Confidence Intervals for the Scale Parameter of Exponential Family of Distributions

Abstract

This article presents a unified approach for computing non-equal tail optimal confidence intervals for the scale parameter of the exponential family of distributions. We prove that there exists a pivotal quantity, as a function of a complete sufficient statistic, with a chi-square distribution. Using the similarity between equations of shortest, unbiased and highest density confidence intervals, all equations are reduced into a system of two equations that can be solved via a straightforward algorithm.KEY WORDS: Equal tail, shortest and unbiased confidence intervals; Highest posterior density credible interval; Pivotal quantity; Scale-exponential family.Hossein Hoshyarmanesh and Amirhossein Karami are graduate students, and Adel Mohammadpour is an associate professor (

On the distribution of the power function for the scale parameter of exponential families

“…The idea, originally proposed by Rufybach et al 3 for tests on the expected value of the normal model, is here extended to the one‐sided testing problem of the scale parameter of exponential families. By exploiting the unifying formulation of Hoshyarmanesh et al, 21 we are able to find expressions of cdf and pdf of the random power of UMP tests that can be specialized to a series of relevant models. Our attention primarily focuses on the choice of the design prior density and on its impact on qualitative features of the resulting distribution of the power.…”

“…The joint density function of $$$ {\mathbf{X}}_n $$$ is $$$$ {f}_{{\mathbf{X}}_n}\left({\mathbf{x}}_n|\theta \right)=\frac{h_n\left({\mathbf{x}}_n\right)}{\theta^{mn}}\exp \left\{-\frac{T\left({\mathbf{x}}_n\right)}{\theta^r}\right\}, $$$$ where $$$ {h}_n\left({\mathbf{x}}_n\right)={\prod}_{i=1}^nh\left({x}_i\right) $$$ and where $$$ T\left({\mathbf{x}}_n\right)={\sum}_{i=1}^n\omega \left({x}_i\right) $$$ is a complete sufficient statistic. Following Hoshyarmanesh et al 21 it can be shown that $$$ T\left({\mathbf{X}}_n\right)\sim \mathrm{Ga}\left(\nu, {\theta}^r\right) $$$ where $$$ \nu = mn/r $$$ and where in general $$$ \mathrm{Ga}\left(a,b\right) $$$ denotes a gamma random variable with shape $$…”

“…All these models belong to the exponential family. The first contribution in our article is then to extend Rufibach et al's approach to one-sided testing for the scale parameter of distributions using the unifying formulation for exponential families given by Hoshyarmanesh et al 21 that incorporates many relevant models as special cases. 22 2.…”

“…All these models belong to the exponential family. The first contribution in our article is then to extend Rufibach et al's approach to one‐sided testing for the scale parameter of distributions using the unifying formulation for exponential families given by Hoshyarmanesh et al 21 that incorporates many relevant models as special cases 22 The analysis is first performed with conjugate design priors, a choice that eases identification of the parameters that play the roles of prior sample size and mode.…”