Unbiased Estimators for the Parameters of the Binomial and Multinomial Distributions

Abstract

The exact expression is derived for the expected value, $< {p_i}> $, for the parameter for any bin $i$ of a histogram following a multinomial distribution derived by sorting $N$ observations into bins of $B$ classes, if $n_i$ of the observations are found to be sorted into bin $i$. This expected value is found to be $ < {p_i}> = \frac {n_i + 1} {N + B}$. The expected value for the variance is found to be $\frac{< p_i > (1-< p_i >)}{N+B+1}$. A general expression is derived to determine $< {p_i}^z > $ for arbitrary values of $B$ and $z$. These expressions hold provided there is no \emph{a priori} reason for $p_i$ associated with any bin to have a value that is exactly equal to 0. For the particular case of the binomial distribution (B=2), these estimators are tested by examining how often the value of $p_{true}$, the value which is used to generate sets of pseudo-random binomial variates, falls within 1.96 estimated standard deviations of the estimated value $< p >$. When compared with the results of identical, earlier reported tests for small sample sizes, the unbiased estimators derived here predictably outperform \emph{asymptotically} unbiased estimators