Journal of Statistical Theory and Practice

, Volume 12, Issue 1, pp 93–99

# Coverage probability and exact inference

• Bikas K. Sinha
Article

## 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 (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 $$CP(n,c) = P[ - c < {\hat \theta _n} - \theta < c],c > 0$$. 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

62F10 62F12

## References

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3. Lehmann, E. L. 1999. Elements of large-sample theory. New York, NY: Springer-Verlag.