Estimation of parameters of the Gompertz distribution using the least ...

134 J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
was first introduced by Gompertz [8]. Recently, many authors have contributed
to the studies of statistical methodology and characterization of this distribution;
for example, Read [15], Makany [13], Rao and Damaraju [14], Franses [6],
Chen [3] and Wu and Lee [17]. Garg et al. [7] studied the properties of the
Gompertz distribution and obtained the maximum likelihood (ML) estimates
for the parameters. Gordon [9] provided the ML estimation for the mixture of
two Gompertz distributions.
Probability plots in their most common form are used with location-scale
parameter models. Parameters were estimated from a probability plot by fitting
a straight line through the points by eye, but it is clear that the line could have
been determined by least squares method. A similar idea can be used more
generally to propose parameter estimates in certain situations. In this paper, we
consider Gompertz model in which the unknown parameters can be related to
some transform of the cumulative distribution function under the complete
data and the first failure-censored data (for example, series system; see [1]).
The remainder of this paper is organized as follows. In Section 2, we propose
unweighted and weighted least squares procedures for estimating the
parameters of the Gompertz distribution for both complete samples and the
first failure-censored samples. Numerical simulation studies are given in Section
3. Some conclusions are presented in Section 4.
2. Least squares estimation of parameters
In this section, we propose both unweighted and weighted least squares
procedures for estimating the parameters c and k of the Gompertz distribution.
For the case of complete sample is discussed in Section 2.1. In Section 2.2, we
derive the unweighted and weighted least squares estimates of c and k under the
first failured-censored sampling plan [1]. The sampling plan proposed by
Balasooriya [1] consists of grouping number of specimens into several sets or
series systems of the same size and testing each of these series systems of
specimens separately until the occurrence of first failure in each series system in
reliability study. Compared to ordinary sampling plans, the first failured-censored
sampling plan has an advantage of saving both test-time and resources.
2.1. Least squares estimates under complete data
The probability density function (p.d.f.) of the Gompertz distribution is
k
f ðxÞ ¼ke cx exp
c ðecx 1Þ ; x > 0; ð1Þ