Design and analysis of shortest two-sided confidence intervals for a probability under prior information
Two-sided confidence intervals for a probability <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>p</mi> </mrow> </math> </EquationSource> </InlineEquation> under a prescribed confidence level <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\gamma $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">γ</mi> </mrow> </math> </EquationSource> </InlineEquation> are an elementary tool of statistical data analysis. A confidence interval has two basic quality characteristics: i) exactness, i. e., whether the actual coverage probability equals or exceeds the prescribed level <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\gamma $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi mathvariant="italic">γ</mi> </mrow> </math> </EquationSource> </InlineEquation>; ii) inferential precision, measured by the length of the confidence interval. The interval provided by Clopper and Pearson (Biometrika 26:404–413, <CitationRef CitationID="CR9">1934</CitationRef>) is the only exact interval actually used in statistical data analysis. Various authors have suggested shorter, i. e., more precise exact intervals. The present paper makes two contributions. i) We provide a general design scheme for minimum volume confidence regions under prior knowledge on the target parameter. ii) We apply the scheme to the problem of confidence intervals for a probability <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$p$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mrow> <mi>p</mi> </mrow> </math> </EquationSource> </InlineEquation> where prior knowledge is expressed in a flexible way by a beta distribution on a subset of the unit interval. Copyright Springer-Verlag Berlin Heidelberg 2014
