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SUMMARY The use of a simple permutation argument to set exact confidence limits for the location parameter of a symmetric distribution is described. The argument is applied to estimates based on original observations and on ranks and to M-estimates. Some key word8: Exact confidence limit; Location estimate; M-estimate; Ranks; Robustness.

SUMMARY The smooth empirical Bayes approach is based on using past observations to estimate a specified type of approximation to the prior distribution. This approach is applied to problems of empirical Bayes hypothesis testing and to the compound decision problem. Problems involving composite hypotheses are considered which have, hitherto, received much less attention than the case of two simple hypotheses. The methods are shown to be asymptotically optimal and results of studies involving finite sample numbers are reported, giving some indication of the rate of approach to optimality in specific cases.

SUMMARY A number of established methods of empirical Bayes estimation for the Poisson distribution are surveyed, and their performances are compared. A new method of smooth empirical Bayes estimation is presented, and its performance is also studied. The question of assessing the effectiveness of the empirical Bayes approach in practice is examined.

SUMMARY A method for setting a distribution-free lower confidence limit for the mean of a nonnegative random variable is developed, and compared with the procedure which relies on asymptotic normality of the distribution of the sample mean. Some key word8: Distribution-free confidence limit; Nonparametric; Robustness.

SUMMARY A study is made of the simple empirical Bayes estimators proposed by Robbins (1956). These estimators are compared with 'best' conventional estimators in terms of their expected squared error 'loss'. The object of the study is to determine the amount of prior data which would be needed for the empirical Bayes estimator to be preferred to the conventional estimator. It is concluded that the required number of previous observations is likely to be too large for the simple empirical Bayes estimators to be useful in practice. Smooth empirical Bayes estimators are proposed, which make more effective use of prior results. A method of smoothing is developed which is based on estimating a step function approximation to the prior distribution of the parameter. Some examples are studied in detail, and the results indicate that these smooth empirical Bayes estimators are potentially useful in practice.

SUMMARY Although the conditions for application of the technique of empirical Bayes estimation of the parameter of a distribution may exist, the form of the prior distribution is generally unknown. This often creates major difficulties in the determination of empirical Bayes estimators. The simple device of approximating the prior distribution by a step function is used to overcome this problem and to obtain smooth empirical Bayes estimators. This paper supplements earlier work by considering continuous distributions and multiple past and current observations. The effectiveness of the proposed smooth empirical Bayes estimators is examined, and the results indicate that they can be substantially better than optimum non-Bayes estimators, even when the amount of past data is small.