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LIFETIME DISTRIBUTION AND ESTIMATION PROBLEMS OF

CONSECUTIVE-*k*-OUT-OF-*n*:F SYSTEMS

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

^{1} *Department of Mathematical Science, Faculty of
Engineering
Science, Osaka University,*

Machikaneyama-cho, Toyonaka, Osaka 560, Japan

^{2} *The 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.

**Source**
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