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. 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 3:

Sample means and standard deviations for Volterra (V) and Dabrowska (D) estimators of the bivariate survival function under a Clayton–Oakes model (9) with γ = 0 with unit exponential marginal distributions and dependency parameter θ. Censoring times were also exponentially distributed and independent of each other with censoring rate c taking values 0 (no censoring), 2 (moderate censoring), or 5 (heavy censoring) at various sample sizes (n), based on 1000 simulations at each configuration

Marginal Survival Joint Survival Volterra n = 100 n = 100 n = 100 n = 250 n = 500
Probabilities Probability Dabrowska c = 0 c = 2 c = 5 c = 5 c = 5
θ=0 (independence)
0.85 0.85 0.723 V 0.723(0.045) 0.723(0.050) 0.723(0.045) 0.722(0.036) 0.722(0.018)
D 0.723(0.045) 0.723(0.050) 0.723(0.045) 0.722(0.036) 0.722(0.018)
0.85 0.70 0.595 V 0.594(0.048) 0.595(0.060) 0.594(0.048) 0.593(0.054) 0.595(0.026)
D 0.594(0.048) 0.594(0.060) 0.594(0.048) 0.593(0.053) 0.595(0.026)
0.85 0.55 0.468 V 0.467(0.050) 0.468(0.072) 0.467(0.050) 0.465(0.089) 0.468(0.043)
D 0.467(0.050) 0.467(0.071) 0.467(0.050) 0.465(0.095) 0.468(0.041)
0.70 0.70 0.490 V 0.491(0.050) 0.491(0.067) 0.491(0.050) 0.492(0.069) 0.489(0.050)
D 0.491(0.050) 0.490(0.066) 0.491(0.050) 0.431(0.069) 0.488(0.047)
0.70 0.55 0.385 V 0.385(0.048) 0.386(0.078) 0.385(0.048) 0.394(0.102) 0.381(0.075)
D 0.385(0.048) 0.385(0.073) 0.385(0.048) 0.387(0.123) 0.3880(0.086)
0.55 0.55 0.303 V 0.302(0.044) 0.306(0.090) 0.302(0.044) 0.311(0.122) 0.298(0.095)
D 0.302(0.044) 0.304(0.085) 0.302(0.044) 0.316(0.142) 0.303(0.117)
θ=2 (cross ratio= 3)
0.85 0.85 0.752 V 0.752(0.043) 0.752(0.047) 0.753(0.054) 0.752(0.034) 0.752(0.025)
D 0.752(0.043) 0.752(0.047) 0.753(0.054) 0.752(0.034) 0.752(0.025)
0.85 0.70 0.642 V 0.641(0.048) 0.641(0.058) 0.642(0.084) 0.641(0.053) 0.642(0.037)
D 0.641(0.048) 0.641(0.058) 0.642(0.087) 0.641(0.052) 0.642(0.036)
0.85 0.55 0.521 V 0.519(0.050) 0.519(0.072) 0.529(0.135) 0.520(0.087) 0.520(0.064)
D 0.519(0.050) 0.519(0.070) 0.522(0.156) 0.519(0.090) 0.520(0.063)
0.70 0.70 0.570 V 0.569(0.049) 0.570(0.064) 0.572(0.112) 0.565(0.070) 0.571(0.050)
D 0.569(0.049) 0.570(0.062) 0.574(0.128) 0.567(0.070) 0.571(0.046)
0.70 0.55 0.480 V 0.478(0.049) 0.480(0.078) 0.493(0.150) 0.483(0.108) 0.479(0.086)
D 0.478(0.049) 0.480(0.071) 0.480(0.183) 0.480(0.136) 0.480(0.084)
0.55 0.55 0.422 V 0.422(0.048) 0.426(0.095) 0.429(0.171) 0.416(0.138) 0.412(0.111)
D 0.422(0.048) 0.427(0.086) 0.427(0.196) 0.421(0.160) 0.422(0.118)