Table 1.

Simulation results for gamma frailty.

CR	σ2	σ̂2	SD	β	NPMLE				J-S1		J-S2
					Bias(%)	SD	SE	CP(%)	Bias(%)	SD	Bias(%)	SD
λ0(t): Weibull type
15%	2	1.98	.30	β̂1	−0.5	.068	.08	97	0.3	.078	0.0	.104
				β̂2	0.5	.061	.07	97	−0.4	.069	0.1	.090
40%	2	1.96	.31	β̂1	−1.6	.085	.09	95	1.2	.090	−1.0	.109
				β̂2	1.5	.076	.08	94	−0.9	.081	1.3	.096
15%	1	0.98	.17	β̂1	−0.5	.069	.07	95	0.3	.075	−0.4	.082
				β̂2	0.4	.060	.06	94	−0.3	.064	0.3	.071
40%	1	0.97	.19	β̂1	−2.2	.082	.10	97	0.4	.089	−1.1	.093
				β̂2	2.3	.071	.08	96	−0.3	.076	1.1	.081
15%	0.5	0.48	.12	β̂1	−0.9	.062	.08	97	−0.1	.069	−0.5	.069
				β̂2	1.1	.055	.07	97	0.1	.061	0.7	.062
40%	0.5	0.47	.14	β̂1	−2.4	.077	.09	96	−0.2	.083	−1.7	.082
				β̂2	2.9	.070	.08	96	0.2	.075	1.9	.074
λ0 (t): reciprocal type
15%	2	2.00	.30	β̂1	−0.3	.154	.17	96	0.6	.161	0.3	.234
				β̂2	1.4	.138	.14	95	0.5	.143	0.8	.208
40%	2	2.00	.32	β̂1	−2.5	.173	.18	95	1.0	.179	0.3	.219
				β̂2	4.0	.154	.16	94	0.8	.157	1.0	.201
15%	1	0.99	.17	β̂1	−1.5	.143	.15	96	−0.5	.144	−0.9	.166
				β̂2	1.4	.125	.13	96	0.8	.128	0.5	.148
40%	1	0.98	.22	β̂1	−4.2	.163	.18	96	−0.8	.160	−1.6	.172
				β̂2	5.2	.146	.16	96	1.0	.145	1.0	.152
15%	0.5	0.48	.11	β̂1	−1.0	.136	.13	94	0.6	.136	0.2	.139
				β̂2	1.1	.124	.12	93	−1.0	.122	0.2	.122
40%	0.5	0.47	.12	β̂1	−5.1	.156	.17	95	0.8	.157	−0.3	.156
				β̂2	5.2	.140	.15	94	−0.9	.139	1.0	.138

^†CR, censoring rate; σ², true variance of frailty; σ̂², estimate for σ²; SD, sample standard deviation; SE, mean of estimated standard errors; CP, empirical coverage probability of 95% Wald-type confidence interval; NPMLE, proposed nonparametric maximum likelihood estimator; J-S¹, smoothed EM-like estimator; J-S², induced smoothing estimator.