An Empirical Likelihood Method for Semiparametric Linear Regression with Right Censored Data

Table 1

Comparison of the coverage probability (CP) and average width (Width) of two empirical likelihood confidence intervals for the slope parameter under four different models with various sample size (𝑛) and censoring rate (CR). Here, ELEE is the proposed method, and ELSD is the method of Li and Wang [15]. Each entry is based on 3,000 Monte Carol samples.

			Nominal level = 90%				Nominal level = 95%
Model	𝑛	CR	CP		width		CP		Width
			ELEE	ELSD	ELEE	ELSD	ELEE	ELSD	ELEE	ELSD
	50	0.75	0.94	0.77	1.59	2.18	0.98	0.84	1.95	2.70
	100	0.75	0.94	0.82	0.85	1.66	0.97	0.89	1.03	2.02
	500	0.75	0.91	0.88	0.31	0.81	0.96	0.94	0.37	0.97
	50	0.3	0.94	0.87	0.69	1.30	0.97	0.92	0.82	1.55
A	100	0.3	0.93	0.89	0.45	0.94	0.97	0.94	0.53	1.12
	500	0.3	0.91	0.90	0.18	0.43	0.96	0.95	0.21	0.51
	50	0.1	0.95	0.88	0.59	1.10	0.98	0.93	0.70	1.30
	100	0.1	0.93	0.89	0.39	0.79	0.97	0.94	0.47	0.94
	500	0.1	0.90	0.90	0.16	0.36	0.95	0.95	0.19	0.43
	50	0.75	0.93	0.83	1.20	2.05	0.95	0.88	1.40	2.50
	100	0.75	0.94	0.87	0.77	1.49	0.97	0.92	0.92	1.80
	500	0.75	0.93	0.89	0.30	0.67	0.96	0.94	0.36	0.80
	50	0.3	0.95	0.88	0.66	1.23	0.98	0.94	0.78	1.48
B	100	0.3	0.94	0.90	0.44	0.88	0.97	0.95	0.52	1.05
	500	0.3	0.92	0.90	0.18	0.39	0.96	0.95	0.21	0.47
	50	0.1	0.94	0.89	0.58	1.08	0.97	0.95	0.69	1.29
	100	0.1	0.94	0.90	0.39	0.77	0.97	0.94	0.46	0.92
	500	0.1	0.91	0.91	0.16	0.35	0.96	0.95	0.19	0.41
	50	0.75	0.93	0.77	1.49	2.01	0.96	0.83	2.01	2.46
	100	0.75	0.93	0.82	0.80	1.56	0.97	0.88	0.97	1.90
	500	0.75	0.92	0.87	0.29	0.76	0.96	0.93	0.35	0.92
	50	0.3	0.93	0.86	0.67	1.21	0.97	0.92	0.81	1.45
C	100	0.3	0.93	0.88	0.44	0.88	0.97	0.93	0.53	1.05
	500	0.3	0.91	0.89	0.18	0.40	0.96	0.94	0.21	0.48
	50	0.1	0.94	0.87	0.60	1.01	0.97	0.93	0.71	1.21
	100	0.1	0.93	0.89	0.39	0.73	0.97	0.94	0.47	0.87
	500	0.1	0.92	0.90	0.16	0.33	0.96	0.95	0.19	0.39
	50	0.75	0.95	0.68	1.74	2.45	0.97	0.77	1.87	2.98
	100	0.75	0.94	0.60	0.83	1.83	0.97	0.69	1.01	2.21
	500	0.75	0.92	0.12	0.30	0.89	0.96	0.18	0.36	1.06
	50	0.3	0.94	0.81	0.68	1.31	0.97	0.88	0.82	1.56
D	100	0.3	0.93	0.76	0.45	0.94	0.97	0.84	0.53	1.12
	500	0.3	0.91	0.39	0.18	0.43	0.96	0.51	0.21	0.51
	50	0.1	0.94	0.86	0.59	1.09	0.97	0.92	0.71	1.30
	100	0.1	0.94	0.87	0.39	0.78	0.97	0.93	0.47	0.93
	500	0.1	0.91	0.78	0.16	0.36	0.95	0.86	0.19	0.42
	50	0.75	0.78	0.76	1.52	2.25	0.85	0.83	1.79	2.75
	100	0.75	0.78	0.80	0.85	1.74	0.85	0.87	0.62	2.11
	500	0.75	0.73	0.85	0.34	0.88	0.81	0.92	0.40	1.06
	50	0.3	0.81	0.85	0.76	1.43	0.87	0.91	0.89	1.71
E	100	0.3	0.79	0.87	0.53	1.05	0.86	0.92	0.62	1.26
	500	0.3	0.77	0.89	0.22	0.50	0.84	0.94	0.26	0.60
	50	0.1	0.81	0.86	0.69	1.24	0.87	0.92	0.81	1.48
	100	0.1	0.80	0.87	0.49	0.91	0.86	0.93	0.57	1.08
	500	0.1	0.76	0.89	0.20	0.42	0.83	0.94	0.23	0.50

Model A: 𝑌=1+𝑋+𝜖, where 𝑋∼𝑁(0,0.5²), 𝜖∼𝑁(0,0.5²), and 𝐶∼𝑁(𝜇,4²); model B: 𝑌=1+𝑋+𝜖, where 𝑋∼ Bernoulli(0.5)−0.5, 𝜖∼𝑁(0,0.5²), and 𝐶∼𝑁(𝜇,4²); model C: 𝑌=𝑋+𝜖, where 𝑋∼𝑁(0,0.5²), 𝜖∼ Weibull (shape =1.843, scale =1), and 𝐶∼𝑁(𝜇,4²); model D (Dependent censoring): 𝑌=1+𝑋+𝜖, where 𝑋∼𝑁(0,0.5²), 𝜖∼𝑁(0,0.5²), and 𝐶∼𝑁(𝜇+2𝑋,15); model E: 𝑌=1+𝑋+𝜖, where 𝑋∼𝑁(0,0.5²), 𝜖∼𝑁(0,𝑋²), and 𝐶∼𝑁(𝜇,4²).