Quality & Quantity 38: 217–233, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
A Note on the Prediction Intervals for a Future
Ordered Observation from a Pareto Distribution
, HAI-LIN LU
, CHONG-HONG CHEN
Department of Statistics, Tamkang University, Tamsui, Taipei, Taiwan, R.O.C.;
Mathematics, National Cheng Kung University, Tainan, Taiwan, R.O.C.
Abstract. In the researching of products’ reliability, the result of life testing is used as the basis for
the evaluation and improvement of reliability. During life testing, however, the future observation in
an ordered sample is often expected to be predicted so as to show how long a sample of units might
run until all fail in life testing. Therefore, we propose ﬁve new pivotal quantities to obtain prediction
intervals of future order statistics based on right type II censored samples from the Pareto distribution
with known shape parameter, then compares the lengths of the prediction intervals when using the
pivotal quantity of Ouyang and Wu (1994) based on best linear unbiased estimator (BLUE) of scale
parameter, and these ﬁve pivotal quantities. An advantage of these ﬁve pivotal quantities is that these
are easier to calculate than the pivotal quantity of Ouyang and Wu (1994) based on BLUE of scale
parameter, since they need to compute the tables of coefﬁcients of BLUE of scale parameter.
Key words: best linear unbiased estimator, Monte Carlo simulation, Pareto distribution, prediction
interval, right type II censored samples.
In most literatures of reliability, the exponential distribution is widely used as a
model of lifetime data. This distribution is characterized by a constant failure rate,
say λ. But in a population of components there could be a ubiquitous variation in
λ-values because of small ﬂuctuations in manufacturing tolerances, so that a com-
ponent selected at random can be regarded as belonging to a random subpopulation
(McNolty and Doyle, 1980).
The lifetime of a particular component have an exponential distribution with
failure rate λ and location parameter (guarantee time) µ;andlettheλ follow a
Gamma distribution with scale parameter σ>0 and shape parameter β>0.
Then the failure time Y of a component selected at random from such a mixed
population has a Pareto distribution of the second kind (Engelhardt et al., 1986)
with the probability density function is given by
Author for correspondence: Jong-Wuu Wu, Department of Statistics, Tamkang University,
Tamsui, Taipei, Taiwan 25137, R.O.C., E-mail: email@example.com.