Unbiased Estimators for the Parameters of the Binomial and Multinomial Distributions

TL;DR

This paper derives exact unbiased estimators for binomial and multinomial distribution parameters, improving estimation accuracy especially for small samples, and validates their performance through simulations.

Contribution

It introduces new unbiased estimators for multinomial parameters and their moments, extending previous methods and providing exact expressions for their expected values.

Findings

Estimators outperform asymptotic unbiased estimators in small samples.

Derived expressions are valid for arbitrary bin counts and moments.

Simulation results confirm improved accuracy of the estimators.

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