0.4
0.6
0.8
1.0
4 3 1 0 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.4
0.6
0.8
N = 100, p 2 = 0.5
N = 100, p 2 = 0.9
0.4
0.6
0.8
1.0
3 0
1
2
^ N N
3 0
1
2
^ N N
3 2 0
0.2
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
p1
N = 500, p 2 = 0.1
N = 500, p 2 = 0.5
N = 500, p 2 = 0.9
0.6
0.8
1.0
3 0
1
2
^ N N
3 0
1
2
^ N N
3 2 1
0.4
1.0
4
p1
4
p1
0.2
1.0
4
N = 100, p 2 = 0.1 4
p1
4
p1
0 0.0
0.2
p1
4
0.0
2
^ N N
3 1 0 0.2
1
^ N N
2
^ N N
3 2 0
1
^ N N
0.0
^ N N
N = 20, p 2 = 0.9
4
N = 20, p 2 = 0.5
4
N = 20, p 2 = 0.1
0.0
0.2
0.4
p1
0.6 p1
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
p1
Figure 2: Relative performance of the Chapman estimator as a function of p1 for fixed N and p2 . The solid line, the upper and lower dotted lines denote the mean, the 97.5% and 2.5% quantiles bC /N . of the simulated distribution of N The performance exposed above is also shown in Figure 3 and Figure 4. For example, for all the values bLP is greater than the variability of N bC , and for the of N reported with p1 = p2 = 0.1, the variability of N Lincoln–Petersen estimator the bias goes from negative to positive and it approaches zero very slowly as N increases, in contrast with the Chapman estimator, which is nearly unbiased for N > 500 with the same record probabilities. In general, for small record probabilities, the performance of the Chapman estimator is better in bias and variance. This also can be shown by the comparison of Figure 3 and Figure 4 for p1 = 0.3, p2 = 0.1. Also for the other configurations of p1 and p2 reported, the variances of the Chapman estimator are lower than the variances of the Lincoln–Petersen estimator. However, the performance of both estimators is very similar for high record probabilities, for all values of N reported, as it can be shown for p1 = 0.7, p2 = 0.9.
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