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Journal of Statistical Theory and Practice

, Volume 12, Issue 1, pp 93–99 | Cite as

Coverage probability and exact inference

  • Gaurangadeb Chattopadhyay
  • 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 

AMS Subject Classification

62F10 62F12 

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References

  1. Brown, L., T.T. Cai, and A. DasGupta. 2001. Interval estimation for a binomial proportion. Statistical Science 16 (2):101–17.MathSciNetMATHGoogle Scholar
  2. Casella, G., and R. L. Berger. 2002. Statistical inference, 2nd ed. San Francisco, CA: Duxbury Press Wadsworth/Brooks Cole.MATHGoogle Scholar
  3. Lehmann, E. L. 1999. Elements of large-sample theory. New York, NY: Springer-Verlag.CrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing, 20 Middlefield Ct, Greensboro, NC 27455 2018

Authors and Affiliations

  1. 1.Department of StatisticsCalcutta UniversityKolkataIndia
  2. 2.Indian Statistical InstituteKolkataIndia

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