AISM 53, 307-324

© 2001 ISM

## Estimation with sequential order statistics from exponential
distributions

### Erhard Cramer and Udo Kamps

Department of Mathematics, University of Oldenburg, D-26111 Oldenburg, Germany

(Received July 21, 1998; revised September 27, 1999)

Abstract.
The lifetime of an ordinary *k*-out-of-*n* system is described by the
$(n-k+1)$-st order statistic from an iid sample. This set-up is based on the assumption
that the failure of any component does not affect the remaining
ones. Since this is possibly not fulfilled in technical systems,
sequential order statistics have been proposed to model a change of the
residual lifetime distribution after the breakdown of some
component. We investigate such sequential *k*-out-of-*n* systems where the
corresponding sequential order statistics, which describe the lifetimes of
these systems, are based on one- and two-parameter exponential
distributions. Given differently structured systems, we focus on
three estimation concepts for the distribution parameters. MLEs,
UMVUEs and BLUEs of the location and scale parameters are
presented. Several properties of these estimators, such as
distributions and consistency, are established. Moreover, we
illustrate how two sequential *k*-out-of-*n* systems based on exponential
distributions can be compared by means of the probability
$P(X < Y)$. Since other models of ordered random variables, such as ordinary
order statistics, record values and progressive type II censored \os s can be
viewed as sequential order statistics, all the results can be applied to
these situations as well.

Key words and phrases:
Sequential *k*-out-of-*n* system, sequential order statistics, order statistics, record values, progressive type II censoring, maximum
likelihood estimation, best linear unbiased estimation, uniformly
minimum variance unbiased estimation, exponential distribution,
Weinman multivariate exponential distribution.

**Source**
( TeX , DVI )