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Nonparametric estimation of cumulative distribution function from noisy data in the presence of Berkson and classical errors
Nonparametric estimation of cumulative distribution function from noisy data in the presence of Berkson and classical errors
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Date
2021
Authors
Cao Xuan Phuong
Le Thi Hong Thuy
Vo Nguyen Tuyet Doan
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Research Projects
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Abstract
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. Supposewe 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.
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Keywords
"Cumulative distribution function,
Deconvolution,
Berkson errors,
Classical errors"