Outliers and credence for location parameter inference. (English) Zbl 0706.62030

O’Hagan, A.

J. Am. Stat. Assoc. 85, No. 409, 172-176 (1990).

Summary: Heavy-tailed distributions are important for modeling problems in which there may be outlying observations of parameters. 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 \(1+d\). The author proves 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. Propagation of credence and outlier rejection are considered briefly in the more complex case of a hierarchical model.

MSC:

62F15	Bayesian inference
62A01	Foundations and philosophical topics in statistics

Keywords:

bimodality; convolution; Student-\(t\) distribution; Heavy-tailed distributions; modeling problems; credence; unknown location parameters; known variances; outlier rejection; hierarchical model