 Lincoln-Petersen estimator

The Lincoln-Petersen estimator is the ratio estimator obtained by matching the observed overlap to its hypergeometric expected value. It is defined when $R>0$. To distinguish that ratio from the exact integer maximization, consecutive positive likelihoods satisfyThis comparison is valid for $N\ge M+C-R$. For the toy counts, the numerator minus the denominator is $500-5(N+1)$. Thus the likelihood function increases up to ninety-nine, ties at ninety-nine and one hundred, then decreases for every larger integer. This independently establishes the global maximum-likelihood estimates, beyond the plotting window. The adjacent-ratio method is given in Guiteras, §8.