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
b
Department of Statistics, Tamkang University, Tamsui, Taipei 25137, Taiwan, ROC 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: [email protected] (J.-W. Wu).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.086
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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 f ðxÞ ¼ kecx exp
k ðecx 1Þ ; c
x > 0;
ð1Þ
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
135
where c > 0 and k > 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 ¼ 1 exp ðecx 1Þ ð2Þ c and F ðxÞ satisfies lnf lnð1 F ðxÞÞg ¼ ln k þ ln
ecx 1 : c
ð3Þ
Now suppose that X1 ; X2 ; . . . ; Xn is a sample of size n from a Gompertz distribution with parameters c and k, and that Xð1Þ < Xð2Þ < < XðnÞ are the order statistics. For observed ordered observations xð1Þ < xð2Þ < < xðnÞ , it follows from (3) that cx e ðiÞ 1 lnf lnð1 F ðxðiÞ ÞÞg ¼ ln k þ ln ; i ¼ 1; 2; . . . ; n: ð4Þ c Let the empirical distribution function of F ðxÞ be denoted by Fb ðxÞ, where b F ðxðiÞ Þ equals i=n. In order to avoid lnð0Þ in (4), we modify Fb ðxðiÞ Þ to be pi ¼
id 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 p1i ¼ i=ðn Pi þ 1Þ, p2i ¼ ði 0:5Þ=n, p3i ¼ ði 0:3Þ=ðn þ 0:4Þ. Alternatively, we use p4i ¼ j¼1 f1=ðn j þ 1Þg to estimate lnð1 F ðxðiÞ ÞÞ in (4) (see [12,16]). First, we estimate c and k by unweighted least squares (UWLS) method. Let cx 2 n X e ðiÞ 1 lnð lnð1 pki ÞÞ ln k ln ; k ¼ 1; 2; 3 Gk ðc; kÞ ¼ c i¼1 and G4 ðc; kÞ ¼
n X
ln p4i ln k ln
i¼1
We solve the normal equations 8 oG ðc; kÞ > < k ¼ 0; oc > : oGk ðc; kÞ ¼ 0 ok
ecxðiÞ 1 c
2 :
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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: ( !) ! n X ^ck xðiÞ e^ck xðiÞ þ e^ck xðiÞ 1 e^ck xðiÞ 1 ^ck ¼ ln ^ck e^ck xðiÞ 1 i¼1 ( !" n X ^ck xðiÞ e^ck xðiÞ þ e^ck xðiÞ 1 lnð lnð1 pki ÞÞ ^ck ðe^ck xðiÞ 1Þ i¼1 !#)1 n n 1X 1X e^ck xðiÞ 1 lnð lnð1 pki ÞÞ þ ln ; ^ck n i¼1 n i¼1 ( k^k ¼
n Y ð lnð1 pki ÞÞ
)1=n (
i¼1
n Y
e^ck xðiÞ 1 ^ck
i¼1
!)ð1=nÞ
and ( ^c4 ¼
n X
^c4 xðiÞ e^c4 xðiÞ þ e^c4 xðiÞ 1 e^c4 xðiÞ 1
i¼1
(
!
n X
^c4 xðiÞ e^c4 xðiÞ þ e^c4 xðiÞ 1 ^c4 ðe^c4 xðiÞ 1Þ i¼1 !#)1 n 1X e^c4 xðiÞ 1 þ ln ; ^c4 n i¼1 ( k^4 ¼
n Y
)1=n ( p4i
i¼1
n Y i¼1
e^c4 xðiÞ 1 ^c4
e^c4 xðiÞ 1 ^c4
ln !"
ln p4i
!)
n 1X ln p4i n i¼1
!)ð1=nÞ :
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 cxðiÞ 2 n X e 1 Kak ðc; kÞ ¼ Waki lnð lnð1 pki ÞÞ ln k ln c i¼1 and K34 ðc; kÞ ¼
n X
W34i ln p4i ln k ln
i¼1
where W1ki ¼ fð1 pki Þ lnð1 pki Þg2 ;
ecxðiÞ 1 c
2 ;
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
( W2ki ¼ ( W34i ¼
3:3pki 27:5½1 ð1 pki Þ n i X j¼1
1 njþ1
0:025
137
)2 ;
)2 :
Solving the normal equations as before, the WLS estimates of c and k satisfy the following normal equations: ! ( !) n X ^cwak xðiÞ e^cwak xðiÞ þ e^cwak xðiÞ 1 e^cwak xðiÞ 1 ^cwak ¼ Waki ln ^cwak e^cwak xðiÞ 1 i¼1 8 2 ! > n <X ^cwak xðiÞ ^cwak xðiÞ ^cwak xðiÞ e þe 1 6 Waki 4 lnð lnð1 pki ÞÞ ^cak xðiÞ > ^cak ðe 1Þ : i¼1 ^c x 391 Pn > e wak ðiÞ 1 = W lnð lnð1 p ÞÞ W ln ki i¼1 aki i¼1 aki ^cwak 7 Pn ; 5 > ; i¼1 Waki
Pn
k^wak
8 ^c x 9 P P > = < ni¼1 Waki lnð lnð1 pki ÞÞ ni¼1 Waki ln e wak^c ðiÞ 1 > wak Pn ¼ exp > > ; : i¼1 Waki
for a ¼ 1, 2 and k ¼ 1, 2, 3 and ! ( n X ^cw34 xðiÞ e^cw34 xðiÞ þ e^cw34 xðiÞ 1 ^cw34 ¼ ln W34i e^cw34 xðiÞ 1 i¼1
8 > n <X > :
W34i
i¼1
2 6 4 ln p4i
k^w34
^cw34 xðiÞ e^cw34 xðiÞ þ e^cw34 xðiÞ 1 ^cw34 ðe^cw34 xðiÞ 1Þ
e^cw34 xðiÞ 1 ^cw34
!)
!
^c x 391 Pn > e w34 ðiÞ 1 = W ln p W ln 34i 4i 34i i¼1 i¼1 ^cw34 7 Pn ; 5 > ; i¼1 W34i
Pn
8 ^c x 9 P P > = < ni¼1 W34i ln p4i ni¼1 W34i ln e w34^c ðiÞ 1 > w34 Pn : ¼ exp > > ; : i¼1 W34i
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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 ( )1 )( n n n n X X X X ^cxðiÞ ^cxðiÞ ^cxðiÞ ^c ¼ n e 1 xðiÞ e 1 n xðiÞ e ; i¼1
i¼1
i¼1
i¼1
ð5Þ ( k^ ¼
^c
n X i¼1
)( xðiÞ
n n X 1X e^cxðiÞ 1 xðiÞ e^cxðiÞ ^c i¼1 i¼1
)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 ^c0 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 Xð1Þ is k cx cx f ðx; c; k Þ ¼ k e exp ðe 1Þ ; c where k ¼ nk. The corresponding c.d.f. is k F ðxÞ ¼ 1 exp ðecx 1Þ : c
ð7Þ
ð8Þ
Suppose Xð1Þ1 ; Xð1Þ2 ; . . . ; Xð1Þm denote the set of first-order statistics of m samples of size n from (1) and let Yð1Þ < Yð2Þ < < YðmÞ be the corresponding order statistics. Clearly, Xð1Þ1 ; Xð1Þ2 ; . . . ; Xð1Þm can also be considered as a random sample from (7). Then F ðxÞ in (8) satisfies cx e 1 lnf lnð1 F ðxÞÞg ¼ ln n þ ln k þ ln : ð9Þ c For observed ordered observations yð1Þ < yð2Þ < < yðmÞ , (9) can be rewritten as cy e ðiÞ 1 lnf lnð1 F ðyðiÞ ÞÞg ¼ ln n þ ln k þ ln ; 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.
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
139
3. Simulation study 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
pi
Weight
1 2 3 4 5 6 7 8 9 10 11 12
UWLSE UWLSE UWLSE UWLSE WLSE WLSE WLSE WLSE WLSE WLSE WLSE MLE
p1i p2i p3i p4i p1i p2i p3i p1i p2i p3i p4i –
– – – – W11i W12i W13i W21i W22i W23i W34i –
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
140
Table 1 The UWLS, WLS and ML estimates of c and k for n ¼ 10 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 2 3 4 5 6 7 8 9 10 11 12
0.00849 0.01107 0.00975 0.00902 0.00860 0.01053 0.00962 0.00941 0.01028 0.00964 0.00970 0.01197
0.01076 (0.00001) 0.00993 (0.00001) 0.01038 (0.00001) 0.01107 (0.00001) 0.01072 (0.00001) 0.01014 (0.00001) 0.01046 (0.00001) 0.01081 (0.00001) 0.01030 (0.00001) 0.01033 (0.00008) 0.01103 (0.00001) 0.00868 (0.00005)
0.08342 0.09827 0.09143 0.08631 0.08820 0.09755 0.09371 0.09573 0.09825 0.09139 0.09591 0.10607
0.01451 (0.00008) 0.01171 (0.00005) 0.01295 (0.00006) 0.01446 (0.00008) 0.01306 (0.00005) 0.01148 (0.00003) 0.01211 (0.00003) 0.01210 (0.00004) 0.01135 (0.00003) 0.01294 (0.00006) 0.01218 (0.00003) 0.00963 (0.00003)
1.8175 2.0194 1.9260 1.8405 1.8410 1.9761 1.9106 1.8970 1.9836 1.9155 1.9473 1.9002
0.02028 (0.00046) 0.01437 (0.00031) 0.01651 (0.00034) 0.02010 (0.00047) 0.01743 (0.00021) 0.01348 (0.00011) 0.01511 (0.00012) 0.01710 (0.00022) 0.01338 (0.00011) 0.01698 (0.00034) 0.01524 (0.00015) 0.01693 (0.00041)
k ¼ 0:02, k^
c ¼ 0:1, ^c
k ¼ 0:02, k^
c ¼ 2:0, ^c
0.01975 (0.00012) 0.01910 (0.00011) 0.01961 (0.00012) 0.02049 (0.00014) 0.02029 (0.00014) 0.01972 (0.00013) 0.02000 (0.00013) 0.02085 (0.00015) 0.02013 (0.00013) 0.01964 (0.00012) 0.02157 (0.00017) 0.01982 (0.00025)
0.08140 0.20326 0.09599 0.08546 0.07862 0.10260 0.09024 0.08610 0.09918 0.09570 0.08812 0.11024
(0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00002) (0.00003)
c ¼ 0:01, ^c
(0.00617) (0.00863) (0.00746) (0.00663) (0.00668) (0.00827) (0.00760) (0.00811) (0.00830) (0.00745) (0.00781) (0.00969)
(0.5894) (0.9584) (0.8598) (0.7915) (0.6668) (0.8937) (0.7724) (0.5641) (0.5666) (0.6334) (0.8621) (0.9065)
k ¼ 0:02, k^
0.02861 (0.00035) 1.7890 (0.4987) 0.03670 0.01638 (0.00009) 1.9950 (0.5874) 0.02680 0.02576 (0.00033) 1.9107 (0.5566) 0.03036 0.03042 (0.00044) 1.8259 (0.4469) 0.03604 0.02900 (0.00035) 1.8330 (0.4258) 0.03198 0.02420 (0.00028) 1.9629 (0.3298) 0.02595 0.02683 (0.00032) 1.9259 (0.3776) 0.02758 0.02948 (0.00041) 1.9498 (0.3951) 0.02930 0.02586 (0.00032) 1.9870 (0.4009) 0.02496 0.02594 (0.00033) 1.9104 (0.3765) 0.03034 0.03129 (0.00049) 1.9225 (0.3912) 0.02873 0.02159 (0.00276) 1.8854 (0.3799) 0.02475 Note. The values in parentheses are sample mean squared error (SMSE) of ^c and k^ and Ô*Õ express SMSE less than Method 12. 1 2 3 4 5 6 7 8 9 10 11 12
0.00972 0.01242 0.01110 0.01039 0.00945 0.01189 0.01079 0.00992 0.01118 0.01107 0.00985 0.01338
(0.00005) (0.00007) (0.00006) (0.00005) (0.00004) (0.00006) (0.00005) (0.00005) (0.00005) (0.00006) (0.00004) (0.00006)
(0.00294) (0.03550) (0.00290) (0.00280) (0.00270) (0.00289) (0.00264) (0.00258) (0.00264) (0.00300) (0.00236) (0.00966)
(0.00162) (0.00080) (0.00108) (0.00157) (0.00087) (0.00055) (0.00062) (0.00078) (0.00046) (0.00107) (0.00064) (0.00035)
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
True value method
Table 2 The UWLS, WLS and ML estimates of c and k for n ¼ 30 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 2 3 4 5 6 7 8 9 10 11 12
0.00837 0.10790 0.00962 0.00892 0.00837 0.01013 0.00938 0.00918 0.01006 0.00955 0.00933 0.01206
0.01087 (0.00001) 0.01012 (0.00001) 0.01053 (0.00001) 0.01120 (0.00001) 0.01089 (0.00001) 0.01037 (0.00001) 0.01060 (0.00001) 0.01093 (0.00001) 0.01049 (0.00001) 0.01058 (0.00001) 0.01123 (0.00001) 0.00863 (0.00002)
0.08422 0.09921 0.09234 0.08730 0.08836 0.09764 0.09386 0.09537 0.09808 0.09231 0.09570 0.10620
0.01440 (0.00008) 0.01164 (0.00005) 0.01288 (0.00006) 0.01436 (0.00008) 0.01314 (0.00004) 0.01156 (0.00003) 0.01218 (0.00003) 0.01228 (0.00004) 0.01148 (0.00003) 0.01287 (0.00006) 0.01235 (0.00003) 0.00883 (0.00003)
1.7930 2.0061 1.9088 1.8271 1.8211 1.9523 1.8902 1.8698 1.9715 1.9052 1.9345 1.9166
0.02093 (0.00051) 0.01422 (0.00022) 0.01697 (0.00032) 0.02048 (0.00049) 0.01784 (0.00021) 0.01448 (0.00014) 0.01568 (0.00016) 0.01778 (0.00024) 0.01365 (0.00011) 0.01694 (0.00027) 0.01508 (0.00013) 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^
0.01004 0.01298 0.01149 0.01082 0.00983 0.01207 0.01159 0.01033 0.01020 0.01039 0.01105 0.01401
0.01985 (0.00012) 0.01901 (0.00011) 0.01950 (0.00012) 0.02048 (0.00014) 0.02007 (0.00013) 0.01952 (0.00012) 0.01758 (0.00013) 0.01942 (0.00012) 0.02136 (0.00016) 0.02069 (0.00014) 0.01982 (0.00012) 0.02023 (0.00418)
0.08460 0.10156 0.09451 0.08836 0.08895 0.10002 0.09551 0.09485 0.10027 0.09449 0.09694 0.10987
0.02580 0.02186 0.02331 0.02588 0.02433 0.02192 0.02285 0.02334 0.02189 0.02330 0.02344 0.02008
1.7850 2.0088 1.9069 1.8219 1.8322 1.9821 1.9264 1.9321 1.9890 1.9066 1.9122 1.8954
0.03770 (0.00175) 0.02664 (0.00082) 0.03122 (0.00116) 0.03705 (0.00170) 0.03232 (0.00090) 0.02576 (0.00058) 0.02805 (0.00069) 0.02980 (0.00083) 0.02536 (0.00053) 0.03119 (0.00115) 0.02933 (0.00066) 0.02386 (0.00125)
1 2 3 4 5 6 7 8 9 10 11 12
(0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00002) (0.00002)
(0.00005) (0.00008) (0.00006) (0.00006) (0.00005) (0.00007) (0.00006) (0.00005) (0.00004) (0.00006) (0.00006) (0.00007)
(0.00630) (0.00879) (0.00760) (0.00677) (0.00669) (0.00827) (0.00760) (0.00803) (0.00826) (0.00760) (0.00778) (0.00971)
(0.00666) (0.00962) (0.00827) (0.00726) (0.00713) (0.00908) (0.00825) (0.00819) (0.00900) (0.00828) (0.00825) (0.01069)
(0.00039) (0.00027) (0.00030) (0.00040) (0.00032) (0.00025) (0.00027) (0.00027) (0.00024) (0.00030) (0.00027) (0.00020)
(0.8687) (0.9555) (0.8121) (0.7256) (0.6249) (0.9625) (0.7805) (0.5978) (0.9474) (0.7459) (0.7954) (0.9012)
(0.5608) (0.4571) (0.5418) (0.4855) (0.4288) (0.4132) (0.6287) (0.6386) (0.4291) (0.5945) (0.7068) (0.5122)
141
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
True value method
142
Table 3 The UWLS, WLS and ML estimates of c and k for m ¼ 10, n ¼ 10 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 2 3 4 5 6 7 8 9 10 11 12
0.04852 0.06270 0.05983 0.05601 0.04327 0.06003 0.04991 0.05072 0.05261 0.05995 0.05596 0.07250
0.00760 (0.00002) 0.00745 (0.00004) 0.00741 (0.00002) 0.00813 (0.00002) 0.00792 (0.00002) 0.00780 (0.00002) 0.00784 (0.00002) 0.00739 (0.00002) 0.00810 (0.00002) 0.00740 (0.00002) 0.00809 (0.00002) 0.00656 (0.00004)
0.10185 0.14774 0.12453 0.11167 0.09508 0.13019 0.11653 0.09787 0.12501 0.12446 0.10484 0.13765
0.01002 (0.00002) 0.00922 (0.00003) 0.00951 (0.00002) 0.01073 (0.00002) 0.01020 (0.00002) 0.00960 (0.00002) 0.00985 (0.00002) 0.01085 (0.00002) 0.00996 (0.00002) 0.00950 (0.00002) 0.01162 (0.00003) 0.00773 (0.00091)
1.6509 2.0926 1.8922 1.7396 1.5860 1.9579 1.8055 1.7376 1.9599 1.8899 1.8824 1.7767
0.01916 (0.00033) 0.01413 (0.00022) 0.01603 (0.00026) 0.01973 (0.00038) 0.01944 (0.00030) 0.01479 (0.00020) 0.01648 (0.00022) 0.01962 (0.00034) 0.01465 (0.00019) 0.01600 (0.00025) 0.01785 (0.00031) 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
0.09970 0.13263 0.11847 0.10560 0.10093 0.11640 0.10976 0.08593 0.10947 0.11648 0.10435 0.09857
0.01427 (0.00007) 0.01407 (0.00008) 0.01445 (0.00008) 0.01608 (0.00007) 0.01485 (0.00007) 0.01516 (0.00007) 0.01487 (0.00007) 0.01633 (0.00006) 0.01536 (0.00007) 0.01442 (0.00007) 0.01669 (0.00007) 0.02968 (0.00122)
0.14746 0.20322 0.17327 0.16110 0.13605 0.17993 0.16093 0.13608 0.17023 0.17707 0.15260 0.11024
1 2 3 4 5 6 7 8 9 10 11 12
(0.00330) (0.00583) (0.00494) (0.00426) (0.00293) (0.00543) (0.00392) (0.00433) (0.00430) (0.00496) (0.00425) (0.00745)
(0.01660) (0.03015) (0.02280) (0.01838) (0.01703) (0.02291) (0.02055) (0.01096) (0.02259) (0.02211) (0.02020) (0.01689)
(0.01523) (0.02963) (0.02174) (0.01791) (0.01307) (0.02247) (0.01847) (0.01288) (0.02118) (0.02177) (0.01581) (0.02303)
(0.01857) (0.03550) (0.02272) (0.01936) (0.01652) (0.02876) (0.02145) (0.01135) (0.02761) (0.02362) (0.02238) (0.00966)
(0.7125) (0.7657) (0.6788) (0.6707) (0.6396) (0.5300) (0.5330) (0.4510) (0.4629) (0.6609) (0.5654) (0.8457)
k ¼ 0:02, k^
0.01666 (0.00006) 1.6248 (0.5007) 0.06124 (0.00610) 0.01638 (0.00009) 2.1286 (0.9220) 0.02497 (0.00050) 0.01677 (0.00007) 1.8284 (0.4664) 0.04996 (0.00478) 0.01848 (0.00007) 1.7102 (0.4756) 0.06049 (0.00640) 0.01800 (0.00006) 1.6052 (0.4363) 0.05903 (0.00490) 0.01755 (0.00008) 1.9063 (0.2940) 0.03777 (0.00199) 0.01766 (0.00008) 1.7556 (0.3227) 0.04613 (0.00271) 0.01900 (0.00006) 1.9563 (0.3035) 0.03605 (0.00199) 0.01817 (0.00008) 1.8855 (0.5413) 0.03825 (0.00254) 0.01670 (0.00008) 1.8295 (0.4590) 0.04910 (0.00442) 0.01979 (0.00008) 1.7849 (0.3257) 0.05092 (0.00331) 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.
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
True value method
Table 4 The UWLS, WLS and ML estimates of c and k for m ¼ 10, n ¼ 30 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 2 3 4 5 6 7 8 9 10 11 12
0.12498 0.17748 0.15482 0.13444 0.14243 0.15687 0.16454 0.13730 0.16120 0.15756 0.13941 0.09990
0.02231 0.02165 0.02203 0.02410 0.02237 0.02278 0.02270 0.02375 0.02315 0.02174 0.02599 0.03139
0.21182 0.27975 0.24466 0.22383 0.19104 0.24524 0.22254 0.18841 0.23236 0.24456 0.20372 0.08483
0.02374 0.02297 0.02368 0.02523 0.02463 0.02457 0.02486 0.02726 0.02553 0.02361 0.02848 0.01801
1.6616 2.1641 1.9169 1.7149 1.5993 2.0068 1.7870 1.8376 2.0356 1.8744 1.5885 1.8946
0.04366 (0.00194) 0.03536 (0.00139) 0.03884 (0.00156) 0.04608 (0.00216) 0.04487 (0.00196) 0.03749 (0.00145) 0.04076 (0.00159) 0.04481 (0.00161) 0.03627 (0.00136) 0.03938 (0.00158) 0.04683 (0.00186) 0.04010 (0.00219)
1 2 3 4 5 6 7 8 9 10 11 12
(0.02916) (0.05726) (0.04213) (0.03400) (0.03706) (0.05748) (0.05346) (0.02728) (0.05008) (0.04412) (0.03891) (0.01699)
(0.00024) (0.00024) (0.00030) (0.00030) (0.00025) (0.00028) (0.00026) (0.00026) (0.00028) (0.00023) (0.00039) (0.00416)
(0.05235) (0.08744) (0.06722) (0.05646) (0.03676) (0.06484) (0.05444) (0.03863) (0.06028) (0.06775) (0.04678) (0.0100)
(0.00028) (0.00029) (0.00031) (0.00035) (0.00031) (0.00033) (0.00034) (0.00043) (0.00036) (0.00030) (0.00049) (0.00073)
(1.0561) (1.2438) (1.0140) (0.9252) (0.9509) (0.8751) (0.8540) (0.6266) (0.7844) (0.9017) (0.5526) (1.0089)
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^
0.31994 0.37894 0.33716 0.32309 0.30241 0.33629 0.33059 0.31525 0.33750 0.35060 0.28843 0.06127
0.04295 0.04328 0.04484 0.04814 0.04457 0.04691 0.04529 0.04942 0.04664 0.04452 0.05238 0.02826
0.33734 0.43907 0.38326 0.34234 0.33130 0.39209 0.34958 0.38639 0.37508 0.40520 0.33549 0.05173
0.04502 0.04501 0.04676 0.05016 0.04800 0.04904 0.04746 0.05118 0.05005 0.04571 0.05502 0.08380
1.7073 2.2012 2.0267 1.8608 1.5958 1.8068 1.8196 1.7541 1.7744 1.9727 1.8865 1.8422
0.07517 (0.00454) 0.06806 (0.00392) 0.06878 (0.00399) 0.07986 (0.00558) 0.07722 (0.00456) 0.08036 (0.00576) 0.07591 (0.00464) 0.08299 (0.00628) 0.07802 (0.00488) 0.06802 (0.00384) 0.07999 (0.00578) 0.01854 (0.00864)
(0.20926) (0.27150) (0.24223) (0.22103) (0.21794) (0.21621) (0.23176) (0.17606) (0.23450) (0.26211) (0.16646) (0.01155)
(0.00086) (0.00098) (0.00108) (0.00125) (0.00099) (0.00121) (0.00104) (0.00132) (0.00116) (0.00107) (0.00163) (0.00121)
(0.16769) (0.27645) (0.19587) (0.17261) (0.15508) (0.22100) (0.17269) (0.20854) (0.19942) (0.21520) (0.16274) (0.01058)
(0.00094) (0.00109) (0.00129) (0.00138) (0.00120) (0.00137) (0.00123) (0.00149) (0.00148) (0.00117) (0.00182) (0.58168)
(1.0609) (1.2478) (1.1409) (1.1185) (1.0209) (1.0586) (0.9108) (1.2386) (0.7914) (1.0945) (0.8574) (1.0958)
143
Note. The values in parentheses are sample mean squared error (SMSE) of ^c and k^ and Ô*Õ express SMSE less than Method 12.
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True value method
144
Table 5 The UWLS, WLS and ML estimates of c and k for m ¼ 30, n ¼ 10 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 2 3 4 5 6 7 8 9 10 11 12
0.02710 0.03325 0.03001 0.02889 0.02597 0.03141 0.02846 0.02731 0.02873 0.03025 0.02461 0.03002
0.00852 (0.00001) 0.01859 (0.00076) 0.00848 (0.00001) 0.00885 (0.00001) 0.00875 (0.00001) 0.00873 (0.00001) 0.00875 (0.00001) 0.00898 (0.00001) 0.00890 (0.00001) 0.00848 (0.00001) 0.00932 (0.00001) 0.00908 (0.00388)
0.08408 0.11010 0.09822 0.09119 0.08556 0.10360 0.09642 0.09506 0.10220 0.09815 0.09604 0.12221
0.01084 (0.00001) 0.00998 (0.00001) 0.01035 (0.00001) 0.01101 (0.00001) 0.01076 (0.00001) 0.01028 (0.00001) 0.01047 (0.00001) 0.01082 (0.00001) 0.01042 (0.00001) 0.01038 (0.00001) 0.01115 (0.00001) 0.00912 (0.00002)
1.7177 1.9839 1.8621 1.7667 1.7934 1.9514 1.8905 1.9157 1.9595 1.8617 1.9146 1.9024
0.01537 (0.00011) 0.01199 (0.00006) 0.01344 (0.00008) 0.01527 (0.00011) 0.01379 (0.00006) 0.01191 (0.00004) 0.01258 (0.00005) 0.01280 (0.00005) 0.01180 (0.00004) 0.01343 (0.00008) 0.01255 (0.00004) 0.01283 (0.00008)
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^
0.01985 (0.00012) 0.01901 (0.00011) 0.01950 (0.00012) 0.02048 (0.00014) 0.02007 (0.00013) 0.01952 (0.00012) 0.01758 (0.00013) 0.01942 (0.00012) 0.02136 (0.00016) 0.02069 (0.00014) 0.01982 (0.00012) 0.02023 (0.00418)
0.08460 0.10156 0.09451 0.08836 0.08895 0.10002 0.09551 0.09485 0.10027 0.09449 0.09694 0.10987
1.7850 2.0088 1.9069 1.8219 1.8322 1.9821 1.9264 1.9321 1.9890 1.9066 1.9122 1.8954
0.03770 (0.00175) 0.02664 (0.00082) 0.03122 (0.00116) 0.03705 (0.00170) 0.03232 (0.00090) 0.02576 (0.00058) 0.02805 (0.00069) 0.02980 (0.00083) 0.02536 (0.00053) 0.03119 (0.00115) 0.02933 (0.00066) 0.02386 (0.00125)
(0.00083) (0.00116) (0.00099) (0.00093) (0.00071) (0.00104) (0.00086) (0.00091) (0.00083) (0.00104) (0.00618) (0.00099)
c ¼ 0:01, ^c
(0.00828) (0.01348) (0.01095) (0.00950) (0.00820) (0.01177) (0.01026) (0.01024) (0.01113) (0.01097) (0.00968) (0.01504)
0.02580 0.02186 0.02331 0.02588 0.02433 0.02192 0.02285 0.02334 0.02189 0.02330 0.02344 0.02008 Note. The values in parentheses are sample mean squared error (SMSE) of ^c and k^ and Ô*Õ 1 2 3 4 5 6 7 8 9 10 11 12
0.01004 0.01298 0.01149 0.01082 0.00983 0.01207 0.01159 0.01033 0.01020 0.01039 0.01105 0.01401
(0.00005) (0.00008) (0.00006) (0.00006) (0.00005) (0.00007) (0.00006) (0.00005) (0.00004) (0.00006) (0.00006) (0.00007)
(0.00666) (0.00962) (0.00827) (0.00726) (0.00713) (0.00908) (0.00825) (0.00819) (0.00900) (0.00828) (0.00825) (0.01069)
(0.00039) (0.00027) (0.00030) (0.00040) (0.00032) (0.00025) (0.00027) (0.00027) (0.00024) (0.00030) (0.00027) (0.00020)
(0.1401) (0.1408) (0.0662) (0.3134) (0.3427) (0.0445) (0.2056) (0.2736) (0.0536) (0.0614) (0.1403) (0.2102)
(0.5608) (0.4571) (0.5418) (0.4855) (0.4288) (0.4132) (0.6287) (0.6386) (0.4291) (0.5945) (0.7068) (0.5122)
express SMSE less than Method 12.
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
True value method
Table 6 The UWLS, WLS and ML estimates of c and k for m ¼ 30, n ¼ 30 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 2 3 4 5 6 7 8 9 10 11 12
0.07070 0.08837 0.08021 0.07410 0.06721 0.08205 0.07771 0.07119 0.07568 0.08053 0.06355 0.07790
0.02505 (0.00026) 0.02469 (0.00026) 0.02479 (0.00026) 0.02618 (0.00030) 0.02559 (0.00028) 0.02549 (0.00028) 0.02539 (0.00027) 0.02636 (0.00031) 0.02569 (0.00028) 0.02478 (0.00026) 0.02729 (0.00034) 0.02332 (0.00241)
0.12255 0.15620 0.13711 0.12839 0.11644 0.14350 0.13003 0.11006 0.13476 0.13647 0.12046 0.13135
0.02807 (0.00038) 0.02736 (0.00036) 0.02784 (0.00037) 0.02921 (0.00043) 0.02858 (0.00039) 0.02823 (0.00038) 0.02856 (0.00039) 0.02688 (0.00030) 0.02879 (0.00041) 0.02782 (0.00037) 0.03059 (0.00048) 0.02921 (0.00265)
1.6523 1.9855 1.8384 1.7218 1.7589 1.9649 1.8826 1.9285 1.9467 1.8397 1.8408 1.7855
0.04146 (0.00141) 0.03401 (0.00090) 0.03707 (0.00108) 0.04141 (0.00142) 0.03775 (0.00104) 0.03363 (0.00080) 0.03518 (0.00088) 0.03494 (0.00085) 0.03473 (0.00086) 0.03700 (0.00108) 0.03758 (0.00098) 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^
0.13062 0.16327 0.14950 0.14173 0.12624 0.15096 0.13699 0.13385 0.13650 0.15190 0.12643 0.09648
0.04956 0.04972 0.04947 0.05114 0.05107 0.05168 0.05190 0.05373 0.05275 0.04943 0.05466 0.02182
0.08106 0.09788 0.09088 0.08439 0.08596 0.09724 0.09265 0.09470 0.09780 0.08999 0.09550 0.10775
0.02680 (0.00042) 0.02262 (0.00028) 0.02449 (0.00034) 0.02705 (0.00044) 0.02488 (0.00032) 0.02226 (0.00023) 0.02329 (0.00027) 0.02328 (0.00026) 0.02220 (0.00023) 0.02460 (0.00035) 0.02360 (0.00027) 0.02358 (0.00028)
1.6299 2.0093 1.8261 1.7083 1.6845 1.9278 1.8587 1.7083 1.8771 1.9733 1.8148 1.7855
0.07580 (0.00531) 0.06577 (0.00408) 0.07067 (0.00466) 0.07666 (0.00550) 0.07323 (0.00480) 0.06673 (0.00395) 0.07121 (0.00366) 0.07666 (0.00550) 0.06947 (0.00424) 0.06508 (0.00371) 0.06998 (0.00439) 0.05620 (0.00652)
1 2 3 4 5 6 7 8 9 10 11 12
(0.00765) (0.00927) (0.00962) (0.00812) (0.00662) (0.00995) (0.00865) (0.00792) (0.00824) (0.00970) (0.00613) (0.00927)
(0.02762) (0.04236) (0.03574) (0.03129) (0.02500) (0.03533) (0.02981) (0.02925) (0.02987) (0.03677) (0.02632) (0.01589)
(0.00170) (0.00174) (0.00171) (0.00184) (0.00183) (0.00190) (0.00191) (0.00208) (0.00199) (0.00171) (0.00215) (0.01922)
(0.02113) (0.03299) (0.02650) (0.02336) (0.01898) (0.02813) (0.02372) (0.01025) (0.02423) (0.02534) (0.01985) (0.02174)
(0.00610) (0.00889) (0.00766) (0.00663) (0.00657) (0.00847) (0.00766) (0.00808) (0.00843) (0.00755) (0.00791) (0.01018)
(0.1005) (0.3238) (0.0492) (0.3381) (0.3585) (0.1467) (0.2017) (0.0286) (0.0398) (0.0564) (0.5426) (0.2101)
(0.1508) (0.2152) (0.1715) (0.4253) (0.2483) (0.1171) (0.1245) (0.1253) (0.0501) (0.1918) (0.1989) (0.1441)
145
Note. The values in parentheses are sample mean squared error (SMSE) of ^c and k^ and Ô*Õ express SMSE less than Method 12.
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True value method
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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).
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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
b
Department of Statistics, Tamkang University, Tamsui, Taipei 25137, Taiwan, ROC 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: [email protected] (J.-W. Wu).
0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2003.08.086
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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 f ðxÞ ¼ kecx exp
k ðecx 1Þ ; c
x > 0;
ð1Þ
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
135
where c > 0 and k > 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 ¼ 1 exp ðecx 1Þ ð2Þ c and F ðxÞ satisfies lnf lnð1 F ðxÞÞg ¼ ln k þ ln
ecx 1 : c
ð3Þ
Now suppose that X1 ; X2 ; . . . ; Xn is a sample of size n from a Gompertz distribution with parameters c and k, and that Xð1Þ < Xð2Þ < < XðnÞ are the order statistics. For observed ordered observations xð1Þ < xð2Þ < < xðnÞ , it follows from (3) that cx e ðiÞ 1 lnf lnð1 F ðxðiÞ ÞÞg ¼ ln k þ ln ; i ¼ 1; 2; . . . ; n: ð4Þ c Let the empirical distribution function of F ðxÞ be denoted by Fb ðxÞ, where b F ðxðiÞ Þ equals i=n. In order to avoid lnð0Þ in (4), we modify Fb ðxðiÞ Þ to be pi ¼
id 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 p1i ¼ i=ðn Pi þ 1Þ, p2i ¼ ði 0:5Þ=n, p3i ¼ ði 0:3Þ=ðn þ 0:4Þ. Alternatively, we use p4i ¼ j¼1 f1=ðn j þ 1Þg to estimate lnð1 F ðxðiÞ ÞÞ in (4) (see [12,16]). First, we estimate c and k by unweighted least squares (UWLS) method. Let cx 2 n X e ðiÞ 1 lnð lnð1 pki ÞÞ ln k ln ; k ¼ 1; 2; 3 Gk ðc; kÞ ¼ c i¼1 and G4 ðc; kÞ ¼
n X
ln p4i ln k ln
i¼1
We solve the normal equations 8 oG ðc; kÞ > < k ¼ 0; oc > : oGk ðc; kÞ ¼ 0 ok
ecxðiÞ 1 c
2 :
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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: ( !) ! n X ^ck xðiÞ e^ck xðiÞ þ e^ck xðiÞ 1 e^ck xðiÞ 1 ^ck ¼ ln ^ck e^ck xðiÞ 1 i¼1 ( !" n X ^ck xðiÞ e^ck xðiÞ þ e^ck xðiÞ 1 lnð lnð1 pki ÞÞ ^ck ðe^ck xðiÞ 1Þ i¼1 !#)1 n n 1X 1X e^ck xðiÞ 1 lnð lnð1 pki ÞÞ þ ln ; ^ck n i¼1 n i¼1 ( k^k ¼
n Y ð lnð1 pki ÞÞ
)1=n (
i¼1
n Y
e^ck xðiÞ 1 ^ck
i¼1
!)ð1=nÞ
and ( ^c4 ¼
n X
^c4 xðiÞ e^c4 xðiÞ þ e^c4 xðiÞ 1 e^c4 xðiÞ 1
i¼1
(
!
n X
^c4 xðiÞ e^c4 xðiÞ þ e^c4 xðiÞ 1 ^c4 ðe^c4 xðiÞ 1Þ i¼1 !#)1 n 1X e^c4 xðiÞ 1 þ ln ; ^c4 n i¼1 ( k^4 ¼
n Y
)1=n ( p4i
i¼1
n Y i¼1
e^c4 xðiÞ 1 ^c4
e^c4 xðiÞ 1 ^c4
ln !"
ln p4i
!)
n 1X ln p4i n i¼1
!)ð1=nÞ :
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 cxðiÞ 2 n X e 1 Kak ðc; kÞ ¼ Waki lnð lnð1 pki ÞÞ ln k ln c i¼1 and K34 ðc; kÞ ¼
n X
W34i ln p4i ln k ln
i¼1
where W1ki ¼ fð1 pki Þ lnð1 pki Þg2 ;
ecxðiÞ 1 c
2 ;
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
( W2ki ¼ ( W34i ¼
3:3pki 27:5½1 ð1 pki Þ n i X j¼1
1 njþ1
0:025
137
)2 ;
)2 :
Solving the normal equations as before, the WLS estimates of c and k satisfy the following normal equations: ! ( !) n X ^cwak xðiÞ e^cwak xðiÞ þ e^cwak xðiÞ 1 e^cwak xðiÞ 1 ^cwak ¼ Waki ln ^cwak e^cwak xðiÞ 1 i¼1 8 2 ! > n <X ^cwak xðiÞ ^cwak xðiÞ ^cwak xðiÞ e þe 1 6 Waki 4 lnð lnð1 pki ÞÞ ^cak xðiÞ > ^cak ðe 1Þ : i¼1 ^c x 391 Pn > e wak ðiÞ 1 = W lnð lnð1 p ÞÞ W ln ki i¼1 aki i¼1 aki ^cwak 7 Pn ; 5 > ; i¼1 Waki
Pn
k^wak
8 ^c x 9 P P > = < ni¼1 Waki lnð lnð1 pki ÞÞ ni¼1 Waki ln e wak^c ðiÞ 1 > wak Pn ¼ exp > > ; : i¼1 Waki
for a ¼ 1, 2 and k ¼ 1, 2, 3 and ! ( n X ^cw34 xðiÞ e^cw34 xðiÞ þ e^cw34 xðiÞ 1 ^cw34 ¼ ln W34i e^cw34 xðiÞ 1 i¼1
8 > n <X > :
W34i
i¼1
2 6 4 ln p4i
k^w34
^cw34 xðiÞ e^cw34 xðiÞ þ e^cw34 xðiÞ 1 ^cw34 ðe^cw34 xðiÞ 1Þ
e^cw34 xðiÞ 1 ^cw34
!)
!
^c x 391 Pn > e w34 ðiÞ 1 = W ln p W ln 34i 4i 34i i¼1 i¼1 ^cw34 7 Pn ; 5 > ; i¼1 W34i
Pn
8 ^c x 9 P P > = < ni¼1 W34i ln p4i ni¼1 W34i ln e w34^c ðiÞ 1 > w34 Pn : ¼ exp > > ; : i¼1 W34i
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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 ( )1 )( n n n n X X X X ^cxðiÞ ^cxðiÞ ^cxðiÞ ^c ¼ n e 1 xðiÞ e 1 n xðiÞ e ; i¼1
i¼1
i¼1
i¼1
ð5Þ ( k^ ¼
^c
n X i¼1
)( xðiÞ
n n X 1X e^cxðiÞ 1 xðiÞ e^cxðiÞ ^c i¼1 i¼1
)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 ^c0 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 Xð1Þ is k cx cx f ðx; c; k Þ ¼ k e exp ðe 1Þ ; c where k ¼ nk. The corresponding c.d.f. is k F ðxÞ ¼ 1 exp ðecx 1Þ : c
ð7Þ
ð8Þ
Suppose Xð1Þ1 ; Xð1Þ2 ; . . . ; Xð1Þm denote the set of first-order statistics of m samples of size n from (1) and let Yð1Þ < Yð2Þ < < YðmÞ be the corresponding order statistics. Clearly, Xð1Þ1 ; Xð1Þ2 ; . . . ; Xð1Þm can also be considered as a random sample from (7). Then F ðxÞ in (8) satisfies cx e 1 lnf lnð1 F ðxÞÞg ¼ ln n þ ln k þ ln : ð9Þ c For observed ordered observations yð1Þ < yð2Þ < < yðmÞ , (9) can be rewritten as cy e ðiÞ 1 lnf lnð1 F ðyðiÞ ÞÞg ¼ ln n þ ln k þ ln ; 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.
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
139
3. Simulation study 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
pi
Weight
1 2 3 4 5 6 7 8 9 10 11 12
UWLSE UWLSE UWLSE UWLSE WLSE WLSE WLSE WLSE WLSE WLSE WLSE MLE
p1i p2i p3i p4i p1i p2i p3i p1i p2i p3i p4i –
– – – – W11i W12i W13i W21i W22i W23i W34i –
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
140
Table 1 The UWLS, WLS and ML estimates of c and k for n ¼ 10 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 2 3 4 5 6 7 8 9 10 11 12
0.00849 0.01107 0.00975 0.00902 0.00860 0.01053 0.00962 0.00941 0.01028 0.00964 0.00970 0.01197
0.01076 (0.00001) 0.00993 (0.00001) 0.01038 (0.00001) 0.01107 (0.00001) 0.01072 (0.00001) 0.01014 (0.00001) 0.01046 (0.00001) 0.01081 (0.00001) 0.01030 (0.00001) 0.01033 (0.00008) 0.01103 (0.00001) 0.00868 (0.00005)
0.08342 0.09827 0.09143 0.08631 0.08820 0.09755 0.09371 0.09573 0.09825 0.09139 0.09591 0.10607
0.01451 (0.00008) 0.01171 (0.00005) 0.01295 (0.00006) 0.01446 (0.00008) 0.01306 (0.00005) 0.01148 (0.00003) 0.01211 (0.00003) 0.01210 (0.00004) 0.01135 (0.00003) 0.01294 (0.00006) 0.01218 (0.00003) 0.00963 (0.00003)
1.8175 2.0194 1.9260 1.8405 1.8410 1.9761 1.9106 1.8970 1.9836 1.9155 1.9473 1.9002
0.02028 (0.00046) 0.01437 (0.00031) 0.01651 (0.00034) 0.02010 (0.00047) 0.01743 (0.00021) 0.01348 (0.00011) 0.01511 (0.00012) 0.01710 (0.00022) 0.01338 (0.00011) 0.01698 (0.00034) 0.01524 (0.00015) 0.01693 (0.00041)
k ¼ 0:02, k^
c ¼ 0:1, ^c
k ¼ 0:02, k^
c ¼ 2:0, ^c
0.01975 (0.00012) 0.01910 (0.00011) 0.01961 (0.00012) 0.02049 (0.00014) 0.02029 (0.00014) 0.01972 (0.00013) 0.02000 (0.00013) 0.02085 (0.00015) 0.02013 (0.00013) 0.01964 (0.00012) 0.02157 (0.00017) 0.01982 (0.00025)
0.08140 0.20326 0.09599 0.08546 0.07862 0.10260 0.09024 0.08610 0.09918 0.09570 0.08812 0.11024
(0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00002) (0.00003)
c ¼ 0:01, ^c
(0.00617) (0.00863) (0.00746) (0.00663) (0.00668) (0.00827) (0.00760) (0.00811) (0.00830) (0.00745) (0.00781) (0.00969)
(0.5894) (0.9584) (0.8598) (0.7915) (0.6668) (0.8937) (0.7724) (0.5641) (0.5666) (0.6334) (0.8621) (0.9065)
k ¼ 0:02, k^
0.02861 (0.00035) 1.7890 (0.4987) 0.03670 0.01638 (0.00009) 1.9950 (0.5874) 0.02680 0.02576 (0.00033) 1.9107 (0.5566) 0.03036 0.03042 (0.00044) 1.8259 (0.4469) 0.03604 0.02900 (0.00035) 1.8330 (0.4258) 0.03198 0.02420 (0.00028) 1.9629 (0.3298) 0.02595 0.02683 (0.00032) 1.9259 (0.3776) 0.02758 0.02948 (0.00041) 1.9498 (0.3951) 0.02930 0.02586 (0.00032) 1.9870 (0.4009) 0.02496 0.02594 (0.00033) 1.9104 (0.3765) 0.03034 0.03129 (0.00049) 1.9225 (0.3912) 0.02873 0.02159 (0.00276) 1.8854 (0.3799) 0.02475 Note. The values in parentheses are sample mean squared error (SMSE) of ^c and k^ and Ô*Õ express SMSE less than Method 12. 1 2 3 4 5 6 7 8 9 10 11 12
0.00972 0.01242 0.01110 0.01039 0.00945 0.01189 0.01079 0.00992 0.01118 0.01107 0.00985 0.01338
(0.00005) (0.00007) (0.00006) (0.00005) (0.00004) (0.00006) (0.00005) (0.00005) (0.00005) (0.00006) (0.00004) (0.00006)
(0.00294) (0.03550) (0.00290) (0.00280) (0.00270) (0.00289) (0.00264) (0.00258) (0.00264) (0.00300) (0.00236) (0.00966)
(0.00162) (0.00080) (0.00108) (0.00157) (0.00087) (0.00055) (0.00062) (0.00078) (0.00046) (0.00107) (0.00064) (0.00035)
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
True value method
Table 2 The UWLS, WLS and ML estimates of c and k for n ¼ 30 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 2 3 4 5 6 7 8 9 10 11 12
0.00837 0.10790 0.00962 0.00892 0.00837 0.01013 0.00938 0.00918 0.01006 0.00955 0.00933 0.01206
0.01087 (0.00001) 0.01012 (0.00001) 0.01053 (0.00001) 0.01120 (0.00001) 0.01089 (0.00001) 0.01037 (0.00001) 0.01060 (0.00001) 0.01093 (0.00001) 0.01049 (0.00001) 0.01058 (0.00001) 0.01123 (0.00001) 0.00863 (0.00002)
0.08422 0.09921 0.09234 0.08730 0.08836 0.09764 0.09386 0.09537 0.09808 0.09231 0.09570 0.10620
0.01440 (0.00008) 0.01164 (0.00005) 0.01288 (0.00006) 0.01436 (0.00008) 0.01314 (0.00004) 0.01156 (0.00003) 0.01218 (0.00003) 0.01228 (0.00004) 0.01148 (0.00003) 0.01287 (0.00006) 0.01235 (0.00003) 0.00883 (0.00003)
1.7930 2.0061 1.9088 1.8271 1.8211 1.9523 1.8902 1.8698 1.9715 1.9052 1.9345 1.9166
0.02093 (0.00051) 0.01422 (0.00022) 0.01697 (0.00032) 0.02048 (0.00049) 0.01784 (0.00021) 0.01448 (0.00014) 0.01568 (0.00016) 0.01778 (0.00024) 0.01365 (0.00011) 0.01694 (0.00027) 0.01508 (0.00013) 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^
0.01004 0.01298 0.01149 0.01082 0.00983 0.01207 0.01159 0.01033 0.01020 0.01039 0.01105 0.01401
0.01985 (0.00012) 0.01901 (0.00011) 0.01950 (0.00012) 0.02048 (0.00014) 0.02007 (0.00013) 0.01952 (0.00012) 0.01758 (0.00013) 0.01942 (0.00012) 0.02136 (0.00016) 0.02069 (0.00014) 0.01982 (0.00012) 0.02023 (0.00418)
0.08460 0.10156 0.09451 0.08836 0.08895 0.10002 0.09551 0.09485 0.10027 0.09449 0.09694 0.10987
0.02580 0.02186 0.02331 0.02588 0.02433 0.02192 0.02285 0.02334 0.02189 0.02330 0.02344 0.02008
1.7850 2.0088 1.9069 1.8219 1.8322 1.9821 1.9264 1.9321 1.9890 1.9066 1.9122 1.8954
0.03770 (0.00175) 0.02664 (0.00082) 0.03122 (0.00116) 0.03705 (0.00170) 0.03232 (0.00090) 0.02576 (0.00058) 0.02805 (0.00069) 0.02980 (0.00083) 0.02536 (0.00053) 0.03119 (0.00115) 0.02933 (0.00066) 0.02386 (0.00125)
1 2 3 4 5 6 7 8 9 10 11 12
(0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00003) (0.00002) (0.00002)
(0.00005) (0.00008) (0.00006) (0.00006) (0.00005) (0.00007) (0.00006) (0.00005) (0.00004) (0.00006) (0.00006) (0.00007)
(0.00630) (0.00879) (0.00760) (0.00677) (0.00669) (0.00827) (0.00760) (0.00803) (0.00826) (0.00760) (0.00778) (0.00971)
(0.00666) (0.00962) (0.00827) (0.00726) (0.00713) (0.00908) (0.00825) (0.00819) (0.00900) (0.00828) (0.00825) (0.01069)
(0.00039) (0.00027) (0.00030) (0.00040) (0.00032) (0.00025) (0.00027) (0.00027) (0.00024) (0.00030) (0.00027) (0.00020)
(0.8687) (0.9555) (0.8121) (0.7256) (0.6249) (0.9625) (0.7805) (0.5978) (0.9474) (0.7459) (0.7954) (0.9012)
(0.5608) (0.4571) (0.5418) (0.4855) (0.4288) (0.4132) (0.6287) (0.6386) (0.4291) (0.5945) (0.7068) (0.5122)
141
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
True value method
142
Table 3 The UWLS, WLS and ML estimates of c and k for m ¼ 10, n ¼ 10 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 2 3 4 5 6 7 8 9 10 11 12
0.04852 0.06270 0.05983 0.05601 0.04327 0.06003 0.04991 0.05072 0.05261 0.05995 0.05596 0.07250
0.00760 (0.00002) 0.00745 (0.00004) 0.00741 (0.00002) 0.00813 (0.00002) 0.00792 (0.00002) 0.00780 (0.00002) 0.00784 (0.00002) 0.00739 (0.00002) 0.00810 (0.00002) 0.00740 (0.00002) 0.00809 (0.00002) 0.00656 (0.00004)
0.10185 0.14774 0.12453 0.11167 0.09508 0.13019 0.11653 0.09787 0.12501 0.12446 0.10484 0.13765
0.01002 (0.00002) 0.00922 (0.00003) 0.00951 (0.00002) 0.01073 (0.00002) 0.01020 (0.00002) 0.00960 (0.00002) 0.00985 (0.00002) 0.01085 (0.00002) 0.00996 (0.00002) 0.00950 (0.00002) 0.01162 (0.00003) 0.00773 (0.00091)
1.6509 2.0926 1.8922 1.7396 1.5860 1.9579 1.8055 1.7376 1.9599 1.8899 1.8824 1.7767
0.01916 (0.00033) 0.01413 (0.00022) 0.01603 (0.00026) 0.01973 (0.00038) 0.01944 (0.00030) 0.01479 (0.00020) 0.01648 (0.00022) 0.01962 (0.00034) 0.01465 (0.00019) 0.01600 (0.00025) 0.01785 (0.00031) 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
0.09970 0.13263 0.11847 0.10560 0.10093 0.11640 0.10976 0.08593 0.10947 0.11648 0.10435 0.09857
0.01427 (0.00007) 0.01407 (0.00008) 0.01445 (0.00008) 0.01608 (0.00007) 0.01485 (0.00007) 0.01516 (0.00007) 0.01487 (0.00007) 0.01633 (0.00006) 0.01536 (0.00007) 0.01442 (0.00007) 0.01669 (0.00007) 0.02968 (0.00122)
0.14746 0.20322 0.17327 0.16110 0.13605 0.17993 0.16093 0.13608 0.17023 0.17707 0.15260 0.11024
1 2 3 4 5 6 7 8 9 10 11 12
(0.00330) (0.00583) (0.00494) (0.00426) (0.00293) (0.00543) (0.00392) (0.00433) (0.00430) (0.00496) (0.00425) (0.00745)
(0.01660) (0.03015) (0.02280) (0.01838) (0.01703) (0.02291) (0.02055) (0.01096) (0.02259) (0.02211) (0.02020) (0.01689)
(0.01523) (0.02963) (0.02174) (0.01791) (0.01307) (0.02247) (0.01847) (0.01288) (0.02118) (0.02177) (0.01581) (0.02303)
(0.01857) (0.03550) (0.02272) (0.01936) (0.01652) (0.02876) (0.02145) (0.01135) (0.02761) (0.02362) (0.02238) (0.00966)
(0.7125) (0.7657) (0.6788) (0.6707) (0.6396) (0.5300) (0.5330) (0.4510) (0.4629) (0.6609) (0.5654) (0.8457)
k ¼ 0:02, k^
0.01666 (0.00006) 1.6248 (0.5007) 0.06124 (0.00610) 0.01638 (0.00009) 2.1286 (0.9220) 0.02497 (0.00050) 0.01677 (0.00007) 1.8284 (0.4664) 0.04996 (0.00478) 0.01848 (0.00007) 1.7102 (0.4756) 0.06049 (0.00640) 0.01800 (0.00006) 1.6052 (0.4363) 0.05903 (0.00490) 0.01755 (0.00008) 1.9063 (0.2940) 0.03777 (0.00199) 0.01766 (0.00008) 1.7556 (0.3227) 0.04613 (0.00271) 0.01900 (0.00006) 1.9563 (0.3035) 0.03605 (0.00199) 0.01817 (0.00008) 1.8855 (0.5413) 0.03825 (0.00254) 0.01670 (0.00008) 1.8295 (0.4590) 0.04910 (0.00442) 0.01979 (0.00008) 1.7849 (0.3257) 0.05092 (0.00331) 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.
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
True value method
Table 4 The UWLS, WLS and ML estimates of c and k for m ¼ 10, n ¼ 30 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 2 3 4 5 6 7 8 9 10 11 12
0.12498 0.17748 0.15482 0.13444 0.14243 0.15687 0.16454 0.13730 0.16120 0.15756 0.13941 0.09990
0.02231 0.02165 0.02203 0.02410 0.02237 0.02278 0.02270 0.02375 0.02315 0.02174 0.02599 0.03139
0.21182 0.27975 0.24466 0.22383 0.19104 0.24524 0.22254 0.18841 0.23236 0.24456 0.20372 0.08483
0.02374 0.02297 0.02368 0.02523 0.02463 0.02457 0.02486 0.02726 0.02553 0.02361 0.02848 0.01801
1.6616 2.1641 1.9169 1.7149 1.5993 2.0068 1.7870 1.8376 2.0356 1.8744 1.5885 1.8946
0.04366 (0.00194) 0.03536 (0.00139) 0.03884 (0.00156) 0.04608 (0.00216) 0.04487 (0.00196) 0.03749 (0.00145) 0.04076 (0.00159) 0.04481 (0.00161) 0.03627 (0.00136) 0.03938 (0.00158) 0.04683 (0.00186) 0.04010 (0.00219)
1 2 3 4 5 6 7 8 9 10 11 12
(0.02916) (0.05726) (0.04213) (0.03400) (0.03706) (0.05748) (0.05346) (0.02728) (0.05008) (0.04412) (0.03891) (0.01699)
(0.00024) (0.00024) (0.00030) (0.00030) (0.00025) (0.00028) (0.00026) (0.00026) (0.00028) (0.00023) (0.00039) (0.00416)
(0.05235) (0.08744) (0.06722) (0.05646) (0.03676) (0.06484) (0.05444) (0.03863) (0.06028) (0.06775) (0.04678) (0.0100)
(0.00028) (0.00029) (0.00031) (0.00035) (0.00031) (0.00033) (0.00034) (0.00043) (0.00036) (0.00030) (0.00049) (0.00073)
(1.0561) (1.2438) (1.0140) (0.9252) (0.9509) (0.8751) (0.8540) (0.6266) (0.7844) (0.9017) (0.5526) (1.0089)
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^
0.31994 0.37894 0.33716 0.32309 0.30241 0.33629 0.33059 0.31525 0.33750 0.35060 0.28843 0.06127
0.04295 0.04328 0.04484 0.04814 0.04457 0.04691 0.04529 0.04942 0.04664 0.04452 0.05238 0.02826
0.33734 0.43907 0.38326 0.34234 0.33130 0.39209 0.34958 0.38639 0.37508 0.40520 0.33549 0.05173
0.04502 0.04501 0.04676 0.05016 0.04800 0.04904 0.04746 0.05118 0.05005 0.04571 0.05502 0.08380
1.7073 2.2012 2.0267 1.8608 1.5958 1.8068 1.8196 1.7541 1.7744 1.9727 1.8865 1.8422
0.07517 (0.00454) 0.06806 (0.00392) 0.06878 (0.00399) 0.07986 (0.00558) 0.07722 (0.00456) 0.08036 (0.00576) 0.07591 (0.00464) 0.08299 (0.00628) 0.07802 (0.00488) 0.06802 (0.00384) 0.07999 (0.00578) 0.01854 (0.00864)
(0.20926) (0.27150) (0.24223) (0.22103) (0.21794) (0.21621) (0.23176) (0.17606) (0.23450) (0.26211) (0.16646) (0.01155)
(0.00086) (0.00098) (0.00108) (0.00125) (0.00099) (0.00121) (0.00104) (0.00132) (0.00116) (0.00107) (0.00163) (0.00121)
(0.16769) (0.27645) (0.19587) (0.17261) (0.15508) (0.22100) (0.17269) (0.20854) (0.19942) (0.21520) (0.16274) (0.01058)
(0.00094) (0.00109) (0.00129) (0.00138) (0.00120) (0.00137) (0.00123) (0.00149) (0.00148) (0.00117) (0.00182) (0.58168)
(1.0609) (1.2478) (1.1409) (1.1185) (1.0209) (1.0586) (0.9108) (1.2386) (0.7914) (1.0945) (0.8574) (1.0958)
143
Note. The values in parentheses are sample mean squared error (SMSE) of ^c and k^ and Ô*Õ express SMSE less than Method 12.
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True value method
144
Table 5 The UWLS, WLS and ML estimates of c and k for m ¼ 30, n ¼ 10 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 2 3 4 5 6 7 8 9 10 11 12
0.02710 0.03325 0.03001 0.02889 0.02597 0.03141 0.02846 0.02731 0.02873 0.03025 0.02461 0.03002
0.00852 (0.00001) 0.01859 (0.00076) 0.00848 (0.00001) 0.00885 (0.00001) 0.00875 (0.00001) 0.00873 (0.00001) 0.00875 (0.00001) 0.00898 (0.00001) 0.00890 (0.00001) 0.00848 (0.00001) 0.00932 (0.00001) 0.00908 (0.00388)
0.08408 0.11010 0.09822 0.09119 0.08556 0.10360 0.09642 0.09506 0.10220 0.09815 0.09604 0.12221
0.01084 (0.00001) 0.00998 (0.00001) 0.01035 (0.00001) 0.01101 (0.00001) 0.01076 (0.00001) 0.01028 (0.00001) 0.01047 (0.00001) 0.01082 (0.00001) 0.01042 (0.00001) 0.01038 (0.00001) 0.01115 (0.00001) 0.00912 (0.00002)
1.7177 1.9839 1.8621 1.7667 1.7934 1.9514 1.8905 1.9157 1.9595 1.8617 1.9146 1.9024
0.01537 (0.00011) 0.01199 (0.00006) 0.01344 (0.00008) 0.01527 (0.00011) 0.01379 (0.00006) 0.01191 (0.00004) 0.01258 (0.00005) 0.01280 (0.00005) 0.01180 (0.00004) 0.01343 (0.00008) 0.01255 (0.00004) 0.01283 (0.00008)
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^
0.01985 (0.00012) 0.01901 (0.00011) 0.01950 (0.00012) 0.02048 (0.00014) 0.02007 (0.00013) 0.01952 (0.00012) 0.01758 (0.00013) 0.01942 (0.00012) 0.02136 (0.00016) 0.02069 (0.00014) 0.01982 (0.00012) 0.02023 (0.00418)
0.08460 0.10156 0.09451 0.08836 0.08895 0.10002 0.09551 0.09485 0.10027 0.09449 0.09694 0.10987
1.7850 2.0088 1.9069 1.8219 1.8322 1.9821 1.9264 1.9321 1.9890 1.9066 1.9122 1.8954
0.03770 (0.00175) 0.02664 (0.00082) 0.03122 (0.00116) 0.03705 (0.00170) 0.03232 (0.00090) 0.02576 (0.00058) 0.02805 (0.00069) 0.02980 (0.00083) 0.02536 (0.00053) 0.03119 (0.00115) 0.02933 (0.00066) 0.02386 (0.00125)
(0.00083) (0.00116) (0.00099) (0.00093) (0.00071) (0.00104) (0.00086) (0.00091) (0.00083) (0.00104) (0.00618) (0.00099)
c ¼ 0:01, ^c
(0.00828) (0.01348) (0.01095) (0.00950) (0.00820) (0.01177) (0.01026) (0.01024) (0.01113) (0.01097) (0.00968) (0.01504)
0.02580 0.02186 0.02331 0.02588 0.02433 0.02192 0.02285 0.02334 0.02189 0.02330 0.02344 0.02008 Note. The values in parentheses are sample mean squared error (SMSE) of ^c and k^ and Ô*Õ 1 2 3 4 5 6 7 8 9 10 11 12
0.01004 0.01298 0.01149 0.01082 0.00983 0.01207 0.01159 0.01033 0.01020 0.01039 0.01105 0.01401
(0.00005) (0.00008) (0.00006) (0.00006) (0.00005) (0.00007) (0.00006) (0.00005) (0.00004) (0.00006) (0.00006) (0.00007)
(0.00666) (0.00962) (0.00827) (0.00726) (0.00713) (0.00908) (0.00825) (0.00819) (0.00900) (0.00828) (0.00825) (0.01069)
(0.00039) (0.00027) (0.00030) (0.00040) (0.00032) (0.00025) (0.00027) (0.00027) (0.00024) (0.00030) (0.00027) (0.00020)
(0.1401) (0.1408) (0.0662) (0.3134) (0.3427) (0.0445) (0.2056) (0.2736) (0.0536) (0.0614) (0.1403) (0.2102)
(0.5608) (0.4571) (0.5418) (0.4855) (0.4288) (0.4132) (0.6287) (0.6386) (0.4291) (0.5945) (0.7068) (0.5122)
express SMSE less than Method 12.
J.-W. Wu et al. / Appl. Math. Comput. 158 (2004) 133–147
True value method
Table 6 The UWLS, WLS and ML estimates of c and k for m ¼ 30, n ¼ 30 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 2 3 4 5 6 7 8 9 10 11 12
0.07070 0.08837 0.08021 0.07410 0.06721 0.08205 0.07771 0.07119 0.07568 0.08053 0.06355 0.07790
0.02505 (0.00026) 0.02469 (0.00026) 0.02479 (0.00026) 0.02618 (0.00030) 0.02559 (0.00028) 0.02549 (0.00028) 0.02539 (0.00027) 0.02636 (0.00031) 0.02569 (0.00028) 0.02478 (0.00026) 0.02729 (0.00034) 0.02332 (0.00241)
0.12255 0.15620 0.13711 0.12839 0.11644 0.14350 0.13003 0.11006 0.13476 0.13647 0.12046 0.13135
0.02807 (0.00038) 0.02736 (0.00036) 0.02784 (0.00037) 0.02921 (0.00043) 0.02858 (0.00039) 0.02823 (0.00038) 0.02856 (0.00039) 0.02688 (0.00030) 0.02879 (0.00041) 0.02782 (0.00037) 0.03059 (0.00048) 0.02921 (0.00265)
1.6523 1.9855 1.8384 1.7218 1.7589 1.9649 1.8826 1.9285 1.9467 1.8397 1.8408 1.7855
0.04146 (0.00141) 0.03401 (0.00090) 0.03707 (0.00108) 0.04141 (0.00142) 0.03775 (0.00104) 0.03363 (0.00080) 0.03518 (0.00088) 0.03494 (0.00085) 0.03473 (0.00086) 0.03700 (0.00108) 0.03758 (0.00098) 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^
0.13062 0.16327 0.14950 0.14173 0.12624 0.15096 0.13699 0.13385 0.13650 0.15190 0.12643 0.09648
0.04956 0.04972 0.04947 0.05114 0.05107 0.05168 0.05190 0.05373 0.05275 0.04943 0.05466 0.02182
0.08106 0.09788 0.09088 0.08439 0.08596 0.09724 0.09265 0.09470 0.09780 0.08999 0.09550 0.10775
0.02680 (0.00042) 0.02262 (0.00028) 0.02449 (0.00034) 0.02705 (0.00044) 0.02488 (0.00032) 0.02226 (0.00023) 0.02329 (0.00027) 0.02328 (0.00026) 0.02220 (0.00023) 0.02460 (0.00035) 0.02360 (0.00027) 0.02358 (0.00028)
1.6299 2.0093 1.8261 1.7083 1.6845 1.9278 1.8587 1.7083 1.8771 1.9733 1.8148 1.7855
0.07580 (0.00531) 0.06577 (0.00408) 0.07067 (0.00466) 0.07666 (0.00550) 0.07323 (0.00480) 0.06673 (0.00395) 0.07121 (0.00366) 0.07666 (0.00550) 0.06947 (0.00424) 0.06508 (0.00371) 0.06998 (0.00439) 0.05620 (0.00652)
1 2 3 4 5 6 7 8 9 10 11 12
(0.00765) (0.00927) (0.00962) (0.00812) (0.00662) (0.00995) (0.00865) (0.00792) (0.00824) (0.00970) (0.00613) (0.00927)
(0.02762) (0.04236) (0.03574) (0.03129) (0.02500) (0.03533) (0.02981) (0.02925) (0.02987) (0.03677) (0.02632) (0.01589)
(0.00170) (0.00174) (0.00171) (0.00184) (0.00183) (0.00190) (0.00191) (0.00208) (0.00199) (0.00171) (0.00215) (0.01922)
(0.02113) (0.03299) (0.02650) (0.02336) (0.01898) (0.02813) (0.02372) (0.01025) (0.02423) (0.02534) (0.01985) (0.02174)
(0.00610) (0.00889) (0.00766) (0.00663) (0.00657) (0.00847) (0.00766) (0.00808) (0.00843) (0.00755) (0.00791) (0.01018)
(0.1005) (0.3238) (0.0492) (0.3381) (0.3585) (0.1467) (0.2017) (0.0286) (0.0398) (0.0564) (0.5426) (0.2101)
(0.1508) (0.2152) (0.1715) (0.4253) (0.2483) (0.1171) (0.1245) (0.1253) (0.0501) (0.1918) (0.1989) (0.1441)
145
Note. The values in parentheses are sample mean squared error (SMSE) of ^c and k^ and Ô*Õ express SMSE less than Method 12.
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True value method
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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).
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