. Author manuscript; available in PMC: 2022 Jan 1.

Published in final edited form as: J Am Stat Assoc. 2020 Feb 10;116(535):1330–1345. doi: 10.1080/01621459.2020.1713792

Table 2:

Simulation summary statistics^* for double failure hazard estimator ${\hat{Λ}}_{11}$ at selected marginal survival probabilities given z, under the Clayton-Oakes model (9) with β₁₀ = β₀₁ = γ = log 2 and dependency parameter $θ = 2,$ at both z = 0 and z = 1.

Sample size (n)		500				1000				250
T_i and T₂ Failing %		22.6				22.6				100
		${\hat{Λ}}_{11}$				${\hat{Λ}}_{11}$				${\hat{Λ}}_{11}$
(T₁, T₂) Marginal	$Λ_{11}$	Sample	Sample	Empirical	95% CI	Sample	Sample	Empirical	95% CI	Sample	Sample	Empirical	95% CI
Survival Rates		Mean	SD	SD	Coverage	Mean	SD	SD	Coverage	Mean	SD	SD	Coverage
	z = 0
(0.85, 0.85)	0.060	0.060	0.021	0.020	0.916	0.060	0.015	0.014	0.921	0.060	0.016	0.015	0.925
(0.85, 0.70)	0.114	0.115	0.041	0.039	0.910	0.115	0.029	0.028	0.934	0.113	0.025	0.024	0.925
(0.85, 0.55)	0.161	0.159	0.070	0.060	0.855	0.163	0.048	0.045	0.898	0.159	0.033	0.032	0.916
(0.70, 0.70)	0.226	0.226	0.083	0.076	0.900	0.226	0.055	0.055	0.927	0.224	0.041	0.040	0.932
(0.70, 0.55)	0.330	0.326	0.151	0.115	0.848	0.331	0.103	0.090	0.901	0.327	0.058	0.055	0.917
(0.55, 0.55)	0.500	0.483	0.274	0.162	0.750	0.497	0.198	0.139	0.843	0.499	0.082	0.078	0.935
	z = 1
(0.85, 0.85)	0.046	0.044	0.016	0.015	0.903	0.045	0.011	0.011	0.927	0.046	0.017	0.017	0.928
(0.85, 0.70)	0.092	0.090	0.027	0.027	0.900	0.091	0.020	0.019	0.928	0.092	0.026	0.026	0.946
(0.85, 0.55)	0.140	0.138	0.043	0.042	0.876	0.139	0.031	0.030	0.921	0.140	0.034	0.035	0.947
(0.70, 0.70)	0.189	0.187	0.050	0.049	0.900	0.186	0.034	0.034	0.926	0.188	0.041	0.041	0.937
(0.70, 0.55)	0.290	0.290	0.082	0.076	0.906	0.287	0.055	0.054	0.927	0.288	0.054	0.054	0.943
(0.55, 0.55)	0.453	0.452	0.129	0.120	0.880	0.447	0.089	0.085	0.924	0.449	0.076	0.074	0.937

Sample mean and standard deviation (SD) based on 1000 simulations at each sample configuration. Empirical SD is the average of SD estimates based on the empirical variance estimator for ${\hat{Λ}}_{11}$ , and 95% Cl coverage is the fraction of the 1000 simulated samples where the asymptotic confidence interval using the empirical SD includes the true $Λ_{11}$ value.