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The proposed estimator is shown to be consistent with respect to the mean integrated squared error under some conditions of the parameters. After that we derive a convergence rate of the estimator under some additional regular assumptions for the density f .
76 SCIENCE AND TECHNOLOGY DEVELOPMENT JOURNAL NATURAL SCIENCES, VOL 2, NO 1, 2018 Estimation of a fold convolution in additive noise model with compactly supported noise density Cao Xuan Phuong Abstract – Consider the model Y X Z , where Y is an observable random variable, X is an unobservable random variable with unknown density f , and Z is a random noise independent of X The density g of Z is known exactly and assumed to be compactly supported We are interested in estimating the m - fold convolution f m f f on the basis of independent and identically distributed (i.i.d.) observations Y1 , , Yn drawn from the distribution of Y Based on the observations as well as the ridge-parameter regularization method, we propose an estimator for the function f m depending on two regularization parameters in which a parameter is given and a parameter must be chosen The proposed estimator is shown to be consistent with respect to the mean integrated squared error under some conditions of the parameters After that we derive a convergence rate of the estimator under some additional regular assumptions for the density f Index Terms – estimator, compactly supported noise density, convergence rate INTRODUCTION I n this paper, we consider the additive noise model Y X Z (1) where Y is an observable random variable, X is an unobservable random variable with unknown density f , and Z is an unobservable random noise with known density g The density g is called noise density We also suppose that X and Z are independent Estimating f on basis of i.i.d Received 06-05-2017; Accepted 15-05-2017; Published 108-2018 Author: Cao Xuan Phuong- Ton Duc Thang University (xphuongcao@gmail.com) observations of Y has been known as the density deconvolution problem in statistics This problem has received much attention during two last decades Various estimation techniques for f can be found in Carroll-Hall [1], Stefanski-Carroll [2], Fan [3], Neumann [4], Pensky-Vidakovic [5], Hall-Meister [6], Butucea-Tsybakov [7], Johannes [8], among others This problem has concerned with many real-life problems in econometrics, biometrics, signal reconstruction, etc For example, when an input signal passes through a filter, output signal is usually disturbed by an additional noise, in which the output signal is observable, but the input signal is not Let Y1 , , Yn be n i.i.d observations of Y In the present paper, instead of estimating f , we focus on the problem of estimating the m -fold convolution fm f f , m , (2) m times based on the observations In the free-error case, i.e Z , there are many papers related to this problem, such as Frees [9], Saavedra-Cao [10], Ahmad-Fan [11], Ahmad-Mugdadi [12], Chesneau et al [13], Chesneau-Navarro [14], and references therein For m , the problem of estimating f m reduces to the density deconvolution problem To the best of our knowledge, for m , m , so far this problem has been only studied by Chesneau et al [15] In that paper, the authors constructed a kernel type of estimator for f m under the assumption that g ft is nonvanishing on , where the function g ft t f x eitx dt is the Fourier transform of g The latter assumption is fulfilled with many usual densities, such as normal, Cauchy, Laplace, gamma, chi-square densities However, there are also several cases of g that cannot be applied to this paper For TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 21, SỐ T1-2018 CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018 instance, the case in which g is a uniform density or a compactly supported density in general In the present paper, as a continuation of the paper of Chesneau et al [15], we consider the case of compactly supported noise density g In fact, the problem was studied by Trong-Phuong [16] in the case of m ; however, the problem has more challenge with m , m The rest of our paper consists of three sections In Section 2, we establish our estimator In Section 3, we state main results of our paper Finally, some conclusions are presented in Section For convenience, we introduce some notations For two sequences un and of positive real 77 replaced by r t g ft (t ) / max g ft (t ) ; t f mft (t ) in the quantity , called the (t ) r t hft (t ) m form Nevertheless, the function (t ) depends on the Fourier transform hft (t ) , which is an unknown quantity, and so, we cannot use (t ) to estimate f mft (t ) Fortunately, from the i.i.d observations , Yn , we can estimate hft (t ) by the empirical Y1 , un / characteristics k- a ridge function Here a 1/ m is a given parameter, and is a regularization parameter that will be chosen according to n later so that as n We then obtain an estimator for numbers, we write un O if the sequence is bounded The number of the hˆft (t ) n1 function n j 1 e itY j combinations from a set of p elements is denoted Hence, another estimator for f mft (t ) is proposed by by C pk The number A is the Lebesgue (t ) r t hˆft (t ) Finally, using the Fourier measure of a set A For a function p Lp , p , the symbol represents the usual Lp function : Z x : x 0 supp -norm , of For a we define \ Z , the closure in L1 ft x 2 x ft L2 ft , inversion formula, we derive an estimator for f m in the final form fˆm , x : 2 of the set and for x , 2 Note 2 , which is called the Parseval identity L e itx (t )dt that the m dt a 1/ m condition (3) implies almost surely Thus, the estimator fˆm, x is well-defined for all values of x , and moreover, fˆm , belongs to L2 moreover, g ft t hˆ ft t itx e a ft max g t ; t and \ Z Regarding the Fourier transform, we recall that m L METHODS RESULTS We now describe the method for constructing an estimator for f m First, from the equation (2) we In this section, we consider consistency and convergence rate of the estimator fˆm , given in (3) f mft (t ) [ f ft (t )]m Also, from the independence of X and Z , we obtain h f g , where h is density of Y The latter equation gives hft t f ft t g ft t , so f mft (t ) [hft (t ) / g ft (t )]m under if g ft (t ) Then applying the Fourier inversion following proposition Proposition Let fˆ have formula, we can obtain an estimator for f m However, it is very dangerous to use [hft (t ) / g ft (t )]m as an estimator for f mft (t ) in case g ft can vanish on In this case, to avoid division by numbers very close to zero, 1/ g ft (t ) is the mean integrated squared error MISE fˆm, , f m fˆm, f m First, a general bound for MISE fˆm, , f m m , is given in the , m , be as in (3) with a 1/ m and Suppose that f L2 Then we have 78 SCIENCE AND TECHNOLOGY DEVELOPMENT JOURNAL NATURAL SCIENCES, VOL 2, NO 1, 2018 MISE fˆm, , f m C 1 Cmk k k 1 n k m m ft 4m2k f ft t max g ft t ; t where Ck 72k 2k a m f ft t 2m dt I1 2 m k a 2k / (2k 1) , k 1, k m ft f t dt (5) dt , 2m m ft ft g t f t a ft max g t ; t 2m max g t ; t g ft t g ft t max g t ; t ,m ft a f ft t m 2m dt , Proof Since f is a density and is in L2 fm L deduce , L f so ft m , we L g ft t 2m Using the Parseval identity, the Fubini theorem and the binomial theorem, we obtain MISE fˆm, , f m 2 2 fˆmft, t f mft t dt m m ft f t dt 2 g ft t a ft max g t ; t C hˆ t h t h t ft ft k ft mk k 0 f ft t dt m Using the inequality z1 z2 z1 z2 z1 , z2 2 MISE fˆm, , f m I1 I , with (4) where I1 I2 g ft t a ft max g t ; t g ft t a ft max g t ; t m ft m ft h t f t dt , m C hˆ t h t h t ft ft k ft m a 2m Cmk m C C k m k 1 g ft t m 4m2k f ft t U j n1 e hˆft t hft t k hft t 2 m k dt k m k 1 max g ft t ; t k 1 itY j 2 m k a 2m n itY e j 1 n j e , j itY j j 1, itY j U e 2k dt ,n satisfies the j 1, , n conditions of Lemma A.1 in Chesneau et al [15], and moreover, U j / n Hence, applying mk 2k n Uj j 1 k 2k 2k 36k 2k k 2k 36k 2k m k k n Uj j 1 k k 1 k 2k 4 72k k : Ck k n n 2k n C I 1 C kk n k 1 dt k 1 Since hft t f ft t g ft t and g ft t g ft t , in which g ft t denotes the conjugate of g ft t , we have max g ft t ; t m m k m 2m 2m Thus, m Lemma A.1 in Chesneau et al [15] with p 2k , we get yields g ft t a k mk m k ˆft ft ft Cm h t h t h t dt k 1 Clearly, the sequence m m Define k m 2m max g ft t ; t g ft t m hˆft t hft t hft t f ft t dt a ft max g t ; t 2m 1 m 2 I2 ft ft g t hˆ t a ft max g t ; t g ft t k m g ft t 4m2k f ft t max g t ; t ft 2 m k a 2m dt (6) From (4) – (6), we obtain the desired conclusion Proposition Let the assumptions of Proposition hold Then there exists a k0 depending only on g such that TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 21, SỐ T1-2018 CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018 m C C kk n k 1 4m2k g ft t k m f ft t max g ft t ; t 22 m 1 k0 t k0 t ma a Proof Since Z g ft and the Lebesgue 2 m k 2m 79 dt dominated convergence theorem, we get m dt Cmk Ck m k 1 n g t ft max g ft t ; t g ft , there is a constant k0 depending only on g such that g ft t 1/ for all t k0 Then for k 1, g ft t max g ft t ; t max g ft t g t ; t g t ft 22 m 1 k0 Hence, m 2m a dt t k0 C Cmk kk n k 1 f t 2m t ma 22 m 1 k0 22 m 1 k0 t k0 dt 2m hft t 2 m k dt max g ft t ; t a m hft t 2 m k dt g t max g t ; t ft a m f ft t 2m dt f ft t max g t ; t ft t k0 In the rest of this section, we study rates of convergence of MISE fˆm, , f m To this, we 2 m k a 2m F , L density on and a 1/ m and is a positive parameter depending on n such that and n m as n Then MISE fˆm, , f m as n with 1/ , L The class F , L contains Z g ft Let fˆm , be as in (3), where ft t 1 t dt L, given in the following theorem u The mean consistency of the estimator fˆm , is f L2 : sup ft u 1 u L Theorem Suppose that need prior information for f and g Concerning the density f , we assume that it belongs to the class dt m kC m Cm kk dt ma k 1 n t m k dt Cm Ck m ma k 1 t n t k0 so Z g ft 4m2k Theorem is satisfied for normal, gamma, Cauchy, Laplace, uniform, triangular densities, among others In particular, if the noise density g is a compactly supported, the Fourier transform g ft can be extended to an analytic function on This implies the set Z g ft is at most countable, The proof of the proposition is completed \Z g ft g ft t 2m ft Remark The condition Z g ft in dt m dt Combining this with Proposition 1, Proposition and the assumptions of the present theorem, we obtain the conclusion 2m ft t k0 a 2 m k 2m as n , m we have 4m2k ft a f ft t m Proof Since the function g ft is continuous on and 2m many important densities, for example, normal and Cauchy densities Note that F , L L1 L2 In fact, for positive integer , if a density is in L2 l weak derivatives , l 1, derivatives are also in L having , , and the weak , then belongs to F , L for L large enough Regarding the noise density g , we consider the following classes of g : 80 SCIENCE AND TECHNOLOGY DEVELOPMENT JOURNAL NATURAL SCIENCES, VOL 2, NO 1, 2018 F M density on : L2 , Theorem Let 1/ , L Assume that g F M with M Let fˆ be as in (3) for supp M , M , m , F c1 , c2 , d , : c1e d t ft t c2 e d t , t density on a known a 1/ m and n with 1/ m Then we have m ˆ sup f F , L MISE f m, , f m O ln n , in which M , c1 , c2 , d , are positive constants Proof The class F M includes compactly supported densities on M , M The class F c1 , c2 , d , contains densities in which Fourier transforms converge to zero with exponential rate of order Normal and Cauchy densities are typical examples of F c1 , c2 , d , In fact, using the Fourier inversion formula and the Lebesgue dominated convergence theorem, one can show that each element of F c1 , c2 , d , is an infinitely 30 2m 1 Me4 2eMR 1 ln R ln 15e ln for small enough we have 30 1 Me4 In addition, BR, t 1/ ln 1 1/ ln n 1/ R 2eM ln n 1 1/ 2 J : g t ft g ft t t t R , g ft t a f ft t 2m f ft t t R , g ft t t t R a m f ft t t R , g ft t t 2m dt dt 2m f ft t a where t 2m max g ft t ; t BR , dt 2m f ft t f ft t 2m dt R 2 m t R f ft t note ft 2m dt dt we : t R, g a t t a 2m dt , that Moreover, since f F , L , we derive t R R0 that Then take for R a , we have Lemma Suppose g F M Given For 1 We , and BR, R 2 m for n large enough Now, convergence rate established in Chesneau et al [15] f F , L there exists an R depending on n such that et al [15] The reason for considering this class in the present paper is that we want to demonstrate that the estimator fˆm , can also be attained the small enough, we choose an depending on such Suppose with 2m , 1/ m and n / Then applying Lemma gives that smooth” densities In fact, the case of g F c1 , c2 , d , has been studied in Chesneau stating main result of our paper, we need the following auxiliary lemma This auxiliary lemma is not a new result It is quite similar to Theorem in Trong-Phuong [16] differentiable function on Hence, F c1 , c2 , d , is often called the class of “super- Now, we consider the case g F M Before | f ft (t ) |2 m dt t R | f ft (t ) |2 (1 t ) [| f ft (t ) |2 (1 t ) ]m 1 (1 t ) m dt Lm R 2 m Hence, J Lm R 2 m R 2eM ln 1 1 we have BR , R , where Lm 30 2m 1 Me O ln n m m ln n m (7) : t R, g ft t Main result of our paper is the following theorem Combining (7) with Proposition and Proposition 2, we obtain MISE( fˆm, , f m ) O (ln n) m (n m )1 Now, we need to choose according to n so that R a , and rate of convergence of (n m )1 is faster than that of (ln n) m A possible choice is TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 21, SỐ T1-2018 CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018 n Then the conclusion of the theorem is followed Remark The parameter in Theorem does not depend on , the prior degree of smoothness of f Therefore, the estimator fˆ x can be m , computed with out any knowledge concerning the degree of smoothness Finally, we consider the case g F c1 , c2 , d , We have Theorem Let 1/ , L Suppose that g F c1 , c2 , d , , where c1 , c2 , d , are the given positive constants Let fˆm , be as in (3) for a a 1/ m known n 18 m / 16m2 1 am / 4m ln n and Then we have 2 m / sup f F , L MISE fˆm, , f m O ln n Proof Suppose f F , L Let T be a positive number that will be selected later Using the inequality max g ft t ; t a m g ft t 2m m t am m C Q : C kk n k 1 Cm nm m 1 f F , L , we have k 1 J : g Q2 2m t 2m t c1 4 m t T ft 2 am t t T f ft t 4m f ft t 2m t a am 1 t a 2m dt e md t f t ft t t T 2m m 1 C k m Ck k n c2 dt dt ft m 1 C k m f t T a 2m dt dt 2m 2 m k f ft t t ; a t 2m dt : Q1 Q2 4m2k 2m k m t T t T dt 4m2k t T 2m e kd t ft f 2 m k am t f e m k d t t ft t f ft 2 m k t t dt dt 2 m k am dt Ck 2 k L c1 e kdT nk m 1 C 2 m k t 2k ft g t ft g ft t L c2 2m Ck 2 k c1 nk dt f ft t t 4m2k max g k 1 Ck L max k 1 4m2k 2m c 2 k e m k dT T 2 am ; c2 4m2k k e kdT k m e m k dT T 2 am n n f ft t dt f ft t 1 t Cmk k 1 f 2m O m e mdT T am T 2 m Also, 1 m am max g ft t ; t g m 1 2m t ; g ft t a 2m k 1 ft ft 2m g ft t Cm Q1 m dt dt am t T n t T g ft t m 2m t T O m e mdT m m T 1 am e 2 mdT , n n 2m max g ft t ; t t T t 2 m k For the quantities Q1 and Q2 , we have the estimates ft Ck nk Cmk f ft t max g ft t ; t g ft t max g 4m2k g ft t k m for and the assumptions g F c1 , c2 , d , , all t 81 m 1 1 t m dt Combining Proposition with the estimates of J , Q1 and Q2 , we get MISE fˆm , , f m O m e mdT T am T 2 m T mdT e m m T 1 am e 2 mdT m n n C C L max c m 1 k k 1 k m 2 k ; c2 4m2k k e kdT k m e m k dT T 2 am n n 82 SCIENCE AND TECHNOLOGY DEVELOPMENT JOURNAL NATURAL SCIENCES, VOL 2, NO 1, 2018 Choosing T ln n / 8md 1/ MISE fˆ m , O n 1/ , fm ln n n m 1/ am / ln n 1/ m 1 C k m Ck L max k 1 m ln n c 2 m / n m 1/ 4 m 2 k ; c2 ln n 1 am / 4m2k n m 1/ am / ln n 1/ m 1 C k m Ck L max k 1 m ln n c 2 k ; c2 O n ln n n 3/ 1/ m am / ln n 2m 1 am / 1 am / 4m2k Choosing n ln n 2 am / ln n 1 O 11/ m 3/ 1/ m m n n 1/ ln n 2 m / n m 1/ 4 m REFERENCES ln n 2 am / 1 am / m ln n [2] L.A Stefanski, R.J Carroll, “Deconvoluting kernel density estimators”, Statistics, 21, pp 169–184, 1990 [3] J Fan, “On the optimal rates of convergence for nonparametric deconvolution problems”, The Annals of Statistics, 19, pp 1257–1272, 1991 [4] M.H Neumann, “On the effect of estimating the error density in nonparametric deconvolution”, Journal of Nonparametric Statistics, 7, pp 307–330, 1997 [5] M Pensky, B Vidakovic, “Adaptive wavelet estimator for nonparametric density deconvolution”, The Annals of Statistics, 27, pp 2033–2053, 1999 [6] P Hall, A Meister, “A ridge-parameter approach to deconvolution”, The Annals of Statistics, 35, pp 1535– 1558, 2007 [7] C Butucea, A.B Tsybakov, “Sharp optimality in density deconvolution with dominating bias”, Theory Probability and Applications, 51, pp 24–39, 2008 [8] J Johannes, “Deconvolution with unknown error distribution”, The Annals of Statistics, 37, pp 2301–2323 2009 [9] E.W Frees, “Estimating densities of functions of observations,” Journal of the American Statistical Association, 89, pp 517–525, 1994 implies the desired conclusion Remark We see that the convergence rate of uniformly over the class MISE fˆm, , f m R.J Carroll, P Hall, “Optimal rates of convergence for deconvolving a density”, Journal of American Statistical Association, vol 83, pp 1184–1186, 1988 2 m / 2m 18 m / 16 m2 [1] 1 O k 11/ m 1/ k 11/ m m n n O n1/ ln n unknown noise density g We leave this problem for our future research yields F , L in Theorem is as same as that of Chesneau et al [15] when g F c1 , c2 , d , In particular, when m , the convergence rate also coincides with the optimal rate of convergence proven in Fan [3] CONCLUSIONS We have considered the problem of nonparametric estimation of the m-fold convolution fm in the additive noise model (1), where the noise density g is known and assumed to be compactly supported An estimator for the function fm has been proposed and proved to be consistent with respect to the mean integrated squared error Under some regular conditions for the density f of X, we derive a convergence rate of the estimator We also have shown that the estimator attains the same rate as the one of Chesneau et al [15] if the density g is supersmooth A possible extension of this work is to study our estimation procedure in the case of [10] A Saavedra, R Cao, “On the estimation of the marginal density of a moving average process”, The Canadian Journal of Statistics, 28, pp.799–815, 2000 [11] I.A Ahmad, Y Fan, “Optimal bandwidth for kernel density estimator of functions of observations”, Statistics & Probability Letters, 51, pp 245–251, 2001 [12] I.A Ahmad, A.R Mugdadi, “Analysis of kernel density estimation of functions of random variables”, Journal of Nonparametric Statistics, vol 15, pp 579–605, 2003 [13] C Chesneau, F Comte, F Navarro, “Fast nonparametric estimation for convolutions of densities”, The Canadian Journal of Statistics, vol 41, pp 617–636, 2013 [14] C Chesneau, F Navarro, “On a plug-in wavelet estimator for convolutions of densities”, Journal of Statistical Theory and Practice, vol 8, pp 653–673, 2014 [15] C Chesneau, F Comte, G Mabon, F Navarro, Estimation of convolution in the model with noise, Journal of Nonparametric Statistics, vol 27, pp 286– 315, 2015 [16] D.D Trong, C.X Phuong, Ridge-parameter regularization to deconvolution problem with unknown error distribution, Vietnam Journal of Mathematics, vol 43, pp 239–256, 2015 TẠP CHÍ PHÁT TRIỂN KH&CN, TẬP 21, SỐ T1-2018 CHUYÊN SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018 83 Ước lượng tự tích chập mơ hình cộng nhiễu với hàm mật độ nhiễu có giá compact Cao Xuân Phương Trường Đại học Tôn Đức Thắng Tác giả liên hệ: xphuongcao@gmail.com Ngày nhận thảo: 06-05-2017, ngày chấp nhận đăng: 15-05-2017, ngày đăng: 10-08-2018 Tóm tắt – Bài báo đề cập mơ hình Y X Z , Y biến ngẫu nhiên quan trắc được, X biến ngẫu nhiên không quan trắc với hàm mật độ f chưa biết, Z nhiễu ngẫu nhiên độc lập với X Hàm mật độ g Z giả thiết biết xác có giá compact Bài báo nghiên cứu vấn đề ước lượng phi tham số cho tự f f f m tích chập m ( lần) sở mẫu Y, ,Y n quan trắc độc lập, phân phối lấy từ phân phối Y Dựa quan trắc phương pháp chỉnh hóa tham số f chóp, ước lượng cho m phụ thuộc vào hai tham số chỉnh hóa đề xuất, tham số cho trước tham số lại chọn sau Ước lượng chứng tỏ vững tương ứng với trung bình sai số tích phân bình phương số điều kiện cho tham số chỉnh hóa Sau đó, nghiên cứu tốc độ hội tụ ước lượng số giả thiết quy bổ sung cho hàm mật độ f Từ khóa – Ước lượng, hàm mật độ nhiễu có giá compact, tốc độ hội tụ ... SAN KHOA HỌC TỰ NHIÊN, TẬP 2, SỐ 1, 2018 instance, the case in which g is a uniform density or a compactly supported density in general In the present paper, as a continuation of the paper of. .. marginal density of a moving average process”, The Canadian Journal of Statistics, 28, pp.799–815, 2000 [11] I .A Ahmad, Y Fan, “Optimal bandwidth for kernel density estimator of functions of observations”,... observations”, Statistics & Probability Letters, 51, pp 245–251, 2001 [12] I .A Ahmad, A. R Mugdadi, “Analysis of kernel density estimation of functions of random variables”, Journal of Nonparametric Statistics,