Outliers and Credence for Location Parameter Inference

Abstract Heavy-tailed distributions are important for modeling problems in which there may be outlying observations or parameters. Examples of their use are given in O'Hagan (1988). This article develops some general theory, based on the notion of credence, for inference about unknown location parameters in the case of known variances. A density on the real line is defined to have credence c if it is bounded above and below by positive multiples of (1 + x 2)-c/2. For instance, a t distribution with d degrees of freedom has credence 1 + d. I prove that the credence of a sum of independent random variables is the minimum of their individual credences, and that the credence of a posterior density of a location parameter is the sum of the credences of the prior and the observations. More generally, when independent information sources are combined, their credences add. When groups of information sources conflict, outlier rejection occurs, with the group having the greatest total credence dominating all others...