Nonparametric estimation of cumulative distribution function from noisy data in the presence of Berkson and classical errors

Cao Xuan Phuong (), Le Thi Hong Thuy () and Vo Nguyen Tuyet Doan ()
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Cao Xuan Phuong: Ton Duc Thang University
Le Thi Hong Thuy: Van Lang University
Vo Nguyen Tuyet Doan: Myongji University

Metrika: International Journal for Theoretical and Applied Statistics, 2022, vol. 85, issue 3, No 2, 289-322

Abstract: Abstract Let X, Y, W, $$\delta $$ δ and $$\varepsilon $$ ε be continuous univariate random variables defined on a probability space such that $$Y = X+\varepsilon $$ Y = X + ε and $$W = X + \delta $$ W = X + δ . Herein X, $$\delta $$ δ and $$\varepsilon $$ ε are assumed to be mutually independent. The variables $$\varepsilon $$ ε and $$\delta $$ δ are called classical and Berkson errors, respectively. Their distributions are known exactly. Suppose we only observe a random sample $$Y_1, \ldots , Y_n$$ Y 1 , … , Y n from the distribution of Y. This paper is devoted to a nonparametric estimation of the unknown cumulative distribution function $$F_W$$ F W of W based on the observations as well as on the distributions of $$\varepsilon $$ ε , $$\delta $$ δ . An estimator for $$F_W$$ F W 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, $$\delta $$ δ and $$\varepsilon $$ 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.

Keywords: Cumulative distribution function; Deconvolution; Berkson errors; Classical errors; 62G05; 62G20. (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s00184-021-00830-5

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