A New Simple Computational Method of Simultaneous Constructing and Comparing Confidence Intervals of Shortest Length and Equal Tails for Making Efficient Decisions Under Parametric Uncertainty

Abstract

A confidence interval is a range of values that provides the user with useful information about how accurately a statistic estimates a parameter. In the present paper, a new simple computational method is proposed for simultaneous constructing and comparing confidence intervals of shortest length and equal tails in order to make efficient decisions under parametric uncertainty. This unified computational method provides intervals in several situations that previously required separate analysis using more advanced methods and tables for numerical solutions. In contrast to the Bayesian approach, the proposed approach does not depend on the choice of priors and is a novelty in the theory of statistical decisions. It allows one to exclude unknown (nuisance) parameters from the problem using the technique of invariant statistical embedding and averaging in terms of pivotal quantities (ISE & APQ). It should be noted that the well-known classical approach to constructing confidence intervals of the shortest length considers at least three versions of possible solutions and is in need of information about the forms of probability distributions of pivotal quantities in order to determine an adequate version of the correct solution. The proposed method does not need such information. It receives this information through the quantiles of the probability distribution of the pivotal quantity. Therefore, the proposed method automatically recognizes an adequate version of the correct solution. To illustrate this method, numerical example is given.