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

138 J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
Garg et al. [7] derived the ML estimates of c and k from
^c ¼
^k ¼
(
(
n Xn
i¼1
^c Xn
i¼1
e^cx ðiÞ
)(
1
x ðiÞ
^c
) (
1
X n
i¼1
X n
i¼1
e^cx ðiÞ
x ðiÞ
X n
i¼1
1
e^cx ðiÞ
X n
i¼1
1
n Xn
i¼1
x ðiÞ e^cx ðiÞ) 1
;
ð5Þ
x ðiÞ e^cx ðiÞ) 1
: ð6Þ
Thus it is only necessary to obtain a solution of Eq. (5) which will be the MLE
of c. An iterative solution to Eq. (5) can be achieved by NewtonÕs method; the
initial estimate ^c 0 may be selected as the LSE of c. The MLE ^k can then be
obtained from Eq. (6).
2.2. Least squares estimates under the first failured-censored sampling plan
The p.d.f. of the first-order statistic X ð1Þ is
k
f ðx; c; k Þ¼k e cx exp
c ðecx 1Þ ; ð7Þ
where k ¼ nk. The corresponding c.d.f. is
k
F ðxÞ ¼1 exp
c ðecx 1Þ : ð8Þ
Suppose X ð1Þ1 ; X ð1Þ2 ; ...; X ð1Þm denote the set of first-order statistics of m
samples of size n from (1) and let Y ð1Þ < Y ð2Þ < < Y ðmÞ be the corresponding
order statistics. Clearly, X ð1Þ1 ; X ð1Þ2 ; ...; X ð1Þm can also be considered as a random
sample from (7). Then F ðxÞ in (8) satisfies
lnf lnð1 F ðxÞÞg ¼ ln n þ ln k þ ln ecx 1
: ð9Þ
c
For observed ordered observations y ð1Þ < y ð2Þ < < y ðmÞ , (9) can be rewritten
as
lnf lnð1 F ðy ðiÞ ÞÞg ¼ ln n þ ln k þ ln ecy ðiÞ
1
; i ¼ 1; ...; m: ð10Þ
c
Proceeding as in Section 2.1, we can obtain the unweighted and weighted
least squares estimates of c and k. Likewise, we can also obtain the ML estimates
of c and k under the first failured-censored sampling plan.