###
SEQUENTIAL ORDER STATISTICS AND *K*-OUT-OF-*N* SYSTEMS

WITH SEQUENTIALLY ADJUSTED FAILURE RATES

###
ERHARD CRAMER AND UDO
KAMPS

*Institute of Statistics, Aachen University of Technology,*

Wüllnerstr. 3, D-52056 Aachen, Germany
(Received January 12, 1995; revised July 11, 1995)

**Abstract.**
*k*-out-of-*n* systems frequently appear
in applications. They consist of *n* components of the same
kind with independent and identically distributed
life-lengths. The life-length of such a system is described by
the (*n-k*+1)-th order statistic in a sample of size *n* when
assuming that remaining components are not affected by
failures. Sequential order statistics are introduced as a more
flexible model to describe `sequential *k*-out-of-*n* systems'
in which the failure of any component possibly influences the
other components such that their underlying failure rate is
parametrically adjusted with respect to the number of
preceding failures. Useful properties of the maximum
likelihood estimators of the model parameters are shown, and
several tests are proposed to decide whether the new model is
the more appropriate one in a given situation. Moreover, for
specific distributions, e.g. Weibull distributions,
simultaneous maximum likelihood estimation of the model
parameters and distribution parameters is considered.

*Key words and phrases*:
Sequential
*k*-out-of-*n*-system, sequential order statistics,
generalized order statistics, type II censoring, maximum
likelihood estimators, extremal quotient, Weibull
distributions.

**Source**
( TeX ,
DVI ,
PS )