LIFETIME DISTRIBUTION AND ESTIMATION PROBLEMS OF CONSECUTIVE-k-OUT-OF-n:F SYSTEMS

SIGEO AKI^{1} AND KATUOMI HIRANO^{2}

1Department of Mathematical Science, Faculty of Engineering Science, Osaka University,

Machikaneyama-cho, Toyonaka, Osaka 560, Japan

2The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu, Minato-ku, Tokyo 106, Japan

(Received February 13, 1995; revised July 13, 1995)

Abstract. Explicit formula is given for the lifetime distribution of a consecutive-k-out-of-n:F system. It is given as a linear combination of distributions of order statistics of the lifetimes of n components. We assume that the lifetimes are independent and identically distributed. The results should make it possible to treat the parametric estimation problems based on the observations of the lifetimes of the system. In fact, we take up, as some examples, the cases where the lifetimes of the components follow the exponential, the Weibull, and the Pareto distributions, and obtain feasible estimators by moment method. In particular, it is shown that the moment estimator is quite good for the exponential case in the sense that the asymptotic efficiency is close to one.

Key words and phrases: Consecutive-k-out-of-n:F system, system reliability, failure time, discrete distributions of order k, order statistics, exponential distribution, Weibull distribution, Pareto distribution, moment estimator.