Journal Articles - Mathematic and Statistic - 2021

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Browsing Journal Articles - Mathematic and Statistic - 2021 by Author "Le Thi Hong Thuy"
  • Publication
    Distribution Estimation of a Sum Random Variable from Noisy Samples
    ( 2021)
    Cao Xuan Phuong
    ;
    Le Thi Hong Thuy
    Let X, Y be independent continuous univariate random variables with unknown distributions. Suppose we observe two independent random samples X′1,…,X′n and Y′1,…,Y′m from the distributions of X′=X+ζ and Y′=Y+η, respectively. Here ζ, η are random noises and have known distributions. This paper is devoted to an estimation for unknown cumulative distribution function (cdf) FX+Y of the sum X+Y on the basis of the samples. We suggest a nonparametric estimator of FX+Y and demonstrate its consistency with respect to the root mean squared error. Some upper and minimax lower bounds on convergence rate are derived when the cdf’s of X, Y belong to Sobolev classes and when the noises are Fourier-oscillating, supersmooth and ordinary smooth, respectively. Particularly, if the cdf’s of X, Y have the same smoothness degrees and n=m, our estimator is minimax optimal in order when the noises are Fourier-oscillating as well as supersmooth. A numerical example is also given to illustrate our method.
  • Publication
    Nonparametric estimation of cumulative distribution function from noisy data in the presence of Berkson and classical errors
    ( 2021)
    Cao Xuan Phuong
    ;
    Le Thi Hong Thuy
    ;
    Vo Nguyen Tuyet Doan
    Let X, Y, W, δ and ε be continuous univariate random variables defined on a probability space such that Y=X+ε and W=X+δ. Herein X, δ and ε are assumed to be mutually independent. The variables ε and δ are called classical and Berkson errors, respectively. Their distributions are known exactly. Suppose we only observe a random sample Y1,…,Yn from the distribution of Y. This paper is devoted to a nonparametric estimation of the unknown cumulative distribution function FW of W based on the observations as well as on the distributions of ε, δ. An estimator for FW depending on a smoothing parameter is suggested. It is shown to be consistent with respect to the mean squared error. Under certain regularity assumptions on the densities of X, δ and ε, we establish some upper and lower bounds on the convergence rate of the proposed estimator. Finally, we perform some numerical examples to illustrate our theoretical results.