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

Applied Mathematics and Computation 158 (2004) 133–147
www.elsevier.com/locate/amc
Estimation of parameters of the
Gompertz distribution using the least
squares method
Jong-Wuu Wu a, *
, Wen-Liang Hung b , Chih-Hui Tsai a
a Department of Statistics, Tamkang University, Tamsui, Taipei 25137, Taiwan, ROC
b Department of Mathematics, National Hsinchu Teachers College, Hsin-Chu, Taiwan, ROC
Abstract
The Gompertz distribution has been used to describe human mortality and establish
actuarial tables. Recently, this distribution has been again studied by some authors. The
maximum likelihood estimates for the parameters of the Gompertz distribution has
been discussed by Garg et al. [J. R. Statist. Soc. C 19 (1970) 152]. The purpose of this
paper is to propose unweighted and weighted least squares estimates for parameters of
the Gompertz distribution under the complete data and the first failure-censored data
(series systems; see [J. Statist. Comput. Simulat. 52 (1995) 337]). A simulation study is
carried out to compare the proposed estimators and the maximum likelihood estimators.
Results of the simulation studies show that the performance of the weighted least
squares estimators is acceptable.
Ó 2003 Elsevier Inc. All rights reserved.
Keywords: Gompertz distribution; Least squares estimate; Maximum likelihood estimate; First
failure-censored; Series system
1. Introduction
The Gompertz distribution plays an important role in modeling human
mortality and fitting actuarial tables. Historically, the Gompertz distribution
* Corresponding author.
E-mail address: jwwu@stat.tku.edu.tw (J.-W. Wu).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved.
doi:10.1016/j.amc.2003.08.086

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

where c > 0andk > 0 are the parameters. It is noted that when c ! 0, the
Gompertz distribution will tend to an exponential distribution. The corresponding
cumulative distribution function (c.d.f.) is
k
F ðxÞ ¼1 exp
c ðecx 1Þ
ð2Þ
and F ðxÞ satisfies
lnf lnð1 F ðxÞÞg ¼ ln k þ ln ecx 1
: ð3Þ
c
Now suppose that X 1 ; X 2 ; ...; X n is a sample of size n from a Gompertz
distribution with parameters c and k, and that X ð1Þ < X ð2Þ < < X ðnÞ are the
order statistics. For observed ordered observations x ð1Þ < x ð2Þ < < x ðnÞ ,it
follows from (3) that
lnf lnð1 F ðx ðiÞ ÞÞg ¼ ln k þ ln ecx ðiÞ
1
; i ¼ 1; 2; ...; n: ð4Þ
c
Let the empirical distribution function of F ðxÞ be denoted by bF ðxÞ, where
bF ðx ðiÞ Þ equals i=n. In order to avoid lnð0Þ in (4), we modify bF ðx ðiÞ Þ to be
p i ¼ i d
n 2d þ 1
for some d ð0 6 d < 1Þ. The reader is referred to Barnett [2] and DÕAgostino
and Stephens [4] for details. In this paper, we only choose three popular
quantities d ¼ 0, 0.5, 0.3 [10,16], and let p 1i ¼ i=ðn þ 1Þ, p 2i ¼ði 0:5Þ=n, p 3i ¼
ði 0:3Þ=ðn þ 0:4Þ. Alternatively, we use p 4i ¼ P i
j¼1 f1=ðn j þ 1Þg to estimate
lnð1 F ðx ðiÞ ÞÞ in (4) (see [12,16]).
First, we estimate c and k by unweighted least squares (UWLS) method. Let
and
G k ðc; kÞ ¼ Xn
i¼1
G 4 ðc; kÞ ¼ Xn
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147 135
i¼1
lnð
ln p 4i
We solve the normal equations
8
oG k ðc; kÞ >< ¼ 0;
oc
>: oG k ðc; kÞ
¼ 0
ok
lnð1 p ki ÞÞ ln k ln ecx ðiÞ 2
1
; k ¼ 1; 2; 3
c
ln k ln ecx ðiÞ 2
1
:
c

136 J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
for each k ¼ 1, 2, 3, 4. Then the corresponding UWLS estimates of c and k
satisfy the following normal equations for k ¼ 1, 2, 3:
( !
!)
^c k ¼
Xn ^c k x e^c kx ðiÞ ðiÞ þ e^c k x ðiÞ
1 e^c kx ðiÞ
1
e^c kx ðiÞ
ln
1
^c k
^k k ¼
(
(
i¼1
X n
Yn
i¼1
i¼1
1 X n
n
i¼1
ð
^c k x ðiÞ e^c kx ðiÞ þ e^c k x ðiÞ
1
^c k ðe^c kx ðiÞ
1Þ
lnð lnð1 p ki ÞÞ þ 1 n
lnð1
!"
lnð
X n
i¼1
ln
lnð1
p ki ÞÞ
e^c kx ðiÞ
1
^c k
!#) 1
;
) 1=n ( !) ð 1=nÞ
Y n e^c kx ðiÞ
1
p ki ÞÞ
^c k
i¼1
and
( !
!)
^c 4 ¼
Xn ^c 4 x e^c 4x ðiÞ ðiÞ þ e^c 4 x ðiÞ
1 e^c 4x ðiÞ
1
e^c 4x ðiÞ
ln
1
^c 4
(
i¼1
X n
i¼1
þ 1 n
i¼1
X n
i¼1
^c 4 x ðiÞ e^c 4x ðiÞ þ e^c 4 x ðiÞ
1
^c 4 ðe^c 4x ðiÞ
1Þ
ln
e^c 4x ðiÞ
1
^c 4
!#) 1
;
!"
ln p 4i
( ) 1=n ( !) ð 1=nÞ
^k 4 ¼
Yn Y n e^c 4x ðiÞ
1
p 4i
:
^c 4
i¼1
1 X n
n
i¼1
ln p 4i
Second, we can also estimate c and k via weighted least squares (WLS)
method (see [5,11]). For a ¼ 1; 2 and k ¼ 1; 2; 3, let
K ak ðc; kÞ ¼ Xn
W aki lnð lnð1 p ki ÞÞ ln k ln ecx ðiÞ 2
1
c
i¼1
and
K 34 ðc; kÞ ¼ Xn
i¼1
W 34i
ln p 4i
ln k ln ecx ðiÞ 2
1
;
c
where
W 1ki ¼fð1 p ki Þ lnð1 p ki Þg 2 ;

J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147 137
( )
W 2ki ¼ 3:3p ki 27:5½1 ð1 p ki Þ 0:025 2
Š
;
n
W 34i ¼
( ) 2 Xi
:
j¼1
i¼1
1
n j þ 1
Solving the normal equations as before, the WLS estimates of c and k satisfy
the following normal equations:
( !
!)
^c wak ¼
Xn ^c wak x e^c wakx ðiÞ ðiÞ þ e^c wak x ðiÞ
1 e^c wakx ðiÞ
1
W aki e^c wakx ðiÞ
ln
1
^c wak
8
><
>:
X n
i¼1
W aki
^c wak x ðiÞ e^c wakx ðiÞ þ e^c wak x ðiÞ
1
^c ak ðe^c akx ðiÞ
1Þ
2
!
6
4 lnð
lnð1
p ki ÞÞ
P n
i¼1 W P 39
n
aki lnð lnð1 p ki ÞÞ
i¼1 W aki ln e^c wak x ðiÞ 1 >=
^c wak
P
7
n
i¼1 W 5
aki
>;
1
;
8
P n
i¼1
^k W P 9
n
aki lnð lnð1 p ki ÞÞ
i¼1 W aki ln e^c wak x ðiÞ ><
1 >=
^c wak
wak ¼ exp
P n
>:
i¼1 W aki
>;
for a ¼ 1, 2 and k ¼ 1, 2, 3 and
( !
!)
^c w34 ¼
Xn ^c w34 x e^c w34x ðiÞ ðiÞ þ e^c w34 x ðiÞ
1 e^c w34x ðiÞ
1
W 34i e^c w34x ðiÞ
ln
1
^c w34
8
><
>:
i¼1
X n
i¼1
!
^c w34 x e^c w34x ðiÞ ðiÞ þ e^c w34 x ðiÞ
1
W 34i
^c ðe^c w34x ðiÞ
w34 1Þ
2
6
4
ln p 4i
P n
i¼1 W P 39
n
34i ln p 4i i¼1 W 34i ln e^c w34 x ðiÞ 1 >=
^c w34
P
7
n
i¼1 W 5
34i
>;
1
;
8
P n
i¼1
^k W P 9
n
34i ln p 4i i¼1 W 34i ln e^c w34 x ðiÞ ><
1 >=
^c w34
w34 ¼ exp
P n
>:
i¼1 W 34i
>; :

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.

3. Simulation study
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147 139
In this section, we compare the 12 estimation methods given in Section 2 in
terms of the mean squared error over the 1000 simulated samples. The 12
methods are as follows:
Method Estimates p i Weight
1 UWLSE p 1i –
2 UWLSE p 2i –
3 UWLSE p 3i –
4 UWLSE p 4i –
5 WLSE p 1i W 11i
6 WLSE p 2i W 12i
7 WLSE p 3i W 13i
8 WLSE p 1i W 21i
9 WLSE p 2i W 22i
10 WLSE p 3i W 23i
11 WLSE p 4i W 34i
12 MLE – –
Tables 1–6 list the results of complete data and the first failured-censored
data, respectively. Based on the results shown in Tables 1–6, our proposed
estimators and ML estimators are biased. For the complete data, by comparing
SMSE of these estimates, we obtain the main conclusions are: (a) the performance
of WLS estimates obtained by Method 11 in c ¼ 0:01, 0.1, 2.0, k ¼ 0:01
is better than ML estimates for n ¼ 10, 30; (b) for n ¼ 10, the WLS estimates
obtained by Methods 6–7, Method 9 and Method 11 are more accurate than
ML estimates in c ¼ 0:01, 0.1, 2.0, k ¼ 0:01; (c) for n ¼ 10, generally the
UWLS and WLS estimates are better than ML estimates in c ¼ 0:01, k ¼ 0:01,
0.02; (d) for n ¼ 10, generally the UWLS and WLS estimates are better than
ML estimates in c ¼ 0:1, k ¼ 0:02; and (e) for n ¼ 30, generally the UWLS and
WLS estimates are better than ML estimates in c ¼ 0:01, k ¼ 0:02. From (a)–
(e), it is suggested that Method 11 is useful for estimating c and k under the
complete data.
For the first failured-censored data, by comparing SMSE of these estimates,
we obtain the main conclusions are: (a) for m ¼ n ¼ 10, the WLS estimates
obtained by Method 6 and Method 9 are better than ML estimates in c ¼ 0:01,
0.1, 2.0, k ¼ 0:01; (b) for m ¼ n ¼ 10, generally the performance of UWLS and
WLS estimates is better than ML estimates in c ¼ 0:01, 0.1, k ¼ 0:01; (c) for
m ¼ n ¼ 10, except Methods 1–2 and Method 4, the UWLS and WLS estimates
are more accurate than ML estimates in c ¼ 2:0, k ¼ 0:02; (d) for
m ¼ 10, n ¼ 30, except Methods 1–3, the UWLS and WLS estimates are better

Table 1
The UWLS, WLS and ML estimates of c and k for n ¼ 10
True
value
method
c ¼ 0:01, ^c k ¼ 0:01, ^k c ¼ 0:1, ^c k ¼ 0:01, ^k c ¼ 2:0, ^c k ¼ 0:01, ^k
1 0.00849 (0.00003) 0.01076 (0.00001) 0.08342 (0.00617) 0.01451 (0.00008) 1.8175 (0.5894) 0.02028 (0.00046)
2 0.01107 (0.00003) 0.00993 (0.00001) 0.09827 (0.00863) 0.01171 (0.00005) 2.0194 (0.9584) 0.01437 (0.00031)
3 0.00975 (0.00003) 0.01038 (0.00001) 0.09143 (0.00746) 0.01295 (0.00006) 1.9260 (0.8598) 0.01651 (0.00034)
4 0.00902 (0.00003) 0.01107 (0.00001) 0.08631 (0.00663) 0.01446 (0.00008) 1.8405 (0.7915) 0.02010 (0.00047)
5 0.00860 (0.00003) 0.01072 (0.00001) 0.08820 (0.00668) 0.01306 (0.00005) 1.8410 (0.6668) 0.01743 (0.00021)
6 0.01053 (0.00003) 0.01014 (0.00001) 0.09755 (0.00827) 0.01148 (0.00003) 1.9761 (0.8937) 0.01348 (0.00011)
7 0.00962 (0.00003) 0.01046 (0.00001) 0.09371 (0.00760) 0.01211 (0.00003) 1.9106 (0.7724) 0.01511 (0.00012)
8 0.00941 (0.00003) 0.01081 (0.00001) 0.09573 (0.00811) 0.01210 (0.00004) 1.8970 (0.5641) 0.01710 (0.00022)
9 0.01028 (0.00003) 0.01030 (0.00001) 0.09825 (0.00830) 0.01135 (0.00003) 1.9836 (0.5666) 0.01338 (0.00011)
10 0.00964 (0.00003) 0.01033 (0.00008) 0.09139 (0.00745) 0.01294 (0.00006) 1.9155 (0.6334) 0.01698 (0.00034)
11 0.00970 (0.00002) 0.01103 (0.00001) 0.09591 (0.00781) 0.01218 (0.00003) 1.9473 (0.8621) 0.01524 (0.00015)
12 0.01197 (0.00003) 0.00868 (0.00005) 0.10607 (0.00969) 0.00963 (0.00003) 1.9002 (0.9065) 0.01693 (0.00041)
c ¼ 0:01, ^c k ¼ 0:02, ^k c ¼ 0:1, ^c k ¼ 0:02, ^k c ¼ 2:0, ^c k ¼ 0:02, ^k
1 0.00972 (0.00005) 0.01975 (0.00012) 0.08140 (0.00294) 0.02861 (0.00035) 1.7890 (0.4987) 0.03670 (0.00162)
2 0.01242 (0.00007) 0.01910 (0.00011) 0.20326 (0.03550) 0.01638 (0.00009) 1.9950 (0.5874) 0.02680 (0.00080)
3 0.01110 (0.00006) 0.01961 (0.00012) 0.09599 (0.00290) 0.02576 (0.00033) 1.9107 (0.5566) 0.03036 (0.00108)
4 0.01039 (0.00005) 0.02049 (0.00014) 0.08546 (0.00280) 0.03042 (0.00044) 1.8259 (0.4469) 0.03604 (0.00157)
5 0.00945 (0.00004) 0.02029 (0.00014) 0.07862 (0.00270) 0.02900 (0.00035) 1.8330 (0.4258) 0.03198 (0.00087)
6 0.01189 (0.00006) 0.01972 (0.00013) 0.10260 (0.00289) 0.02420 (0.00028) 1.9629 (0.3298) 0.02595 (0.00055)
7 0.01079 (0.00005) 0.02000 (0.00013) 0.09024 (0.00264) 0.02683 (0.00032) 1.9259 (0.3776) 0.02758 (0.00062)
8 0.00992 (0.00005) 0.02085 (0.00015) 0.08610 (0.00258) 0.02948 (0.00041) 1.9498 (0.3951) 0.02930 (0.00078)
9 0.01118 (0.00005) 0.02013 (0.00013) 0.09918 (0.00264) 0.02586 (0.00032) 1.9870 (0.4009) 0.02496 (0.00046)
10 0.01107 (0.00006) 0.01964 (0.00012) 0.09570 (0.00300) 0.02594 (0.00033) 1.9104 (0.3765) 0.03034 (0.00107)
11 0.00985 (0.00004) 0.02157 (0.00017) 0.08812 (0.00236) 0.03129 (0.00049) 1.9225 (0.3912) 0.02873 (0.00064)
12 0.01338 (0.00006) 0.01982 (0.00025) 0.11024 (0.00966) 0.02159 (0.00276) 1.8854 (0.3799) 0.02475 (0.00035)
Note. The values in parentheses are sample mean squared error (SMSE) of ^c and ^k and Ô*Õ express SMSE less than Method 12.
140 J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147

Table 2
The UWLS, WLS and ML estimates of c and k for n ¼ 30
True
value
method
c ¼ 0:01, ^c k ¼ 0:01, ^k c ¼ 0:1, ^c k ¼ 0:01, ^k c ¼ 2:0, ^c k ¼ 0:01, ^k
1 0.00837 (0.00003) 0.01087 (0.00001) 0.08422 (0.00630) 0.01440 (0.00008) 1.7930 (0.8687) 0.02093 (0.00051)
2 0.10790 (0.00003) 0.01012 (0.00001) 0.09921 (0.00879) 0.01164 (0.00005) 2.0061 (0.9555) 0.01422 (0.00022)
3 0.00962 (0.00003) 0.01053 (0.00001) 0.09234 (0.00760) 0.01288 (0.00006) 1.9088 (0.8121) 0.01697 (0.00032)
4 0.00892 (0.00003) 0.01120 (0.00001) 0.08730 (0.00677) 0.01436 (0.00008) 1.8271 (0.7256) 0.02048 (0.00049)
5 0.00837 (0.00003) 0.01089 (0.00001) 0.08836 (0.00669) 0.01314 (0.00004) 1.8211 (0.6249) 0.01784 (0.00021)
6 0.01013 (0.00003) 0.01037 (0.00001) 0.09764 (0.00827) 0.01156 (0.00003) 1.9523 (0.9625) 0.01448 (0.00014)
7 0.00938 (0.00003) 0.01060 (0.00001) 0.09386 (0.00760) 0.01218 (0.00003) 1.8902 (0.7805) 0.01568 (0.00016)
8 0.00918 (0.00003) 0.01093 (0.00001) 0.09537 (0.00803) 0.01228 (0.00004) 1.8698 (0.5978) 0.01778 (0.00024)
9 0.01006 (0.00003) 0.01049 (0.00001) 0.09808 (0.00826) 0.01148 (0.00003) 1.9715 (0.9474) 0.01365 (0.00011)
10 0.00955 (0.00003) 0.01058 (0.00001) 0.09231 (0.00760) 0.01287 (0.00006) 1.9052 (0.7459) 0.01694 (0.00027)
11 0.00933 (0.00002) 0.01123 (0.00001) 0.09570 (0.00778) 0.01235 (0.00003) 1.9345 (0.7954) 0.01508 (0.00013)
12 0.01206 (0.00002) 0.00863 (0.00002) 0.10620 (0.00971) 0.00883 (0.00003) 1.9166 (0.9012) 0.01493 (0.00035)
c ¼ 0:01, ^c k ¼ 0:02, ^k c ¼ 0:1, ^c k ¼ 0:02, ^k c ¼ 2:0, ^c k ¼ 0:02, ^k
1 0.01004 (0.00005) 0.01985 (0.00012) 0.08460 (0.00666) 0.02580 (0.00039) 1.7850 (0.5608) 0.03770 (0.00175)
2 0.01298 (0.00008) 0.01901 (0.00011) 0.10156 (0.00962) 0.02186 (0.00027) 2.0088 (0.4571) 0.02664 (0.00082)
3 0.01149 (0.00006) 0.01950 (0.00012) 0.09451 (0.00827) 0.02331 (0.00030) 1.9069 (0.5418) 0.03122 (0.00116)
4 0.01082 (0.00006) 0.02048 (0.00014) 0.08836 (0.00726) 0.02588 (0.00040) 1.8219 (0.4855) 0.03705 (0.00170)
5 0.00983 (0.00005) 0.02007 (0.00013) 0.08895 (0.00713) 0.02433 (0.00032) 1.8322 (0.4288) 0.03232 (0.00090)
6 0.01207 (0.00007) 0.01952 (0.00012) 0.10002 (0.00908) 0.02192 (0.00025) 1.9821 (0.4132) 0.02576 (0.00058)
7 0.01159 (0.00006) 0.01758 (0.00013) 0.09551 (0.00825) 0.02285 (0.00027) 1.9264 (0.6287) 0.02805 (0.00069)
8 0.01033 (0.00005) 0.01942 (0.00012) 0.09485 (0.00819) 0.02334 (0.00027) 1.9321 (0.6386) 0.02980 (0.00083)
9 0.01020 (0.00004) 0.02136 (0.00016) 0.10027 (0.00900) 0.02189 (0.00024) 1.9890 (0.4291) 0.02536 (0.00053)
10 0.01039 (0.00006) 0.02069 (0.00014) 0.09449 (0.00828) 0.02330 (0.00030) 1.9066 (0.5945) 0.03119 (0.00115)
11 0.01105 (0.00006) 0.01982 (0.00012) 0.09694 (0.00825) 0.02344 (0.00027) 1.9122 (0.7068) 0.02933 (0.00066)
12 0.01401 (0.00007) 0.02023 (0.00418) 0.10987 (0.01069) 0.02008 (0.00020) 1.8954 (0.5122) 0.02386 (0.00125)
Note. The values in parentheses are sample mean squared error (SMSE) of ^c and ^k and Ô*Õ express SMSE less than Method 12.
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147 141

Table 3
The UWLS, WLS and ML estimates of c and k for m ¼ 10, n ¼ 10
True
value
method
c ¼ 0:01, ^c k ¼ 0:01, ^k c ¼ 0:1, ^c k ¼ 0:01, ^k c ¼ 2:0, ^c k ¼ 0:01, ^k
1 0.04852 (0.00330) 0.00760 (0.00002) 0.10185 (0.01523) 0.01002 (0.00002) 1.6509 (0.7125) 0.01916 (0.00033)
2 0.06270 (0.00583) 0.00745 (0.00004) 0.14774 (0.02963) 0.00922 (0.00003) 2.0926 (0.7657) 0.01413 (0.00022)
3 0.05983 (0.00494) 0.00741 (0.00002) 0.12453 (0.02174) 0.00951 (0.00002) 1.8922 (0.6788) 0.01603 (0.00026)
4 0.05601 (0.00426) 0.00813 (0.00002) 0.11167 (0.01791) 0.01073 (0.00002) 1.7396 (0.6707) 0.01973 (0.00038)
5 0.04327 (0.00293) 0.00792 (0.00002) 0.09508 (0.01307) 0.01020 (0.00002) 1.5860 (0.6396) 0.01944 (0.00030)
6 0.06003 (0.00543) 0.00780 (0.00002) 0.13019 (0.02247) 0.00960 (0.00002) 1.9579 (0.5300) 0.01479 (0.00020)
7 0.04991 (0.00392) 0.00784 (0.00002) 0.11653 (0.01847) 0.00985 (0.00002) 1.8055 (0.5330) 0.01648 (0.00022)
8 0.05072 (0.00433) 0.00739 (0.00002) 0.09787 (0.01288) 0.01085 (0.00002) 1.7376 (0.4510) 0.01962 (0.00034)
9 0.05261 (0.00430) 0.00810 (0.00002) 0.12501 (0.02118) 0.00996 (0.00002) 1.9599 (0.4629) 0.01465 (0.00019)
10 0.05995 (0.00496) 0.00740 (0.00002) 0.12446 (0.02177) 0.00950 (0.00002) 1.8899 (0.6609) 0.01600 (0.00025)
11 0.05596 (0.00425) 0.00809 (0.00002) 0.10484 (0.01581) 0.01162 (0.00003) 1.8824 (0.5654) 0.01785 (0.00031)
12 0.07250 (0.00745) 0.00656 (0.00004) 0.13765 (0.02303) 0.00773 (0.00091) 1.7767 (0.8457) 0.01966 (0.00021)
c ¼ 0:01, ^c k ¼ 0:02, ^k c ¼ 0:1, ^c k ¼ 0:02, ^k c ¼ 2:0, ^c k ¼ 0:02, ^k
1 0.09970 (0.01660) 0.01427 (0.00007) 0.14746 (0.01857) 0.01666 (0.00006) 1.6248 (0.5007) 0.06124 (0.00610)
2 0.13263 (0.03015) 0.01407 (0.00008) 0.20322 (0.03550) 0.01638 (0.00009) 2.1286 (0.9220) 0.02497 (0.00050)
3 0.11847 (0.02280) 0.01445 (0.00008) 0.17327 (0.02272) 0.01677 (0.00007) 1.8284 (0.4664) 0.04996 (0.00478)
4 0.10560 (0.01838) 0.01608 (0.00007) 0.16110 (0.01936) 0.01848 (0.00007) 1.7102 (0.4756) 0.06049 (0.00640)
5 0.10093 (0.01703) 0.01485 (0.00007) 0.13605 (0.01652) 0.01800 (0.00006) 1.6052 (0.4363) 0.05903 (0.00490)
6 0.11640 (0.02291) 0.01516 (0.00007) 0.17993 (0.02876) 0.01755 (0.00008) 1.9063 (0.2940) 0.03777 (0.00199)
7 0.10976 (0.02055) 0.01487 (0.00007) 0.16093 (0.02145) 0.01766 (0.00008) 1.7556 (0.3227) 0.04613 (0.00271)
8 0.08593 (0.01096) 0.01633 (0.00006) 0.13608 (0.01135) 0.01900 (0.00006) 1.9563 (0.3035) 0.03605 (0.00199)
9 0.10947 (0.02259) 0.01536 (0.00007) 0.17023 (0.02761) 0.01817 (0.00008) 1.8855 (0.5413) 0.03825 (0.00254)
10 0.11648 (0.02211) 0.01442 (0.00007) 0.17707 (0.02362) 0.01670 (0.00008) 1.8295 (0.4590) 0.04910 (0.00442)
11 0.10435 (0.02020) 0.01669 (0.00007) 0.15260 (0.02238) 0.01979 (0.00008) 1.7849 (0.3257) 0.05092 (0.00331)
12 0.09857 (0.01689) 0.02968 (0.00122) 0.11024 (0.00966) 0.02159 (0.00276) 1.7858 (0.4688) 0.05685 (0.00587)
Note. The values in parentheses are sample mean squared error (SMSE) of ^c and ^k and Ô*Õ express SMSE less than Method 12.
142 J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147

Table 4
The UWLS, WLS and ML estimates of c and k for m ¼ 10, n ¼ 30
True
value
method
c ¼ 0:01, ^c k ¼ 0:01, ^k c ¼ 0:1, ^c k ¼ 0:01, ^k c ¼ 2:0, ^c k ¼ 0:01, ^k
1 0.12498 (0.02916) 0.02231 (0.00024) 0.21182 (0.05235) 0.02374 (0.00028) 1.6616 (1.0561) 0.04366 (0.00194)
2 0.17748 (0.05726) 0.02165 (0.00024) 0.27975 (0.08744) 0.02297 (0.00029) 2.1641 (1.2438) 0.03536 (0.00139)
3 0.15482 (0.04213) 0.02203 (0.00030) 0.24466 (0.06722) 0.02368 (0.00031) 1.9169 (1.0140) 0.03884 (0.00156)
4 0.13444 (0.03400) 0.02410 (0.00030) 0.22383 (0.05646) 0.02523 (0.00035) 1.7149 (0.9252) 0.04608 (0.00216)
5 0.14243 (0.03706) 0.02237 (0.00025) 0.19104 (0.03676) 0.02463 (0.00031) 1.5993 (0.9509) 0.04487 (0.00196)
6 0.15687 (0.05748) 0.02278 (0.00028) 0.24524 (0.06484) 0.02457 (0.00033) 2.0068 (0.8751) 0.03749 (0.00145)
7 0.16454 (0.05346) 0.02270 (0.00026) 0.22254 (0.05444) 0.02486 (0.00034) 1.7870 (0.8540) 0.04076 (0.00159)
8 0.13730 (0.02728) 0.02375 (0.00026) 0.18841 (0.03863) 0.02726 (0.00043) 1.8376 (0.6266) 0.04481 (0.00161)
9 0.16120 (0.05008) 0.02315 (0.00028) 0.23236 (0.06028) 0.02553 (0.00036) 2.0356 (0.7844) 0.03627 (0.00136)
10 0.15756 (0.04412) 0.02174 (0.00023) 0.24456 (0.06775) 0.02361 (0.00030) 1.8744 (0.9017) 0.03938 (0.00158)
11 0.13941 (0.03891) 0.02599 (0.00039) 0.20372 (0.04678) 0.02848 (0.00049) 1.5885 (0.5526) 0.04683 (0.00186)
12 0.09990 (0.01699) 0.03139 (0.00416) 0.08483 (0.0100) 0.01801 (0.00073) 1.8946 (1.0089) 0.04010 (0.00219)
c ¼ 0:01, ^c k ¼ 0:02, ^k c ¼ 0:1, ^c k ¼ 0:02, ^k c ¼ 2:0, ^c k ¼ 0:02, ^k
1 0.31994 (0.20926) 0.04295 (0.00086) 0.33734 (0.16769) 0.04502 (0.00094) 1.7073 (1.0609) 0.07517 (0.00454)
2 0.37894 (0.27150) 0.04328 (0.00098) 0.43907 (0.27645) 0.04501 (0.00109) 2.2012 (1.2478) 0.06806 (0.00392)
3 0.33716 (0.24223) 0.04484 (0.00108) 0.38326 (0.19587) 0.04676 (0.00129) 2.0267 (1.1409) 0.06878 (0.00399)
4 0.32309 (0.22103) 0.04814 (0.00125) 0.34234 (0.17261) 0.05016 (0.00138) 1.8608 (1.1185) 0.07986 (0.00558)
5 0.30241 (0.21794) 0.04457 (0.00099) 0.33130 (0.15508) 0.04800 (0.00120) 1.5958 (1.0209) 0.07722 (0.00456)
6 0.33629 (0.21621) 0.04691 (0.00121) 0.39209 (0.22100) 0.04904 (0.00137) 1.8068 (1.0586) 0.08036 (0.00576)
7 0.33059 (0.23176) 0.04529 (0.00104) 0.34958 (0.17269) 0.04746 (0.00123) 1.8196 (0.9108) 0.07591 (0.00464)
8 0.31525 (0.17606) 0.04942 (0.00132) 0.38639 (0.20854) 0.05118 (0.00149) 1.7541 (1.2386) 0.08299 (0.00628)
9 0.33750 (0.23450) 0.04664 (0.00116) 0.37508 (0.19942) 0.05005 (0.00148) 1.7744 (0.7914) 0.07802 (0.00488)
10 0.35060 (0.26211) 0.04452 (0.00107) 0.40520 (0.21520) 0.04571 (0.00117) 1.9727 (1.0945) 0.06802 (0.00384)
11 0.28843 (0.16646) 0.05238 (0.00163) 0.33549 (0.16274) 0.05502 (0.00182) 1.8865 (0.8574) 0.07999 (0.00578)
12 0.06127 (0.01155) 0.02826 (0.00121) 0.05173 (0.01058) 0.08380 (0.58168) 1.8422 (1.0958) 0.01854 (0.00864)
Note. The values in parentheses are sample mean squared error (SMSE) of ^c and ^k and Ô*Õ express SMSE less than Method 12.
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147 143

Table 5
The UWLS, WLS and ML estimates of c and k for m ¼ 30, n ¼ 10
True
value
method
c ¼ 0:01, ^c k ¼ 0:01, ^k c ¼ 0:1, ^c k ¼ 0:01, ^k c ¼ 2:0, ^c k ¼ 0:01, ^k
1 0.02710 (0.00083) 0.00852 (0.00001) 0.08408 (0.00828) 0.01084 (0.00001) 1.7177 (0.1401) 0.01537 (0.00011)
2 0.03325 (0.00116) 0.01859 (0.00076) 0.11010 (0.01348) 0.00998 (0.00001) 1.9839 (0.1408) 0.01199 (0.00006)
3 0.03001 (0.00099) 0.00848 (0.00001) 0.09822 (0.01095) 0.01035 (0.00001) 1.8621 (0.0662) 0.01344 (0.00008)
4 0.02889 (0.00093) 0.00885 (0.00001) 0.09119 (0.00950) 0.01101 (0.00001) 1.7667 (0.3134) 0.01527 (0.00011)
5 0.02597 (0.00071) 0.00875 (0.00001) 0.08556 (0.00820) 0.01076 (0.00001) 1.7934 (0.3427) 0.01379 (0.00006)
6 0.03141 (0.00104) 0.00873 (0.00001) 0.10360 (0.01177) 0.01028 (0.00001) 1.9514 (0.0445) 0.01191 (0.00004)
7 0.02846 (0.00086) 0.00875 (0.00001) 0.09642 (0.01026) 0.01047 (0.00001) 1.8905 (0.2056) 0.01258 (0.00005)
8 0.02731 (0.00091) 0.00898 (0.00001) 0.09506 (0.01024) 0.01082 (0.00001) 1.9157 (0.2736) 0.01280 (0.00005)
9 0.02873 (0.00083) 0.00890 (0.00001) 0.10220 (0.01113) 0.01042 (0.00001) 1.9595 (0.0536) 0.01180 (0.00004)
10 0.03025 (0.00104) 0.00848 (0.00001) 0.09815 (0.01097) 0.01038 (0.00001) 1.8617 (0.0614) 0.01343 (0.00008)
11 0.02461 (0.00618) 0.00932 (0.00001) 0.09604 (0.00968) 0.01115 (0.00001) 1.9146 (0.1403) 0.01255 (0.00004)
12 0.03002 (0.00099) 0.00908 (0.00388) 0.12221 (0.01504) 0.00912 (0.00002) 1.9024 (0.2102) 0.01283 (0.00008)
c ¼ 0:01, ^c k ¼ 0:02, ^k c ¼ 0:1, ^c k ¼ 0:02, ^k c ¼ 2:0, ^c k ¼ 0:02, ^k
1 0.01004 (0.00005) 0.01985 (0.00012) 0.08460 (0.00666) 0.02580 (0.00039) 1.7850 (0.5608) 0.03770 (0.00175)
2 0.01298 (0.00008) 0.01901 (0.00011) 0.10156 (0.00962) 0.02186 (0.00027) 2.0088 (0.4571) 0.02664 (0.00082)
3 0.01149 (0.00006) 0.01950 (0.00012) 0.09451 (0.00827) 0.02331 (0.00030) 1.9069 (0.5418) 0.03122 (0.00116)
4 0.01082 (0.00006) 0.02048 (0.00014) 0.08836 (0.00726) 0.02588 (0.00040) 1.8219 (0.4855) 0.03705 (0.00170)
5 0.00983 (0.00005) 0.02007 (0.00013) 0.08895 (0.00713) 0.02433 (0.00032) 1.8322 (0.4288) 0.03232 (0.00090)
6 0.01207 (0.00007) 0.01952 (0.00012) 0.10002 (0.00908) 0.02192 (0.00025) 1.9821 (0.4132) 0.02576 (0.00058)
7 0.01159 (0.00006) 0.01758 (0.00013) 0.09551 (0.00825) 0.02285 (0.00027) 1.9264 (0.6287) 0.02805 (0.00069)
8 0.01033 (0.00005) 0.01942 (0.00012) 0.09485 (0.00819) 0.02334 (0.00027) 1.9321 (0.6386) 0.02980 (0.00083)
9 0.01020 (0.00004) 0.02136 (0.00016) 0.10027 (0.00900) 0.02189 (0.00024) 1.9890 (0.4291) 0.02536 (0.00053)
10 0.01039 (0.00006) 0.02069 (0.00014) 0.09449 (0.00828) 0.02330 (0.00030) 1.9066 (0.5945) 0.03119 (0.00115)
11 0.01105 (0.00006) 0.01982 (0.00012) 0.09694 (0.00825) 0.02344 (0.00027) 1.9122 (0.7068) 0.02933 (0.00066)
12 0.01401 (0.00007) 0.02023 (0.00418) 0.10987 (0.01069) 0.02008 (0.00020) 1.8954 (0.5122) 0.02386 (0.00125)
Note. The values in parentheses are sample mean squared error (SMSE) of ^c and ^k and Ô*Õ express SMSE less than Method 12.
144 J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147

Table 6
The UWLS, WLS and ML estimates of c and k for m ¼ 30, n ¼ 30
True
value
method
c ¼ 0:01, ^c k ¼ 0:01, ^k c ¼ 0:1, ^c k ¼ 0:01, ^k c ¼ 2:0, ^c k ¼ 0:01, ^k
1 0.07070 (0.00765) 0.02505 (0.00026) 0.12255 (0.02113) 0.02807 (0.00038) 1.6523 (0.1005) 0.04146 (0.00141)
2 0.08837 (0.00927) 0.02469 (0.00026) 0.15620 (0.03299) 0.02736 (0.00036) 1.9855 (0.3238) 0.03401 (0.00090)
3 0.08021 (0.00962) 0.02479 (0.00026) 0.13711 (0.02650) 0.02784 (0.00037) 1.8384 (0.0492) 0.03707 (0.00108)
4 0.07410 (0.00812) 0.02618 (0.00030) 0.12839 (0.02336) 0.02921 (0.00043) 1.7218 (0.3381) 0.04141 (0.00142)
5 0.06721 (0.00662) 0.02559 (0.00028) 0.11644 (0.01898) 0.02858 (0.00039) 1.7589 (0.3585) 0.03775 (0.00104)
6 0.08205 (0.00995) 0.02549 (0.00028) 0.14350 (0.02813) 0.02823 (0.00038) 1.9649 (0.1467) 0.03363 (0.00080)
7 0.07771 (0.00865) 0.02539 (0.00027) 0.13003 (0.02372) 0.02856 (0.00039) 1.8826 (0.2017) 0.03518 (0.00088)
8 0.07119 (0.00792) 0.02636 (0.00031) 0.11006 (0.01025) 0.02688 (0.00030) 1.9285 (0.0286) 0.03494 (0.00085)
9 0.07568 (0.00824) 0.02569 (0.00028) 0.13476 (0.02423) 0.02879 (0.00041) 1.9467 (0.0398) 0.03473 (0.00086)
10 0.08053 (0.00970) 0.02478 (0.00026) 0.13647 (0.02534) 0.02782 (0.00037) 1.8397 (0.0564) 0.03700 (0.00108)
11 0.06355 (0.00613) 0.02729 (0.00034) 0.12046 (0.01985) 0.03059 (0.00048) 1.8408 (0.5426) 0.03758 (0.00098)
12 0.07790 (0.00927) 0.02332 (0.00241) 0.13135 (0.02174) 0.02921 (0.00265) 1.7855 (0.2101) 0.03561 (0.00545)
c ¼ 0:01, ^c k ¼ 0:02, ^k c ¼ 0:1, ^c k ¼ 0:02, ^k c ¼ 2:0, ^c k ¼ 0:02, ^k
1 0.13062 (0.02762) 0.04956 (0.00170) 0.08106 (0.00610) 0.02680 (0.00042) 1.6299 (0.1508) 0.07580 (0.00531)
2 0.16327 (0.04236) 0.04972 (0.00174) 0.09788 (0.00889) 0.02262 (0.00028) 2.0093 (0.2152) 0.06577 (0.00408)
3 0.14950 (0.03574) 0.04947 (0.00171) 0.09088 (0.00766) 0.02449 (0.00034) 1.8261 (0.1715) 0.07067 (0.00466)
4 0.14173 (0.03129) 0.05114 (0.00184) 0.08439 (0.00663) 0.02705 (0.00044) 1.7083 (0.4253) 0.07666 (0.00550)
5 0.12624 (0.02500) 0.05107 (0.00183) 0.08596 (0.00657) 0.02488 (0.00032) 1.6845 (0.2483) 0.07323 (0.00480)
6 0.15096 (0.03533) 0.05168 (0.00190) 0.09724 (0.00847) 0.02226 (0.00023) 1.9278 (0.1171) 0.06673 (0.00395)
7 0.13699 (0.02981) 0.05190 (0.00191) 0.09265 (0.00766) 0.02329 (0.00027) 1.8587 (0.1245) 0.07121 (0.00366)
8 0.13385 (0.02925) 0.05373 (0.00208) 0.09470 (0.00808) 0.02328 (0.00026) 1.7083 (0.1253) 0.07666 (0.00550)
9 0.13650 (0.02987) 0.05275 (0.00199) 0.09780 (0.00843) 0.02220 (0.00023) 1.8771 (0.0501) 0.06947 (0.00424)
10 0.15190 (0.03677) 0.04943 (0.00171) 0.08999 (0.00755) 0.02460 (0.00035) 1.9733 (0.1918) 0.06508 (0.00371)
11 0.12643 (0.02632) 0.05466 (0.00215) 0.09550 (0.00791) 0.02360 (0.00027) 1.8148 (0.1989) 0.06998 (0.00439)
12 0.09648 (0.01589) 0.02182 (0.01922) 0.10775 (0.01018) 0.02358 (0.00028) 1.7855 (0.1441) 0.05620 (0.00652)
Note. The values in parentheses are sample mean squared error (SMSE) of ^c and ^k and Ô*Õ express SMSE less than Method 12.
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147 145

146 J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
than ML estimates in c ¼ 2:0, k ¼ 0:01; (e) for m ¼ 10, n ¼ 30, the estimates
obtained by Method 1, Methods 5–7 and Methods 9–11 are better than ML
estimates in c ¼ 2:0, k ¼ 0:02; (f) for m ¼ 30, n ¼ 10, the performance of
UWLS and WLS estimates is better than ML estimates in c ¼ 0:1, k ¼ 0:01 and
c ¼ 0:01, k ¼ 0:02, respectively; (g) for m ¼ 30, n ¼ 10, the UWLS estimates
obtained by Method 3 and the WLS estimates obtained by Method 7 and
Method 9 are better than ML estimates in c ¼ 0:01, 0.1, 2.0, k ¼ 0:01; (h) for
m ¼ 30, n ¼ 30, the WLS estimates obtained by Method 8 are better than ML
estimates in c ¼ 0:01, 0.1, 2.0, k ¼ 0:01; (i) for m ¼ 30, n ¼ 30, generally the
UWLS and WLS estimates are better than ML estimates in c ¼ 0:01, k ¼ 0:01;
(j) for m ¼ 30, n ¼ 30, the WLS estimates obtained by Methods 6–10 are better
than ML estimates in c ¼ 2:0, k ¼ 0:01 and (k) for m ¼ 30, n ¼ 30, the WLS
estimates obtained by Methods 6–9 are better than ML estimates in c ¼ 0:1,
2.0, k ¼ 0:02. In addition, from the above results, it is suggested that Method 9
is useful for estimating c and k under the first failured-censored data.
4. Concluding remarks
In summary, least squares methods often provide simple and fairly effective
ways of obtaining estimates with complete data and the first failured-censored
data. The procedures described were based on transform of F ðxÞ, which is the
Gompertz cumulative distribution function. Results from simulation studies
illustrate the performance of the WLS estimates is acceptable.
Acknowledgement
This research was partially supported by the National Science Council,
ROC (Plan No. NSC 89-2118-M-032-013).
References
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[4] R.B. DÕAgostino, M.A. Stephens, Goodness-of-fit Techniques, Marcel Dekker, New York,
1986.
[5] B. Faucher, W.R. Tyson, On the determination of Weibull parameters, Journal of Materials
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