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

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