Deconvolution of Cumulative Distribution Function with Unknown Noise Distribution

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Deconvolution of Cumulative Distribution Function with Unknown Noise Distribution Cao Xuan Phuong1

Received: 23 January 2020 / Accepted: 18 June 2020 © Springer Nature B.V. 2020

Abstract Let Y , X and ε be continuous univariate random variables satisfying the model Y = X + ε. Herein X is of interest, Y is a noisy version of X, and ε is a random noise independent of X. This paper is devoted to a nonparametric estimation of cumulative distribution  ) function FX of X on the basis of independent random samples (Y1 , . . . , Yn ) and (ε1 , . . . , εm drawn from the distributions of Y and ε, respectively. We provide an estimator for FX based on a direct inversion formula and the ridge-parameter regularization. Our estimator is shown to be mean consistency with respect to the mean squared error whenever the set of all zeros of the characteristic function of ε has Lebesgue measure zero. We then derive some convergence rates of the mean squared error uniformly on a nonparametric class for FX and on some different regular classes for the density of ε. A numerical example is performed to illustrate the efficiency of our method. Keywords Cumulative distribution function · Non-standard noise · Mean consistency · Convergence rate Mathematics Subject Classification (2010) 62G07 · 62G20

1 Introduction Let X be a continuous univariate random variable defined on a probability space (, F , P). In the present paper, we consider the problem of estimating unknown cumulative distribution function (cdf for short) FX , defined by FX (x) := P(X ≤ x) with x ∈ R, of X. This problem has been encountered in many fields of science and engineering, such as medicine, climatological studies, seismology, environmental sciences and so on. In medicine, for example, people with blood pressure greater than 140 mmHg are considered as high blood pressure or hypertension. If X represents the blood pressure, then estimating FX (140) helps to determine the hypertension prevalence in a population. In climatological studies, X can

B C.X. Phuong

[email protected]

1

Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

C.X. Phuong

be considered as hurricane wind speed, and an estimation of FX (x) with a high value x is related to the determination of the occurrence probability of a hurricane wind with speed exceed x. A detailed review on this problem and its applications can be found in Quintela-del Río and Estévez-Pérez [27]. In a statistical context, the problem of estimating FX is usually conducted on the basis of a sample of independent and identically distributed (i.i.d.) observations X1 , . . . , Xn taken from the distribution of X. Various estimates of FX have been developed in the literature, such as empirical cdf, kernel-based estimates (see, e.g., Nadaraya [25], Yamato [34], Azzalini [1], Reiss [28], Lejeune and Sarda [20], Bowman et al. [2], Dutta [11]), smooth monotone polynomial spline estimation (see, e.g., Xue and Wang [33]). However, there are many situations where a random variable of in