Estimation

Abstract

In Chapter 3, we discussed methods for choosing decision rules in problems with specified loss functions. In Section 3.3, we gave an axiomatic derivation of some of those methods. This derivation led to the conclusion that there is a probability and a loss function, and one should minimize the expected loss. There are decision problems in which ℵ and Ω are the same (or nearly the same) space and the loss function L(θ,a) is an increasing function of some measure of distance between θ and a. Such problems are often called point estimation problems. The classical framework makes no use of the probability over the parameter space provided by the axiomatic derivation. One can also try to ignore the loss function as well. To estimate θ without a specific loss function, one can adopt ad hoc criteria to decide if an estimator is good. In this chapter, we will study some of these criteria as well as some criteria for the problem of set estimation. In set estimation, the action space is a collection of subsets of the parameter space (or the closure of the parameter space). The idea is to find a set that is likely to contain the parameter without being “too big” in some sense.