Quality & Quantity 38: 753–769, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Optimal Estimation of the Parameters of the
Gompertz Distribution Based on the Doubly Type II
and PAI-LING LI
Department of Statistics, Tamkang University, Tamsui, Taipei, Taiwan, R.O.C.
Abstract. In this paper, we develop exact conﬁdence intervals and exact joint conﬁdence regions
for the parameters of the Gompertz distribution under the doubly type II censored sample. We also
provide optimal criteria for ﬁnding a best exact conﬁdence interval and a best exact joint conﬁdence
region among these interval estimations. Finally, we give a numerical example to illustrate our pro-
posed method. Furthermore, when compared to estimation of Chen (1997), our proposed method can
get a better parameter estimation.
Key words: doubly type II censored sample, Gompertz distribution, joint conﬁdence region.
The Gompertz distribution gives good ﬁt to data from clinical trials on older
subjects and is also useful in the construction of life tables. Gompertz (1825)
derived this probability model for human mortality. Many authors have discussed
the problems about parameter estimation of this distribution. For instance, Grag et
al. (1970) obtained the maximum likelihood estimation of the parameters of the
Gompertz distribution, and Gordon (1990) provided the maximum likelihood es-
timation for mixtures of two Gompertz distributions under censoring. Chen (1997)
developed an exact conﬁdence interval and joint conﬁdence region for the para-
meters of the Gompertz distribution under type II censoring, but he only studied
the parameter estimations based on a complete or right censored sample. Hence,
in this article, we modify Chen’s idea to obtain the exact conﬁdence intervals and
joint conﬁdence regions for the parameters of the Gompertz distribution based on
doubly type II censored sample. Moreover, we also provide the optimal criterions
to ﬁnd a best estimation among these parameter estimations by using the minimum
tolerance length of a conﬁdence interval and the minimal area of a joint conﬁdence
region, respectively. Finally, we give a numerical example to illustrate our proposed
method. Furthermore, when compared to estimation of Chen (1997), our proposed
method can get a better parameter estimation.
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