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

Author

Listed:
  • Cao Xuan Phuong

    (Ton Duc Thang University)

  • Le Thi Hong Thuy

    (Van Lang University)

  • Vo Nguyen Tuyet Doan

    (Myongji University)

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.

Suggested Citation

  • Cao Xuan Phuong & Le Thi Hong Thuy & Vo Nguyen Tuyet Doan, 2022. "Nonparametric estimation of cumulative distribution function from noisy data in the presence of Berkson and classical errors," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(3), pages 289-322, April.
  • Handle: RePEc:spr:metrik:v:85:y:2022:i:3:d:10.1007_s00184-021-00830-5
    DOI: 10.1007/s00184-021-00830-5
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    References listed on IDEAS

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    3. Aurore Delaigle & Peter Hall & Peihua Qiu, 2006. "Nonparametric methods for solving the Berkson errors‐in‐variables problem," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(2), pages 201-220, April.
    4. Wand, M. P., 1998. "Finite sample performance of deconvolving density estimators," Statistics & Probability Letters, Elsevier, vol. 37(2), pages 131-139, February.
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