2013/02/23 by Jonathan M. Friedman, Friedman, Jonathan M.
Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Statistical Distribution Estimation and Applications #Statistical Methods and Bayesian Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1302.5749
openalex publication_date 2013/02/23 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
The exact expression is derived for the expected value, < pi> , 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 ni of the observations are found to be sorted into bin i. This expected value is found to be < pi> = \frac ni + 1 N + B. The expected value for the variance is found to be (< pi > (1-< pi >))/(N+B+1). A general expression is derived to determine < piz > for arbitrary values of B and z. These expressions hold provided there is no a priori reason for pi 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 ptrue, 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 asymptotically unbiased estimators