A0712
Title: Estimation of $n$ in the binomial $(n,p)$ distribution with both parameters unknown
Authors: Yoshiji Takagi - Nara University of Education (Japan) [presenting]
Abstract: Estimation of the population size $n$ in the binomial $(n,p)$ distribution with unknown success probability $p$ has a long history of over eighty years, but easily computable and easily motivated estimators are still generally lacking. When $p$ is apparently near zero, the familiar estimators obtained by frequentist methods are confronted by two difficulties, instability and underestimation. Indeed, the maximum likelihood estimator and the moment estimator can be extremely unstable in the sense that changing an observed success count $s$ to $s+1$ can result in a massive change in the estimate of $n$. The sample maximum strongly underestimates the true $n$ even for large sample sizes. The Bayesian approach is useful to overcome these difficulties, and several Bayesian estimators have been proposed. However, it is debatable how one should choose the parameters in the prior distribution. Here, the frequentist methods are reconsidered, and a new estimator constructed by four values is proposed: the sample minimum, the sample maximum, the sample mean and the sample variance. Last, some properties of this estimator are examined, including stability and less underestimation.
