Journal of Science and Technology, Vol 52B, 2021 NONPARAMETRIC ESTIMATION OF X Y WITH LAPLACE ERROR DENSITIES DANG DUC TRONG1, TON THAT QUANG NGUYEN1 Faculty of Mathematics and Computer Science, VNUHCM-University of Science Faculty of Fundamental Science, Industrial University of Ho Chi Minh City tonthatquangnguyen@iuh.edu.vn Abstract We survey the nonparametric estimation of the probability : X Y when two random variables X and Y are observed with additional errors Specifically, from the noise versions X1, , X n of X and Y1, , Ym of Y , we introduce an estimator ˆ of and then establish the mean consistency for the suggested estimator when the error random variables have the Laplace distribution Next, using some further assumption about the condition of the densities f X of X and fY of Y , we then derive the convergence rate of the root mean square error for the estimator Keywords Nonparametric, error density, estimator, convergence rate INTRODUCTION Let X1 , , X n be i.i.d random variables from an unknow density function f X of X and Y1 , , Ym be i.i.d random variables from an unknow density function fY of Y We concern the problem of estimating the quantity : X Y (1) from given the two independent samples X j X j j , Yk Yk k , j 1, , n; k 1, , m (2) Here, one observes X j from f X j , j 1, , n and Yk from fYk , k 1, , m The random variables j and k are known as error ones The random variables X j , j , Yk , k are assumed to be mutually independent for j, j n, k , k m In addition, assume that each j has its own known density g , j and each k has its own known density g ,k The densities g , j and g ,k are also called error densities The quantity has many applicabilities in various fields For instance, is equal to the area under ROC curve which is used as a graphical tool for evaluation of the performance of diagnostic tests (see Metz [1], Bamber [3], Hughes et al [11], Kim-Gleser [17], Coffin-Sukhatme [20], Zhou [27]) Besides, the quantity plays an important role in biostatistics (see Pepe [21]) and in engineering (see Kotz et al [24]) Additionally, the quantity is also applied in agriculture (see Dewdney et al [22]) In the context of error free data, i.e., j and k 0, there are many papers researching in both parametric and nonparametric approaches (see Kundu-Gupta [7, 8], DeLong et al [9], Wilcoxon [10], Mann-Whitney [12], Tong [13], Montoya- Rubio [16], Constantine et al [18], Huang et al [19], Kotz et al [24], Woodward-Kelley [26], among others) However, for the problem of estimating the quantity from given contaminated observations as in (2), the problem has not been studied much For a nonparametric framework, there are a few papers related to the problem in Coffin-Sukhatme [20], with contaminated observations, the Wilcoxon-Mann-Whitney estimator was used to survey the bias of the estimator In KimGleser [17], the authors used the SIMEX method, proposed by Cook-Stefanski [15], to construct an estimator of , in which the measurement errors have the standard normal distribution Applying nonparametric deconvolution tools and basing on the contaminated samples, Dattner [14] developed an © 2021 Industrial University of Ho Chi Minh City 104 X Y WITH LAPLACE ERROR DENSITIES NONPARAMETRIC ESTIMATION OF optimal estimator of when error density functions g , j and g ,k are assumed to be supersmooth Herein a density is called supersmooth if its Fourier transform decays with an exponential rate at infinity Next, Trong et al [4] considered the problem in the case where g , j and g ,k are compactly supported ones Following the latter paper, Trong et al [5] considered the problem with heteroscedastic errors This means that j and k have different distributions for j n, k m Recently, Phuong-Thuy [2] concentrated on the case where the distribution of the random errors is unknown but symmetric around zero and can be estimated from some additional samples To the best of our knowledge, so far the problem of estimating the quantity when error densities g , j and g ,k are ordinary smooth has not been considered in any research yet This is a popular standard condition where the error densities have the Fourier transform decaying with polynomial rate at infinity Therefore, in our current work, we fill partially the gap by considering the problem in the setting where error densities g , j and g ,k are the Laplace density, which is a specific case of ordinary smooth density This is also the condition about the problem that has never been considered before Moreover, it is also known that the Laplace distribution plays an important role in many scientific fields It has attracted interesting applications in the modeling of detector relative efficiencies, measurement errors, extreme wind speeds, position errors in navigation, the Earth’s magnetic field, wind shear data and stock return An indepth survey of the Laplace distribution including various properties and applications is provided by Kotz et al [23] For convenience, we introduce some notations The convolution of two functions f and g is denoted by f * g The notation h t eitxh x dx denotes the Fourier transform of a function h( x), i 1 The notations z and z denote the imaginary part and conjugate of z, respectively The number A 1 is a is the Lebesgue measure of a measurable set A For two sequences of positive real numbers an ,m and bn,m , the notation an,m b means a n, m n ,m const bn,m for large n, m The notation positive constant which is independent of n, m MAIN RESULTS We know that, for a continuous distribution function F , one has F x 1 1 eitx f t t dt , x , where f is the density function corresponding to F Let Z X Y Then Z 0 FZ 0 , where FZ is the distribution function of Z In addition, since f Z f X fY , we get that 1 1 1 f Z t dt t FZ f X t fY t t dt (3) From (3), in the present paper, we suggest an estimator of in the form ˆ : n m j ,k , nm j 1 k 1 (4) in which g , j (t )g ,k (t ) it ( X j Yk ) e (5) dt , 0 t ( t s | g , j (t ) |2 )( t s | g ,k (t ) |2 ) where s and the number 0,1 plays a role as a regularization parameter and must be selected j ,k © 2021 Trường Đại học Cơng nghiệp thành phố Hồ Chí Minh NONPARAMETRIC ESTIMATION OF X Y WITH LAPLACE ERROR DENSITIES 105 according to the sample sizes n, m later The estimator ˆ was also considered in Trong et al [5] Now, for error random variables, we assume that j and k , j 1,, n; k 1,, m have the Laplace distribution where the densities of j and k have the form g L x 1/ e x with g L t 1/ 1 t It is well-known that in the additive measurement error model, the class of the ordinary smooth error densities is a popular standard class where the error densities have the Fourier transform decaying with polynomial rate at infinity and the Laplace density is a famous example belonging to this class In order to prove some below results, we need the the following specific quality of the Laplace density, g L t 2t t 1 t 2 2t t 4t , t 1/ Proposition 2.1 Let the observations be given by model (2) Let the quantity be defined as in (1) and the estimator ˆ be as in (4) with (0,1), s Suppose that f X fY L1 ( ) Besides, suppose that g , j and g ,k are the Laplace density, j 1,, n; k 1, , m Then, we have (ˆ ) s 1 n m t f X t fY t C0 dt nm j 1 k 1 1/2 t s g t ,j s 1 n m t f X t fY t dt , nm j 1 k 1 1/2 t s g t ,k where the constant C0 only depends on s Proposition 2.2 Let the observations be given by model (2) Let the quantity be defined as in (1) and the estimator ˆ be given by (4) with (0,1), s Suppose that g , j and g ,k are the Laplace density, j 1,, n; k 1,, m; along with f X fY L1 ( ) Then, we get | ˆ |2 C s 1 s 1 n m t f t f t t f t f t X Y X Y dt dt 2 1/2 1/ 2 s s nm j 1 k 1 t g , j t t g , k t 1 , n m where the constant C1 only depends on s Next, the following theorem represents the mean consistency of the estimator ˆ Theorem 2.3 The assumptions are the same as in Proposition 2.2 Besides, suppose that is a parameter depending on the sample sizes n, m such that 0, n , m as n, m Then, | ˆ |2 as n, m Now, in order to obtain the rate of the convergence of the estimator ˆ , we need the following definition For and C 0, we consider the class © 2021 Industrial University of Ho Chi Minh City 106 X Y WITH LAPLACE ERROR DENSITIES NONPARAMETRIC ESTIMATION OF F ,C , : , are densities on t t 1 t , dt C The class F ,C is quite usual It is used in Trong et al [4, 5] We can see some examples to see its usual quality, if and are in Sobolev class, then the couple , belongs to F ,C Moreover, if is a normal density or the Cauchy density and is any density, then the couple , is in F ,C | ˆ |2 is provided by the following important theorem An upper bound for convergence rate of , C Let the observations be given by model are the Laplace density, j 1,, n; k 1,, m By choosing Theorem 2.4 Given and g ,k (2) Suppose that g , j 1 1/3 d 2 1 6 s /3 1 3 , n m where d with D 2 1 6 2s / 3, we obtain 2 1 D | ˆ |2 sup ( f X , fY )F ,C (1) n 1 /62 1 D m 1/62 1 D CONCLUSIONS We have considered the problem of nonparametric estimation of the probability : X Y when two random variables X and Y are observed with additional errors We use noise versions X 1, , X n of X and Y1, , Ym of Y to introduce an estimator ˆ of and then establish the mean consistency for ˆ when the error random variables have f X , fY F ,C , : , are densities on polynomial convergence rate of sup ( f X , fY )F ,C the Laplace , | ˆ |2 1/ distribution t t 1 t Proof of Proposition 2.1 Using the Fubini theorem, we get g , j t g ,k t f X j t fYk t ( j ,k ) t t s g t t s g t ,j ,k 2 g , j t g , k t f X t f Y t t t s g t t s g t ,j ,k dt dt From (3) and (4), combining the latter equality, we obtain (ˆ ) nm n m j 1 k 1 2 g , j t g ,k t 1 1 2 t t s g , j t t s g ,k t © 2021 Trường Đại học Công nghiệp thành phố Hồ Chí Minh f X t fY for dt C , we derive the PROOFS Finally, t dt X Y WITH LAPLACE ERROR DENSITIES NONPARAMETRIC ESTIMATION OF nm n m g , j , g ,k 1/2 0 j 1 k 1 Since g , j t 2s s t t g , j t g ,k t t t s g t t s g t ,j ,k 4t and g ,k t 2 f X t t dt fY 107 (6) j 1,, n, k 1, , m, we have 4t , t 1/ 2; 1/ 2, for all t 0,1/ 2 Therefore, we get that f t 2s s t t g , j t g ,k t t t s g t t s g ,j ,k 1/2 2 X t fY t dt (1/ 2)2 s 2 (1/ 2) s 2t s 2 t s dt t g t g t 2s s ,j ,k (1/ 2) s 2(1/ 2) s 2s s Also, we get 1/2 2 (7) f t 2s s t t g , j t g ,k t t t s g t ,j 2 t s g ,k t s 1 f X t fY t t s g , j t 1/2 2 dt X t t dt fY t s 1 f X t fY t t s g ,k t 1/2 2 dt (8) From (6), (7), and (8), we get the conclusion of the proposition To prove Proposition 2.2, we need to prove Lemma and Lemma Lemma Let the observations be given by model (2) Suppose that g , j and g ,k are the Laplace density, j 1,, n, k 1,, m Let j ,k be given as in (5) with s and 0,1 Then, we have max [ j ,k ]2 ; j ,k j,k ; j ,k j ,k 11 * s , 2s s 1 ,1/ , j 1, , n; k 1,, m 1/ min2, s 6 where * : Proof For all t 0, * , since g , j t 4t and g ,k t 4t , j 1,, n, k 1,, m and (0,1), we have g , j t t s g , j t t s 4t t s 2 Therefore, t s g , j t Likewise, t s g ,k t all r , we obtain 1 g , j t t s k 1 g ,k t t s k 1 5t min2, s :1 Z k ,j :1 Z k ,k t t for all t (0, * ) Hence, for © 2021 Industrial University of Ho Chi Minh City 108 X Y WITH LAPLACE ERROR DENSITIES NONPARAMETRIC ESTIMATION OF S (r ) : * t t s g , j t sin tr * t s g , k t dt * Z , j t * Z , k t * Z , j t Z , k t sin tr dt dt dt dt 0 t t t t : S1 S2 S3 S4 From Lemma 2.6.2, Section 2.6 in Kawata [25], we obtain S1 a1 a2 k r* u 2k 1 a1k a2k for all a1 , a2 , k 1, we get that 1 g t Z , j t ,j k 1 ts k 2 k 1 k 1 sin u g , j t k du Using the inequality 2k 1 t s k k 1 k 1 8t k t sk k 1 k 1 (9) for any t (0, * ) With the same argument as above, we also have k 8t 2k t sk , t (0, * ) k 1 k 1 Z ,k t From (9), we obtain S2 * 1 k sk k t t dt t k 1 k 1 * k k 1 t k t sk 1 dt k 1 k 1 k 8* 2k *sk k 1 k sk k 1 1 2s 8 k 2 s k * * k 1 s k 1 1 Next, we have 2s k 2k t sk 8t 2k t sk dt k 1 k 1 k 1 Similarly, the inequality (10) results in S3 S4 * k 1 8t t k 1 k 1 1 * k 8*2 k *sk 8 t k 1 2k t sk 1 dt k 1 2s k 1 k 1 k 1 11 From the bounds of S1 , S , S3 and S , these imply S (r ) for all r 2s Next, for convenience, we denote it X j Yk e g , j t g , k t dt , a b , j , k 2 a t s s t g , j t t g ,k t Saj,,bk b © 2021 Trường Đại học Cơng nghiệp thành phố Hồ Chí Minh (10) X Y WITH LAPLACE ERROR DENSITIES NONPARAMETRIC ESTIMATION OF 11 , we obtain 2s By the Fubini theorem and the inequality S (r ) S0,j ,k* * sin t X j Yk u v t t s g , j t 109 t s g ,k t dt g , j u g , k v dudv 11 2s Additionally, using the inequalities z z for z a1 a2a3 and a a a a 2 for a1 , a2 , a3 0, we have g , j t g ,k t * t s g t g t ,j ,k Sj*,k, * Therefore, j ,k j , k S S0,j ,k Sj ,k, * * * s dt * * t s s 1 dt Sj*,,k j , k 0,* 11 * s , 2s s 1 2 j ,k j ,k S S j , k 0, , * * 2 S Sj*,k, j ,k 0, * 2 11 * s , 2s s 1 j , k j , k S j ,k 0,* S 2 j ,k 0,* Sj*,k, S Sj*,k, j , k 0,* S Sj*,,k j , k 0,* Sj*,,k 11 * s 2s s 1 2 Finally, from the bounds of j ,k ; j , k j,k ; j ,k j ,k , we get the conclusion of the lemma Lemma Let the observations be given by model (2) Suppose that g , j and g ,k are the Laplace density, j 1, , n; k 1, , m Let the estimator ˆ be given by (4) with (0,1) and s Then, we have 1 n m Var(ˆ ) Proof Since j ,k Var(ˆ ) j ,k j ,k j,k for any j j , k k , we have 2 nm n m j ,k j 1 k 1 2 nm n m k 1 j 1 k 2 j 1 2 nm n m j 1 k 1 j ,k j , j 2 nm j ,k n m k 1 j 1 k 2 j 1 j ,k j , j © 2021 Industrial University of Ho Chi Minh City 110 X Y WITH LAPLACE ERROR DENSITIES NONPARAMETRIC ESTIMATION OF 2 nm n j 1 m j ,k j,k j k 1 j 1 2 nm n j 1 m j k 1 j 1 j ,k j ,k (11) 1 by using Lemma and (11) This completes the proof of n m Hence, we obtain Var(ˆ ) the lemma Proof of Proposition 2.2 Using Proposition 2.1, Lemma 2, and the common variance-bias decomposition | ˆ |2 | (ˆ ) |2 Var(ˆ ), we get the result of the proposition Proof of Theorem 2.3 From the assumptions of the theorem and from Proposition 2.2, we only need to prove that 1/ 2 t s 1 f X t fY t dt 0, t s | g , j t |2 t s 1 f X t fY t t s g ,k t 1/2 2 dt as n, m Indeed, using the Lebesgue dominated convergence theorem, we get the result of the theorem To prove Theorem 2.4, we use the following lemma, the statement and the proof for the general case of which are presented in Trong-Phuong [6] Lemma Suppose that the error density g satisfies the condition sin t 1 t For R and 0, put Bg , R , t g t 0t R: g Then there exists a constant K ( g ) depending on g such that Bg , R , K ( g ) R (12) Proof of Theorem 2.4 Given ( f X , fY ) F ,C , we set Q , j : t s 1 f X t fY t 1/2 t s g , j t Q ,k : dt , t s 1 f X t fY t t s g ,k t 1/2 2 dt For 1 small enough and let R1 1, we write Q , j Q , j ,1 Q , j ,2 Q , j ,3 , where Q , j ,1 : t R1 t s 1 f X t fY t t s g , j t dt , t s 1 f X t fY t Q , j ,2 : 1/2 t R1 , g , j t 1 t s g , j t t s 1 f X t fY t Q , j ,3 : 1/2 t R1 , g , j t 1 t s g , j t dt , dt We have Q , j ,1 f X t fY R1 t R1 Using (12), we infer Q , j ,2 2 (Bg t 1 t , j , R1 ,1 1 t dt C R12 1 ) 2K ( g , j )1R13 In addition, applying the Cauchy © 2021 Trường Đại học Cơng nghiệp thành phố Hồ Chí Minh NONPARAMETRIC ESTIMATION OF inequality, we get that t s 1 f X t fY t Q , j ,3 X Y WITH LAPLACE ERROR DENSITIES g , j t t s 1/2 t R1 , g , j t 1 2 R1s 1 1/2 f X t fY t dt 111 dt R1s C 1 From these upper bounds of Q , j ,1 , Q , j ,2 , and Q , j,3 , we obtain Q , j C R1s C 2 K ( g ) R ,j 1 R12 1 21 (13) With the same argument, we also have Q ,k C R1s C 2 K ( g ) R ,k 1 R12 1 21 (14) Using Proposition 2.2, (13), (14), and the inequality b1 b2 b3 b4 4(b12 b22 b32 b42 ), we deduce | ˆ |2 1 R122 1 R1s 1 12 R16 1 n m s 1 3 2 13 1 R1 and 1 R1 results in n m 1 2/3 22 1 62 s /3 ˆ | | 1 R1 n m Choosing 2 1 R 1/3 2 1 6 s /3 1 R13 n m d 1 , we obtain n m Choosing R1 2 1 d 1 (1) (1) n2 1d m 22 1d n m c1 c2 for all c1 , c2 0, we get the conclusion of the Finally, applying the inequality c1 c2 | ˆ |2 theorem ACKNOWLEDGMENT The authors would like to thank the reviewers for careful reading, helpful comments 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Kozubowski, and K Podgórski, The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance, Birkhäuser, Boston, 2001 [24] S Kotz, S Lumelskii, and M Pensky, The Stress-Strength Model and its Generalizations Theory and Applications, Singapore: World Scientific, 2003 [25] T Kawata, Fourier Analysis in Probability Theory, Press, New York, Lon Don, 1972 [26] W A Woodward and G D Kelley, Minimum variance unbiased estimation of P[Y < X] in the normal case, Technometrics, vol 19, no 1, pp 95-98, 1977 [27] W Zhou, Statistical inference for P(X < Y ), Statistics in Medicine, vol 27, no 2, pp 257-279, 2008 ƯỚC LƯỢNG PHI THAM SỐ CỦA X Y VỚI CÁC HÀM MẬT ĐỘ SAI SỐ LAPLACE Tóm tắt Chúng tơi khảo sát ước lượng phi tham số xác suất : X Y hai biến ngẫu nhiên X Y quan trắc có tính đến sai số Cụ thể, từ phiên nhiễu X 1, , X n X Y1, , Ym Y , giới thiệu ước lượng ˆ sau đó, thiết lập tính vững theo trung bình ước lượng đề nghị biến ngẫu nhiên sai số có phân phối Laplace Tiếp theo, sử dụng giả thiết thêm vào điều kiện hàm mật độ f X X fY Y , rút tốc độ hội tụ bậc hai trung bình sai số bình phương ước lượng ˆ Từ khóa Phi tham số, mật độ sai số, ước lượng, tốc độ hội tụ Received on: 26/05/2021 Accepted on: 16/08/2021 © 2021 Industrial University of Ho Chi Minh City ...104 X Y WITH LAPLACE ERROR DENSITIES NONPARAMETRIC ESTIMATION OF optimal estimator of when error density functions g , j and g ,k are assumed... phố Hồ Chí Minh NONPARAMETRIC ESTIMATION OF X Y WITH LAPLACE ERROR DENSITIES 113 [22] M M Dewdney, A R Biggs, and W W Turechek, A statistical comparison of the reliability of the blossom... problem of nonparametric estimation of the probability : X Y when two random variables X and Y are observed with additional errors We use noise versions X 1, , X n of X and Y1, , Ym of