4.1 Basic Concepts

In some application, formulation of a problem as a hypothesis-testing one would be artificial. In some of these instances, estimation seems to be more appropriate. In the fixed-sample size situation there is a close connection between acceptance regions and confidence regions, whereas that analogy does not hold in the sequential situation. Hence, there is a need for a theory of sequential estimation. The stopping rules in sequential testing may not be meaningful in sequential estimation.

In this section we formulate (1) the general loss function involved in sequential estimation and (2) certain stopping rules.

Let X ₁, X ₂,..., be a sequence of independent random variables having common pdf f(x; ?). Let r[ ?(x); ?] denote the loss resulting from making a terminal decision ?(x) when ? is the true value of the parameter. We should add to the loss the cost of experimentation, namely C(N), the cost of taking N observations (where N is a random variable). The statistician's task lies in choosing a stopping rule and a terminal decision rule (estimate for ?). Then according to Lehmann (1950, Section 2.4) the statistician might be faced with the following situations:

1. Limited resources (forcing a bound on expectation of total cost of experimentation). He then seeks to minimize the risk function (expectation of the loss function) subject to an upper bound n ₀ on the expectation of C(N).

2. Limit on...