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.
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References
Boente G, Fraiman R (1988) Consistency of a nonparametric estimate of a density function for dependent variables. J Multivar Anal 25(1):90–99
Chai G (1989) Consistency of nearest neighbor density estimator of stationary processes. Acta Math Sin 32(3):423–432
Chen XR (1981) The rate of consistency of nearest neighbor density estimator. Sci China Ser A 12:1419–1428
Chen Y, Wang L, Wang YB (2013) Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models. J Math Anal Appl 401:114–129
Chen W, Wang YB, Cheng DY (2016) An inequality of widely dependent random variables and its applications. Lith Math J 56(1):16–31
Devroye LP, Wagner TJ (1977) The strong uniform consistency of nearest neighbor density estimates. Ann Stat 5:536–540
He W, Cheng DY, Wang YB (2013) Asymptotic lower bounds of precise large deviations with nonnegative and dependent random variables. Stat Probab Lett 83:331–338
Li YM (2017) On the rate of strong convergence for a recursive probability density estimator of END samples and its applications. J Math Inequal 11(2):335–343
Li YM, Zhou Y, Liu C (2018) On the convergence rates of kernel estimator and hazard estimator for widely dependent samples. J Inequal Appl 2018:71
Liu Y, Zhang Y (2010) The consistency and asymptotic normality of nearest neighbor density estimator under \(\varphi \)-mixing condition. Acta Math Sci 30(3):733–738
Liu XJ, Gao QW, Wang YB (2012) A note on a dependent risk model with constant interest rate. Stat Probab Lett 82:707–712
Loftsgaarden DO, Quesenberry CP (1965) A nonparametric estimate of a multivariate density function. Ann Math Stat 36(3):1049–1051
Moore DS, Henrichon EG (1969) Uniform consistency of some estimates of a density function. Ann Math Stat 40(4):1499–1502
Shen AT (2013) Bernstein-type inequality for widely dependent sequence and its application to nonparametric regression models. Abstr Appl Anal 2013:862602
Wagner TJ (1973) Strong consistency of a nonparametric estimate of a density function. IEEE Trans Syst Man Cybernet 3:289–290
Wang XJ, Hu SH (2015) The consistency of the nearest neighbor estimator of the density function based on WOD samples. J Math Anal Appl 429:497–512
Wang YB, Cui Z, Wang K, Ma X (2012) Uniform asymptotics of the finite-time ruin probability for all times. J Math Anal Appl 390:208–223
Wang K, Wang Y, Gao Q (2013) Uniform asymptotics for the finite-time ruin probability of a new dependent risk model with a constant interest rate. Methodol Comput Appl Probab 15:109–124
Wang XJ, Xu C, Hu TC, Volodin A, Hu SH (2014) On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models. TEST 23:607–629
Wang XJ, Wu Y, Rosalsky A (2017) Complete convergence for arrays of rowwise widely orthant dependent random variables and its applications. Stochastics 89:1228–1252
Xia H, Wu Y, Tao XR, Wang XJ (2018) The consistency for the weighted estimator of non-parametric regression model based on widely orthant-dependent errors. Probab Eng Inf Sci 32:469–481
Yang SC (2003) Consistency of nearest neighbor estimator of density function for negative associated samples. Acta Math Appl Sin 26(3):385–395
Zheng X, Wu QY (2012) The rate of strong consistency of the nearest neighbor density estimator for negatively dependent random variables. ISRN Appl Math 2012:569547
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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).
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Shen, A., Tao, H. & Wang, X. The asymptotic properties for the estimators of the survival function and failure rate function based on WOD samples. Stat Papers 61, 2671–2684 (2020). https://doi.org/10.1007/s00362-018-01064-w
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DOI: https://doi.org/10.1007/s00362-018-01064-w