Statistica Sinica 13(2003), 19-49

INTERVAL ESTIMATION IN EXPONENTIAL FAMILIES

Lawrence D. Brown$^1$, T. Tony Cai$^1$ and Anirban DasGupta$^2$

$^1$University of Pennsylvania and $^2$Purdue University

Abstract: In this paper we consider interval estimation of the mean in the natural Exponential family with a quadratic variance function; the family comprises the binomial, Poisson, negative binomial, normal, gamma, and a sixth distribution.

For the three discrete cases, the Wald confidence interval and three alternative intervals are examined by means of two term Edgeworth expansions of the coverage probability and a two term expansion of the expected length. The results and additional computation suggest that the equal tailed Jeffreys interval and the likelihood ratio interval are the best overall alternatives to the Wald interval. We also show that the poor performance of the Wald interval is not limited to the discrete cases, and a serious negative bias occurs in the nonnormal continuous cases as well. The results are complemented by various illustrative examples.

Key words and phrases: Bayes, binomial distribution, confidence intervals, coverage probability, edgeworth expansion, expected length, Jeffreys prior, natural exponential family, negative binomial distribution, normal approximation, Poisson distribution, quadratic variance function.