Bayesian Estimation of Two-Parameter Weibull Distribution Using Extension of Jeffreys' Prior Information with Three Loss Functions

Table 2

Estimated values for shape parameter (𝛽).

𝑛	𝛼	𝑐	𝛽	𝛽ML	𝛽BS	𝛽BL	𝛽BG	𝛽BL	𝛽BG	𝛽BL	𝛽BG	𝛽BL	𝛽BG
						𝑎=𝑘=0.6		𝑎=𝑘=−0.6		𝑎=𝑘=1.6		𝑎=𝑘=−1.6
	0.5	0.4	0.8	1.0454	1.0477	0.8564	1.0281	0.7159	1.1591	0.7062	0.8973	0.8018	0.8057
	0.5	0.4	1.2	1.3514	1.3527	1.1073	1.0796	1.1406	1.1407	1.1272	1.1940	1.0460	1.4267
	0.5	1.4	0.8	0.6586	0.6389	0.8242	1.0641	1.0882	0.7452	0.8399	0.8443	0.9755	0.6734
25	0.5	1.4	1.2	1.4325	1.4535	1.0676	1.3325	1.1003	1.2631	1.1017	1.5879	1.7119	1.4090
	1.5	0.4	0.8	0.8348	0.8372	0.9252	0.7872	0.8233	0.7591	0.6915	0.8214	1.0681	0.7146
	1.5	0.4	1.2	1.5022	1.5066	1.6441	1.1465	1.0843	1.2717	0.9345	1.0699	1.3695	1.1185
	1.5	1.4	0.8	0.7619	0.7479	0.9579	0.9326	1.0925	0.7201	0.7907	0.8520	0.9334	0.8799
	1.5	1.4	1.2	1.2474	1.2634	1.1192	0.9848	1.3929	1.4066	0.9187	1.7935	1.3334	1.3515
	0.5	0.4	0.8	0.9557	0.9539	0.7057	0.8352	0.7332	0.8519	1.0029	0.9683	0.8278	0.8372
	0.5	0.4	1.2	0.9728	0.9722	1.1239	1.3097	1.1101	1.2972	1.3873	1.2662	1.3432	1.1437
	0.5	1.4	0.8	0.8172	0.8122	0.9569	1.1437	0.8258	0.6818	0.6581	0.8804	0.7442	0.9061
50	0.5	1.4	1.2	1.1071	1.1099	1.0502	1.0761	1.2463	1.4298	0.9602	1.3399	1.0862	1.2761
	1.5	0.4	0.8	0.8729	0.8732	0.8280	0.7585	0.8472	0.8479	0.8013	0.6405	0.8036	0.7602
	1.5	0.4	1.2	0.9723	0.9718	1.2733	1.1256	1.0164	1.0970	1.2592	1.2069	1.4058	1.4118
	1.5	1.4	0.8	0.9268	0.9252	0.6871	0.8229	0.9727	0.7801	0.8241	0.8163	0.8887	1.9312
	1.5	1.4	1.2	1.5180	1.5311	1.3820	1.2674	1.1753	1.0914	1.1521	1.0809	1.1595	1.7258
	0.5	0.4	0.8	0.8278	0.8278	0.8695	0.8295	0.8808	0.8804	0.8652	0.8872	0.7676	0.8867
	0.5	0.4	1.2	1.4511	1.4516	1.0680	1.1749	1.1831	1.0743	1.1896	1.1903	1.2274	1.2721
	0.5	1.4	0.8	0.8587	0.8570	0.8663	0.8560	0.7669	0.7513	0.7885	0.6868	0.9070	0.7282
100	0.5	1.4	1.2	1.2865	1.2901	1.1739	1.2018	1.2087	1.1995	1.0343	1.2006	1.1511	1.1779
	1.5	0.4	0.8	0.8042	0.8031	0.8366	0.7696	0.8478	0.7576	0.8695	0.7714	0.8057	0.7576
	1.5	0.4	1.2	1.2398	1.2401	1.1890	1.4574	1.1682	1.1526	1.1927	1.1173	1.3274	1.1577
	1.5	1.4	0.8	0.7971	0.7939	0.7608	0.7807	0.8389	0.7186	0.7693	0.7596	0.7385	0.9887
	1.5	1.4	1.2	1.1083	1.1099	1.2435	1.1113	1.4072	1.1371	1.1197	1.1448	1.2652	1.1783

ML: maximum likelihood, BG: general entropy loss function, BL: LINEX loss function, BS: squared error loss function.