Estimation of the Probability That an Observation Will Fall into a Specified Class

When observations in a random sample of n are classified into k categories with n i falling into a given class, the usual estimate of the corresponding probability is ni/n. When a priori information about the distribution of the probabilities is available, more precise estimates can be derived from data in any one sample of n. When the a priori distribution is not specified completely, but its general form can be inferred, the parameters of that distribution can be estimated from the average of the ni/n and their estimated sampling variances. The computations are analogous to those that arise in regression theory with the bivariate normal frequency distribution when Y = X + e and the expected value of X for a given Y is estimated from the regression of X on Y. The parameters of the distribution of X have to be estimated from the observed distribution of Y and the sampling errors in the individual values of those observations.