Coverage probability and exact inference

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Research Article

Publication Title

Journal of Statistical Theory and Practice


With reference to “point estimation” of a real-valued parameter Θ involved in the distribution of a real-valued random variable X, we consider a sample size n and an underlying unbiased estimator (Formula presented.) of Θ for every n = k, k + 1, k + 2, . . . ; where k is the minimum sample size needed for existence of unbiased estimator(s) of Θ based on (X1, X2, .... Xk).. We wish to investigate exact small-sample properties of the sequence of estimators considered here. This we study by considering what is termed “coverage probability” (CP) and defined as (Formula presented.). For Θ > 0, we may redefine CP(n,c) as (Formula presented.) since (Formula presented.). When Θ > 0 serves as a scale parameter, bounds to the ratio are more meaningful than bounds to the difference. In either case, it is desired that the sequence CP(n, c) n = k, k + 1, k + 2, ......] behaves like an increasing sequence for every c > 0. We may note in passing that we are asking for a property beyond “consistency” of a sequence of unbiased estimators. As is well known, consistency is a large-sample property. In this article we discuss several interesting features of the behavior of the CP(n,c) in the exact sense.

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