= Chapman estimator
{c}
{title2=$\widehat N_{\mathrm{Ch}}=\frac{(M+1)(C+1)}{R+1}-1$}
The <Chapman estimator> is a correction intended to reduce the small-sample <bias> of the <Lincoln-Petersen estimator>. Under the fixed-size <hypergeometric> sampling assumptions it is exactly an <unbiased estimator> when $M+C\geq N$; it can have nonzero <bias> otherwise. This qualification and the formula are stated in https://www.maths.otago.ac.nz/~rbarker/Thesis/thesis.pdf[Barker, §1.1.1, equation (1.3), citing Chapman's work through Seber]. For the toy observations,
$$
\widehat N_{\mathrm{Ch}}=\frac{(20+1)(25+1)}{5+1}-1
=\frac{546}{6}-1=90.
$$
It is an alternative <estimator>, not a <maximum-likelihood estimate> for this <capture-recapture model>. The <Chapman estimator> remains finite when $R=0$, but a finite output does not supply the missing overlap information or repair unequal catchability, lost marks, or population change.
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