Nonparametric sequential estimation of the cumulative function by orthogonal series and application in reliability

Abstract

This article proposes the orthogonal series method for the cumulative distribution function estimation based on a random number of observations N_t, which may be recorded in time $(0,t].$ We give the statistical properties of the estimator (bias, variance, mean square error, mean square integrated error, and the rate of convergence) and some asymptotic properties. We consider that N_t is independent of the observations and $N_t\stackrel{P}{\to }\infty$ as $t\to \infty .$ The choice of the smoothing parameter (bandwidth) of the estimator is investigated by two methods: The Kronmal–Tarter method and a new technique proposed by Saadi et al. adapted to the random sample size case. A detailed study with a Dirichlet basis is presented. The obtained estimator is asymptotically unbiased and consistent. An application to the reliability field is investigated with simulations in order to study the behavior of the reliability and failure rate estimators, from different life distributions, namely: Lognormal, Gamma, and Weibull distributions. A real lifetime data sets are analyzed by our proposed approach. The results show that the proposed estimators are performant.

Keywords:

• Cumulative distribution function
• Dirichlet basis
• failure rate
• Nonparametric estimation
• Orthogonal method
• Random sample size
• Reliability

Acknowledgements

We sincerely thank the editor and two anonymous referees for their valuable comments.