are unobserved error random variables, distributed with the density function g and independent of X j. We know that if h is the probability density function of Y j, then we have the relation h = f g where the symbol denotes the convolution of two functions f and g, f gx = f x tg tdt, x R. This paper is ported by National Foundation of Scientific and Technology Development NAFOSTED.

2 We denote the Fourier transform of the function f by Put f ft t = NZ g = f xe itx dx, t R. 3 { } t R : g ft t 0. Informally, if h is known, we can apply the Fourier transform to both sides of to get f ft = hft g ft for all t NZ g. 4 Then using the inverse Fourier transform, we can find f. This is a classical problem in Analysis. In practical situations, we do not have the density function h. We only have the observations Y j, j =,..., n. The problem of recovering f from observations Y j is called the deconvolution problem in statistics or deconvolution problem for short. Euation is an integral euation and solving is a typically ill-posed problem. A specific deconvolution problem is the one of consistency. To prove a deconvolution problem is consistent, we have to show that there exists a seuence of estimators {f n } such that lim E f n ; Y,..., Y n f n + X = 0 where X is an appropriate Banach space. In fact, the simplest case is NZ g = R. In this case, there are many methods to construct the estimator f n x; Y,..., Y n. Kernel estimation is one of the most popular approach to deal with the deconvolution problem. In this method, one estimates the density function f by the estimator f n x; Y,..., Y n = π e itxkft tb g ft t n n e ityj dt, 5 where K is a kernel function and K ft is compactly ported. This method was first introduced in the papers of Carroll and Hall [9], Stefanski and Carroll [], Fan [], [3]. The estimator 5 has known as the standard deconvolution kernel density. We note that the estimator 5 is defined as g ft t 0 for all t R, and so the condition NZ g = R has become common in deconvolution topics. In fact, the density functions g often satisfy g ft t C + t α { exp C0 t γ} where C, C 0 > 0, α 0, γ 0 and α + γ > 0. Similarly, in case of bivariate random variables, Goldenshluger [] also assumed that min g ft t C exp { C 0 v }, v > 0. t v j=

3 However, there are many important density functions which do not satisfy NZ g = R, e.g., the uniform densities, the self-convolved uniform densities or the convolution of an arbitrary density function with a uniform density. Deconvolution problem in the case NZ g R is very difficult. According to our knowledge, there is only a few articles mentioning this case. The first paper which considered this problem is Devroye [0]. The consistency was established with respect to the L R- norm. Using the techniue of truncation, he constructed a consistent estimator f n for the target density f when the Fourier transform g ft vanishes on a Lebesgue-zero-set, f n x; Y,..., Y n = π Re itxk th n g ft e ityj dt, x < T, t n R\A r e j= = 0, x T, where A r = { t R : g ft t < r }, r > 0, h > 0 and K ft is compactly ported. However, no convergence rate is provided in Devroye [0]. In Meister [3], the density deconvolution is also considered in case the target density f is contained in the class of densities which satisfy S S f xdx = and f ft t + t β dt C with S, C, β > 0 whereas the error density belongs to the class of densities which has g ft t µ for t [ ν, ν] and g C. The rate of uniform convergence of MISEf n, f is O lnn β δ lnlnn β with δ [0,. This rate is only derived [ Olnn when the sample size n is chosen such that S δ ; O lnn δ]. Actually, this O condition is difficult to verify because S is not known exactly and so we cannot choose n exactly in general. In Groeneboom and Jongbloed [4], the authors focused on considering the deconvolution problem in a uniform density model. By choosing a suitable bandwidth, they proved that it is possible to construct a kernel estimator of target density f if f has a finite left endpoint. In Hall and Meister [5], the authors have given an approach to solve the deconvolution problem in case NZ g R. To avoid division by zero, the authors replaced g ft t by the maximum of g ft t and h n t = n ξ t ρ with ξ > 0, ρ > 0. The function h n t as above is called the ridge function. An estimator for the density f is defined by f n x = Re π e itx g ft t g ft t r { max g ft t ; hn t } r+ n n e ityj dt, 6 j= 3

4 with r 0. The optimal rates of estimation are provided. Recently, using a modified kernel method, Delaigle and Meister [5] also gave a similar result. We see that the condition imposed on g ft is very strict. In these papers, the density function g is assumed to satisfy { g t ft c sinkt ν + t α exp d t β}, t R, 7 with k > 0, c > 0, d > 0, α 0, β 0, α + β > 0, ν > 0. In this case, R\NZ g { nπ k : n Z }. In other words, the positions of zeros of g ft are fixed. If g is a uniform density then 7 holds, but if g is an arbitrary density function then the condition 7 often does not satisfy. Motivated by this problem, in the present paper, we shall consider the deconvolution problem in case the Fourier transform of error distribution has zeros on the real line, with no specific constraints on NZ g. Applying Tikhonov s regularization, we introduce an estimation procedure for the target density function. Using properties of entire functions and some results from harmonic analysis, we consider the low level sets of the function g and give the convergence rate of our procedure. The rest of our paper consists of three sections. In Section, we shall present the Tikhonov s regularization; and use it to give a result of consistency and an estimation for probability density functions. In Section 3, we state and prove approximation results and provide a lower bound for error of the estimator. In Section 4, by estimating of Lebesgue measure of the low level sets of the Fourier transform of g, we prove Lemma 3. which is stated in Section 3. Tikhonov s regularization As discussed in Section, we know that Problem is typically ill-posed and a regularization is reuired. In the theory of ill-posed problems, a method of regularization which is often used for the deconvolution problem is the Tikhonov s regularization. In this method, we shall approximate f ft by a function having the form ϕg ft where ϕ is often called the filter function. In fact, we consider the linear operator A : L R L R, A ϕ = ϕg ft for all ϕ L R. For each δ > 0, we consider the Tikhonov s functional J δ ϕ = Aϕ h ft + δ L R ϕ L R, ϕ L R. 8 We shall find the function ϕ minimizing J δ. As known, J δ attains its minimum at a uniue minimum function ϕ δ L R. This minimum ϕ δ is the uniue solution of the euation δϕ δ + A Aϕ δ = A h ft, 9 4

5 where A : L R L R is the adjoint operator of A see Theorem., Section. in [6]. From 9, we get and so we have the approximation δϕ δ + g ft ϕδ = g ft h ft ϕ δ = g ft δ + g ft to the Fourier transform f ft of the density function f. After that, using the inverse Fourier transform gives f δ x = π This can be seen as an estimator for the density function f. e itx gft th ft t δ + g ft t dt, x R. 0 As mentioned, in practical situations we do not have the density function h, we have only the observations Y,..., Y n. Thus, we cannot use directly the formula 0 to make an approximation for f. However, in case we have the i.i.d. observations Y,..., Y n, since E n e ityj = n E e ityj = h ft t, n n j= we can replace h ft t in 0 by the uantity j= Ψt; Y,..., Y n = n n e ityj. It suggests an approximation for the density function f based on Y,..., Y n as follows L δ,g x; Y,..., Y n = e itx g ft t π δ + g ft t n e ityj dt. n We have the following general estimate for error E Lδ,g f L R. j= j= Lemma. Let δ > 0, g L R L R be the density function of error random variables and f L R L R be the solution of Problem. Then E Lδ,g f L R = g ft t πn δ + g ft t f ft tg ft t dt + π f ft t δ δ + g ft t dt. 3 5

6 Proof. From, we get L ft δ,g t = g ft t δ + g ft t n n e ityj, t R. Applying the Parseval s identity, the Fubini s theorem and the euality EL ft δ,g t fft t = VarL ft EL δ,g t + ft δ,g t f ft t, we derive E Lδ,g f L R = π = π E L ft j= δ,g t fft t dt VarL ft δ,g tdt + π EL ft δ,g t fft t dt. As the Y, Y,..., Y n are i.i.d. random variables, we get VarL ft δ,g tdt = g ft t n n δ + g ft t Var e ityj dt j= = g ft t n δ + g ft t Var e ity dt = g ft t n δ + g ft t E E e ity e ity dt and = n = n EL ft δ,g t fft t dt = g ft t δ + g ft t g ft t δ + g ft t = = = h ft t dt f ft tg ft t dt g ft t e δ + g ft t E ity f ft t dt g ft t δ + g ft t hft t f ft t dt g ft t δ + g ft t fft t f ft t dt f ft t δ δ + g ft t dt. Combining the above eualities, we get the conclusion of Lemma. 6

7 The consistency of the problem with respect to the L R-norm was studied in Devroye [0]. Meister [4] also gave a very general consistency result in L R-weighted norm NZ g is assumed to be dense in R. Now, we shall give a consistency result in L R-norm with a simple estimator and an easy proof. Theorem. Let g L R L R be the density function of the error random variables and f L R L R be the solution of Problem. Assume that m R\NZ g = 0 where m is the Lebesgue measure on R. Let δ n be a positive seuence such that δ n 0, nδ n + as n +. Then lim E L δn,g f n + L R = 0. Proof. Applying the result 3 of Lemma., we have E L δn,g f + g ft t = L R πn δ n + g ft t + π πnδ n f ft tg ft t dt f ft t δ n δ n + g ft t dt g ft t dt + + f ft t δ n π δ n + g ft t dt f ft t δ n δ n + g ft t dt. = nδn g L R + π Using the Lebesgue s dominated convergence theorem, we get E Lδn,g f L R 0 as n +. The proof of the theorem is completed. The following theorem will be used in Section to get the main result of our paper. Theorem.3 Let ρ > 0, δ > 0, R > 0, let g L R L R be the density function of error random variables and f L R L R be the solution of Problem. Then E Lδ,g f L R C m B ρ,r + f ft t dt + δ ρ 4 + nδ, 4 where C = B ρ,r = { π max { t R : Proof. From 3, we get the estimate E Lδ,f f L R πnδ t >R ; f ft g ; ft L R } g ft t < ρ, t < R. g ft t dt + π L R }, f ft t δ δ + g ft t dt. 7

8 Now we write + f ft t δ δ + g ft t dt = t <R, g ft t <ρ m B ρ,r + Therefore, dt + t >R t >R, g ft t <ρ + + t <R, g ft t <ρ t >R, g ft t <ρ g ft t >ρ f ft t dt + f ft t dt + δ f ft ρ 4 E Lδ,g f L R C L R. m B ρ,r + f ft t δ δ + g ft t dt f ft t δ δ + g ft t dt f ft t δ δ + g ft t dt δ ρ t >R g ft t >ρ f ft t dt f ft t dt + δ ρ 4 + nδ, where C = π {; max f ft L R ; } g ft. The proof of the theorem is completed. L R 3 Approximation results The difficulty of applying directly the formula 4 arises in two aspects: one, in reality we cannot have the density function h, and two, even we have h, we cannot compute efficiently the inverse Fourier transform of f ft if the function g ft has zeros on the real axis. Hence, a regularization is in order. For each entire function ψ, low level sets of ψ are defined by {z C : ψ z < }, > 0. In case g is compactly ported, the set of zeros g ft t affects heavily the recovering of the function f from its Fourier transform. For actually computing the solution f and for the regularization of euation, we must know more about the Lebesgue measure of the low level sets of g ft. The latter goes back to the well-known theorem of Cartan about the size of the low level sets A = {z C : P z < }, > 0, where Pz is a polynomial. He proved that A is contained in a finite set of disks whose sum of radius is less than C n, where n is the degree of Pz and C is a constant that depends only on the leading coefficient of Pz and n see Theorem 3 of. in [7]. In particular, we have lim m {z C : P z < } = 0 0 8

9 where m is the Lebesgue measure on C. We shall use the Cartan s theorem to give an asymptotic estimate on the low level sets { t R : g ft t <, t < R } of the function g, and shall apply this estimate to the Tikhonov s regularization of Problem. To get an explicit estimate for E Lδ,g f, some prior information to f and g L R must be assumed. From now on, we assume that f is contained in the set F,K = { f L R L R : f 0, where K and >. The condition f ft t f xdx =, f ft t K +t K + t }, 5 imposed on the density f F,K as in 5 is uite natural. It is euivalent to the condition that the density function f is in the Sobolev space W, R. Moreover, let s 0 > 0, γ,, M and T >, we assume that the density function g of the error random variables belongs to the class of functions G s0,γ,m,t = { g L R : g 0, g xdx =, g te s0 t γ dt M, g L R T }. 6 We note that the density function g x = e x of Gauss distribution and compactly π ported density functions are in G s0,γ,m,t. Not all of such functions satisfies NZ g = R. For each > 0, we put { s = inf s > 0 : t s g t dt }. 7 Lemma 3. Let s 0 > 0, λ,, M, T >, β 0,, > and let g G s 0,γ,M,T be the density function of error random variables. For > 0 small enough, choose R to satisfy If > 0 is small enough then [ es R + lnr + ln 5e 3] = ln β +. 8 where B ρ,r is defined in Theorem.3. m B β,r R +, We shall prove Lemma 3. in Section 4 see also [8]. Our main result is Theorem 3. Let s 0 > 0, γ,, α 0,, β 0,, αβ < 4, ν = 4 + αβ and let K, M, T >, >. Choosing = n α, δ = n ν and denoting f n x = L δ,g x; Y,..., Y n, 9

10 we have the estimate g G s0,γ,m,t E f n f L R f F,K C 3 s 0 + γ 30 + e 4 lnn α γ + + lnn α ν α for all n N large enough, where C 3 > 0 depends on, K, T. Proof. Let f F,K and g G s0,γ,m,t. We have f ft t Kdt Kdt dt + t t >R t t >R t >R K R + for all > 0 small enough. Combining this with Theorem.3 and Lemma 3., we get E L δ,g f [ C L R + K R + + δ 4β + ] nδ C R + + δ 4β + nδ for all > 0 small enough, where C = C + K enough, from 5 we have the estimate Therefore R + E L δ,g f L R C s 0 γ 30 + e 4 s 0 γ 30 + e 4 + ln + ln. Moreover, for all > 0 small γ +. γ + + δ 4β + nδ Replacing = n α, δ = n ν to the right hand side of the latter ineuality, we obtain E f n f L R C s 0 + γ 30 + e 4 lnn α γ + + n ν for all n N large enough.. Furthermore, we have n ν = n α ν α lnn α ν α. Hence, E f n f L R C for all n N large enough. s 0 γ 30 + e 4 + lnn α γ + + lnn α ν α 0

11 Because f F,K and g G s0,γ,m,t, we have C = { π max ; f ft } g ; ft L R L R So, for all n N large enough, we have π max { ; π } Kdt + t ; πt. E f n f L R C 3 s 0 γ 30 + e 4 + lnn α γ + + lnn α ν α, 9 where C 3 = + K { } Kdt max ; π π + t ; πt. We note that the right hand side of 9 is independent of f and g. Therefore, g G s0,γ,m,t E f n f L R f F,K C 3 s 0 + γ 30 + e 4 lnn α γ + + lnn α ν α The proof of the theorem is completed. In case the error density function g is compactly ported, we get the following result Theorem 3.3 Let assumptions be as in Theorem 3.. Moreover, assume that the density function g has p g [ L; L] where p g is the port of g. Then g G s0,γ,m,t f F,K E f n f L R C 3 [ 30L + e 4 where constant C 3 > 0 depends on, K, T. + 4 lnn α lnn α ν α ], Proof. For each > 0, from the definition of s in 7, we get t s g t dt. Moreover, from the property of infimum, we have t s η g t dt > for all η > 0, which implies t s g t dt. So g t dt =. t s If s > L then t s g t dt = 0 which is a contradiction. So s L for all > 0 small enough.

12 From, for all > 0 small enough, we have [ ] R 5L + e 4 β ln + ln + β It follows that R + 30L + e 4 30L + e 4 ln + 4 ln Therefore, combining with the proof of Theorem 3., we derive E L δ,g f [ L R C + 4 ] 4 3 ln + 4 δ + 30L + e 4 4β + nδ. Replacing = n α, δ = n ν, we get [ ] E f n f L R C lnn α 30L + e lnn α ν α for all n N large enough. The proof of the theorem is completed. Thus, from Theorem 3. we see that the MISE of our estimator attains the logarithmic rate. The following theorem will show that the logarithmic rates are unavoidable. Theorem 3.4 Let s 0 > 0, γ,, K, M, T >, > and m is an integer greater than. Then, for all δ > 0 small enough, we have m 4 m ln. 8πm δ g G s0,γ,m,t f F,K E L δ,g f L R Proof. We consider the density function g 0 x = π e x. The function g 0 G s0,γ,m,t and has g0 ft t t = e. With the density function ψ x = e x of Laplace distribution also known as double-exponential density; See page 35, Section.4 in [], the function f 0 = ψ ψ... ψ }{{} mtimes is also a density function. This function has f0 ft t = ψ ft t m = +t m. Since f ft 0 t = we have f 0 F,K. We denote H δ = + t m + t m g G s0,γ,m,t K + t, f F,K E Lδ,g f L R.

13 From the euality 3 of Lemma., for all f F,K and g G s0,γ,m,t, we have Therefore, E Lδ,g f L R π f ft t δ δ + g ft t dt. H δ E L δ,g0 f L f F R E L δ,g0 f 0,K f ft 0 π t δ δ + g ft = π π = π 4π 4π e t δ ln δ ln δ ln δ By direct computations, we derive H δ 8πm + ln 0 t dt. δ + t m δ + e t dt δ + t m δ + e t dt δ + t m δ + e t dt + t mdt t + t m+dt. δ m 4 m ln 8πm L R m δ for all δ > 0 small enough. The proof of the theorem is completed. Choosing δ = n ν with ν = 4 + αβ, α 0,, β 0,, αβ < 4 and denoting f n x = L δ,g x; Y,..., Y n, we get as in Theorem 3. Corollary 3.5 Let assumptions be as in Theorem 3. and Theorem 3.4. Then 4 m ν lnn m 8πm g G s0,γ,m,t for all n N large enough. f F,K E f n f L R 4 Proof of Lemma 3. To estimate the Lebesgue measure of low level sets, we shall use the following result see Theorem 4, Section.3 in [7]. 3

14 Lemma 4. Let fz be an analytic function in the disk {z : z er}, f0 =, and let η be an arbitrary small positive number. Then the estimate 5e 3 ln f z > ln. lnm f er η is valid everywhere in the disk {z : z R} except a set of disks D j with sum of radius rj ηr, where M f r = max f z. z =r Using the latter lemma, we now state and prove an estimate for low level sets. Theorem 4. Let the density function g be in G s0,λ,m,t where s 0 > 0, λ,, M, T > and let β 0,, >. For > 0 small enough, we choose s as in 7 and choose R to satisfy [ es R + lnr + ln 5e 3] = ln β +. 0 Then lim R = +. 0 Moreover, if is small enough, we have m D β + R +, where with } D β + = {z R : Φ z < β +, z < R Φ z = s s g te zt dt, z C. Proof. The proof of the theorem is divided into steps. In Step, we give the existence of R and prove that lim R = +. In Step, we shall estimate m D β 0 +. Step. We consider the function ψ R = es R [ + lnr + ln 5e 3] + ln β +, R 0. We have ψ R + as R + and ψr ln β + < 0 as R 0 for > 0 small enough. So there exists an R > 0 such that ψr = 0, i.e, R satisfies 0. Also from 0, we get + er ln 5e 3 R ln s β. + In view of the ineuality lnx x for all x > 0, we have R β ln + ln + β 5e 4. + s 4

15 From the definition of s in 7, we get t s g t dt =. Thus Me s0sγ e s0sγ e s0 t γ g t dt e s0sγ The latter ineuality implies From and, we get R 5e 4 + s s 0 s 0 γ β γ ln M ln ln M γ t s e s0 t γ g t dt. γ. ln + s 0 γ. ln M + β γ. 3 As lim 0 ln ln M γ : M γ ln = +, we obtain ln ln M γ M γ ln, 4 for all > 0 small enough. Furthermore, since ln ln lim 0 ln M = +, lim γ 0 ln M γ + β = 0, we have R for all > 0 small enough. s 0 γ M γ 30 + βe 4 ln s 0 γ 30 + e 4 ln γ 5 From the ineuality 5, we obtain R + as 0. Step. We see that Φ is an entire function, i.e, a complex function analytic on C, because Φ z = g te zt s dt g t e z t dt e z s, z C. 6 s s s Since g is a density function, the function g ft is a non-trivial function of L R. Hence, Φ is a non-trivial entire function, and so there exists an x 0 R such that Φ x 0 = C 4 > 0. Changing variable if necessary, we may assume that Φ 0 = if > 0 small enough. For all z = er, from 6, we get Φ z e esr and that lnm Φ er ln max Φ z es R. z =er 5

16 We choose η = R. Then, for all z R, applying Lemma 4., we have the estimate { } 5e 3 Φ z exp ln lnm Φ er exp = β + R { es R [ + lnr + ln 5e 3]} except a set of disks {D z j, r j } j J whose sum of radius is less than η R = R +. This implies } D β + {z R : Φ z < β +, z < R D z j, r j R 7 j J where we recall From 7 we derive D z j, r j = {z C : z z j < r j }, j J. m D β + m D z j, r j R m D z j, r j R j J j J j J r j R + for all > 0 small enough. This completes the proof of Step and the proof of our theorem. Finally, we turn to the Proof of Lemma 3. For each > 0, we put g t = { g t, t s, 0, t > s. We see that g ft x = Φ ix if x R. For all x R, we have g ft x g ft s x = g te itx dt g te itx dt g t dt =. s t s Thus, if g ft x < β, x < R then g ft x < β +. This implies Φ ix < β +. Applying Theorem 4., we get m B β,r m Dβ + R + for all > 0 small enough. The proof of the lemma is completed. Acknowledgments. We thank the referees for their kind and careful reading of the paper and for helpful comments and suggestions leading to the improvement version of our paper. 6

17 References [] A. Goldenshluger, Density Deconvolution in the Circular Structural Model, Journal of Multivariate Analysis 8 00, [] A. Meister, Deconvolution Problems in Nonparametric Statistics, Springer-Verlag, Berlin Heidelberg, 009. [3] A. Meister, Deconvolving Compactly Supported Densities, Mathematical Methods of Statistics, Vol. 6, No. 007, [4] A. Meister, Non-estimability in spite of identifiability in density deconvolution, Mathematical Methods of Statistics, Vol 4. No , [5] A. Delaigle and A. Meister, Nonparametric function estimation under Fourieroscillating noise, Statistics Sinica 0, [6] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer- Verlag, New York, 996. [7] B. Y. Levin, Lectures on Entire Functions, Trans. Math. Monographs, Vol. 50, AMS, Providence, Rhole Island, 996. [8] D. D. Trong and T. T. Tuyen, Error of Tikhonov s regularization for integral convolution euation, Version, Oct 006. [9] R. Carroll and P. Hall, Optimal Rates of Convergence for Deconvolving a Density, Journal of American Statistical Association, Vol. 83, No , [0] L. Devroye, Consistent Deconvolution in Density Estimation, The Canadian Journal of Statistics, No. 989, [] L. Stefanski and R. Carroll, Deconvoluting Kernel Density Estimators, Statistics 990, [] J. Fan, On the optimal rates of convergence for nonparametric deconvolution problems, The Annals of Statistics, Vol. 9, No. 3 99, [3] J. Fan, Asymptotic normality for deconvolution kernel density estimators, Sankhya 53 99, [4] P. Groeneboom and G. Jongbloed, Density estimation in the uniform deconvolution model, Stat. Neerlandica , [5] P. Hall and A. Meister, A ridge-parameter approach to deconvolution, The Annals of Statistics, Vol. 35, No ,

18 Dang Duc Trong, Faculty of Mathematics and Computer Science, Ho Chi Minh City National University, 7 Nguyen Van Cu, District 5, Ho Chi Minh City, Viet Nam. address: ddtrong@hcmus.edu.vn Cao Xuan Phuong, Faculty of Mathematics and Statistics, Ton Duc Thang University, Nguyen Huu Tho, District 7, Ho Chi Minh City, Viet Nam. address: caoxuanphuong@tdt.edu.vn Truong Trung Tuyen, Department of Mathematics, Indiana University, Bloomington IN 47405, USA. address: truongt@indiana.edu Dinh Ngoc Thanh, Faculty of Mathematics and Computer Science, Ho Chi Minh City National University, 7 Nguyen Van Cu, District 5, Ho Chi Minh City, Viet Nam. address: dnthanh@hcmus.edu.vn 8