Detailed Statistical Inference – From Multiple Decisions to Confidence Sets – A Non­Bayesian Approach

A new approach to detailed statistical inference for the two-decision and multiple decision problem was developed by Chatterjee and Chattopadhyay (1992, 1993). The procedure developed for the multipledecision case is used here to find a solution to, what may be called, the problem of estimation by a ‘confidence set’ when the parameter space is finite. The confidence set problem in which we are required to state a datadependent measure of ‘confidence’ which can be interpreted in terms of betting odds that are ‘optimal’ under repeated experimentation, is considered. A truncated form of utility function is introduced. The case of the truncated form of logarithmic utility function is considered in detail. AMS {2000} Subject Classification: 62A01, 62F99.

Detailed Statistical Inference-multiple Decision Problem

The procedure for detailed statistical inference developed by the authors in an earlier paper (1992) for the two-decision case, is extended here to the case of several decisions. Alongwith the rule for choosing the decision, the problem of stating data dependent measures of confidence in terms of betting odds, is considered. The extension involves generalization oftlie coneept of legitimacy of betting odds introduced in the earlier paper and the choice of a suitable utility function for bets. The actual solution is worked out in the case of logarithmic utility. A rather intricate mathematical result requires to be established to prove the existence of an optimum rule in this case. Application of the procedure is illustrated through some numerical examples.