On an Asymptotic Theory of Conditional and Unconditional Coverage Probabilities of Empirical Bayes Con®dence Intervals GAURI SANKAR DATTA University of Georgia MALAY GHOSH University of Florida DAVID DANIEL SMITH U.S. Bureau of the Census PARTHASARATHI LAHIRI University of Nebraska ABSTRACT. Empirical Bayes (EB) methodology is now widely used in statistics. However, construction of EB con®dence intervals is still very limited. Following Cox (1975), Hill (1990) and Carlin & Gelfand (1990, 1991), we consider EB con®dence intervals, which are adjusted so that the actual coverage probabilities asymptotically meet the target coverage probabilities up to the second order. We consider both unconditional and conditional coverage, conditioning being done with respect to an ancillary statistic. Key words: ancillary statistic, bias correction, calibration 1. Introduction Empirical Bayes (EB) methodology is now widely used in statistics for solving problems of both theoretical and applied interest. Indeed, the research explosion in this area during the past two decades has been simply phenomenal. Despite this wide popularity, the EB methods are still primarily con®ned to the development of point estimators. Construction of EB con®dence intervals has been very sparse. This can partly be attributed to the fact that naõÈve EB con®dence intervals fail to take into account the uncertainty due to estimation of prior parameters. Consequently, naõÈve EB estimators of the posterior variances are usually smaller than what they should actually be. This, in turn, usually leads to con®dence intervals which are too short, thereby failing to meet the target coverage probabilities. Cox (1975) initiated the discussion on construction of EB con®dence intervals in a simple normal setting. His intervals meet the target coverage probabilities asymptotically up to Om 1 , when m, the number of parameters, tends to in®nity. He suggested also construction of EB con®dence intervals conditional on some statistic. A dierent approach towards the construction of EB con®dence intervals is given in Morris (1983a, b). Morris's method consists essentially in ®nding a hierarchical Bayesian (HB) con®dence interval for the parameter of interest, approximating this interval with estimates of the prior parameters only at the last stage. However, since the posteriors are typically skewed, Morris's choice of normal percentiles for construction of EB con®dence intervals is basically an approximation. Later Laird & Louis (1987) proposed EB bootstrap con®dence intervals in the spirit of Morris (1983a, b), while Carlin & Gelfand (1990), following a suggestion of Efron (1987), proposed calibrating the naõÈve EB con®dence intervals by correcting the bias. They suggested also Ó Board of the Foundation of the Scandinavian Journal of Statistics 2002. Published by Blackwell Publishers Ltd, 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA Vol 29: 139±152, 2002