A New Unified Computational Method for Finding Confidence Intervals of Shortest Length and/or Equal Tails under Parametric Uncertainty

Abstract

In statistics, confidence intervals are used to represent a range of values that is likely to contain a population parameter with a certain level of confidence. Confidence intervals allow us to generalize our findings from the samples from which our data were taken to the population from which our sample was drawn. For example, this ability to summarize one’s findings is often very helpful in the following areas: 1) Manufacturing, where confidence intervals are often used by engineers in manufacturing plants to determine if some new process, technique, method, etc. causes a meaningful change in the number of defective products produced by the plant, 2) Clinical trials, where confidence intervals are often used to determine the mean change in blood pressure, heart rate, cholesterol, etc. produced by some new drug or treatment, 3) Hypothesis testing, where (in general) for every test of hypothesis there is an equivalent statement about whether the hypothesized parameter value is included in a confidence interval. In the present paper, a new unified computational method for finding confidence intervals of shortest length and/or equal tails under parametric uncertainty is proposed. The unified computational technique yields intervals in several situations which have previously required separate analyses using more advanced techniques and tables for numerical solutions. Unlike the Bayesian approach, the proposed approach is independent of the choice of priors and represents a novelty in the theory of statistical decisions. To illustrate the proposed approach, numerical examples are given.