# Coverage probability and exact inference

## Abstract

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 \({\hat \theta _n}\) of *θ* for every *η* = *k, k + l, k +* 2,..., where *k* is the minimum sample size needed for existence of unbiased estimator(s) of *θ* based on (*X*_{1}, *X*_{2},...,*X*_{k}). 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 \(CP(n,c) = P[ - c < {\hat \theta _n} - \theta < c],c > 0\). For *θ >
* 0, we may redefine *CP(n, c)* as \(CP(n,c) = P[1 - c < {\hat \theta _n}/\theta < 1 + c]\) since \(E({\hat \theta _n}/\theta ) = 1\). 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.

## Keywords

Point estimation unbiasedness exact inference consistency## AMS Subject Classification

62F10 62F12## Preview

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## References

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