×

Group Bayes estimation of the exponential mean: A retrospective view of the Wald theory. (English) Zbl 0793.62006

Gupta, Shanti S. (ed.) et al., Statistical decision theory and related topics V. Proceedings of the fifth Purdue international symposium on statistical decision theory and related topics held at Purdue University, West Lafayette, IN (USA), June 14-19, 1992. New York: Springer-Verlag. 35-49 (1994).
Summary: Statistical decision theory, as formulated by Abraham Wald, has undergone more or less half a century of development. But although this development has occupied some of this century’s leading statistical researchers and despite its early promise, this theory does not appear to have anything like the impact of, say, the theory of experimental design including randomization which was propounded just shortly before Wald’s theory. Indeed, one might well ask if the theory has any significant value.
Our thesis is positive. Our argument for the value of this theory will go beyond those of its intrinsic merit and its role in uniting the diverse sub-theories of statistics such as hypothesis testing, within a comprehensive framework. Such arguments as those could really serve as a retrospective, terminal view of the subject.
Our support for the theory will be based on its applicability to dynamically evolving areas of current research which lie well beyound the domain for which the theory was originally conceived, namely, robust- and its relative, group-Bayesian analysis. The role of randomized decision rules is rediscovered, albeit in a subjectivist context. And the role of the criteria and solution concepts of Wald will be seen to be valuable in identifying rules of good quality. Fundamental theorems of the Wald theory become applicable in this new context.
Our approach will be to view the theory from the perspective of a problem of interest in its own right, the estimation of the mean of the (negative) exponential distribution, \(\lambda\), when \(\lambda\) has the conjugate prior distribution with not fully specified hyperparameters. We will consider appropriate loss functions for this problem and explore not only squared error loss but a bounded loss as well; the results obtained for these losses are qualitatively very dissimilar. We will consider the issue of specifying the parameter of the loss, a topic which seems to have been relatively neglected both in the classical Wald theory as well as in robust Bayesian analysis. We will consider both posterior and pre- posterior analysis, and by applying new and available theory, determine various good, or classes of good, decision rules. We will, in particular, consider the case of a bounded (hyperparameter) space.
The extended domain considered in this paper for the application of Wald’s theory presents many conceptual as well as technical challenges which would not naturally arise in the domain for which the theory was originally formulated. Some of these will be highlighted. Also, from the perspective of the specific problem explored in this paper can be seen some of the unevenness and the gaps in the way the Wald theory has developed. And this in turn leads to a host of unsolved problems for the future.
For the entire collection see [Zbl 0782.00068].

MSC:

62C10 Bayesian problems; characterization of Bayes procedures
62F15 Bayesian inference
01A60 History of mathematics in the 20th century
62-03 History of statistics
62C05 General considerations in statistical decision theory
PDFBibTeX XMLCite