The asymptotic properties for the estimators of the survival function and failure rate function based on WOD samples
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The asymptotic properties for the estimators of the survival function and failure rate function based on WOD samples Aiting Shen1 · Huiling Tao1 · Xuejun Wang1 Received: 1 February 2018 / Revised: 18 September 2018 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper, the consistency for the estimators of the survival function and failure rate function in reliability theory is investigated. The strong consistency and the convergence rate for the estimators of the survival function and failure rate function based on widely orthant dependent (WOD, in short) samples are established. Our results established in the paper generalize the corresponding ones for independent samples and some negatively dependent samples. Keywords Survival function · Failure rate function · Reliability theory · WOD random variables · Strong consistency Mathematics Subject Classification 62G07
1 Introduction Suppose that the population X has the unknown density function f (x) and X 1 , X 2 , . . . , X n are samples of X . Let {kn , n ≥ 1} be a sequence of positive integers such that 1 ≤ kn < n. For fixed x and n, denote an (x) = min{a : a > 0 such that [x − a, x + a] contains at least kn observations}. Loftsgaarden and Quesenberry (1965) introduced the nearest neighbor estimator of f (x) for multivariate samples X 1 , X 2 , . . . , X n as follows:
Supported by the National Natural Science Foundation of China (11871072, 11671012, 11501004, and 11701005), the Natural Science Foundation of Anhui Province (1508085J06) and the Key Projects for Academic Talent of Anhui Province (gxbjZD2016005).
B 1
Aiting Shen [email protected] School of Mathematical Sciences, Anhui University, Hefei 230601, People’s Republic of China
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f n (x) =
kn . 2nan (x)
(1.1)
Let F(x) stand for the distribution function of the population X , which is continuous. In reliability theory and related problems, the survival function F(x) and the failure rate function r (x) play important roles, which are defined as follows: F(x) = 1 − F(x), r (x) =
f (x) f (x) = . 1 − F(x) F(x)
n Let X 1 , X 2 , . . . , X n be the samples from the population X . Let Fn (x) = n1 i=1 I (X i < x) be the empirical distribution function of X 1 , X 2 , . . . , X n , where I (A) stands for the indicator function of the event A. We consider the estimators of F(x) and r (x) as follows: F n (x) = 1 − Fn (x) =
n f n (x) 1 f n (x) = I (X i ≥ x), rn (x) = , n 1 − Fn (x) F n (x) i=1
where f n (x) is defined by (1.1). Since the concept of the nearest neighbor estimator of the density function was introduced by Loftsgaarden and Quesenberry (1965), many authors devoted to the study of the asymptotic properties of the nearest neighbor estimator of the density function. For independent samples, Wagner (1973) established the strong consistency; Moore and Henrichon (1969) and Devroye and Wagner (1977) obtained the uniform consistency and strong uniform consistency, respectively; Chen (1981) established the convergence rate of the consistency of the
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