EMPIRICAL BAYES ESTIMATES OF DOMAIN SCORES UNDER BINOMIAL AND HYPERGEOMETRIC DISTRIBUTIONS FOR TEST-SCORES

We introduce two simple empirical approximate Bayes estimators (EABEs) -d(N)(x) and delta(N)(x)-for estimating domain scores under binomial a nd hypergeometric distributions, respectively. Both EABEs (derived fro m corresponding marginal distributions of observed test score x withou t relying on knowledge of prior domain score distributions) have been proven to hold DELTA-asymptotic optimality in Robbins' sense of conver gence in mean. We found that, where d(N) and delta(N)* are the monoto nized versions of d(N) and delta(N) under Van Houwelingen's monotoniza tion method, respectively, the convergence rate of the overall expecte d loss of Bayes risk in either d(N) or delta(N)* depends on test leng th, sample size, and ratio of test length to size of domain items. In terms of conditional Bayes risk, d(N) and delta(N)* outperform their maximum likelihood counterparts over the middle range of domain scales . In terms of mean-squared error, we also found that: (a) given a unim odal prior distribution of domain scores, delta(N) performs better th an both d(N) and a linear EBE of the beta-binomial model when domain item size is small or when test items reflect a high degree of heterog eneity; (b) d(N) performs as well as delta(N)* when prior distributio n is bimodal and test items are homogeneous; and (c) the linear EBE is extremely robust when a large pool of homogeneous items plus a unimod al prior distribution exists.