Strategies for Choosing an Estimator
How can we resolve the ambiguity of the second kind and choose between two admissible estimators, T and W, in Example 7.2.1?
Subjective strategy. One strategy is to compare the graphs of the mean squared errors for T and W in Figure 7.5 and to choose one after considering the a priori likely values of p. For example, suppose we believe a priori that any value of p is equally likely and express this situation by a uniform density over the interval [0, 1]. We would then choose the estimator which has the minimum area under the mean squared error function. In our example, T and W are equally good by this criterion. This strategy is highly subjective; therefore, it is usually not discussed in a textbook written in the framework of classical statistics. It is more in the spirit of Bayesian statistics, although, as we shall explain in Chapter 8, a Bayesian would proceed in an entirely different manner, rather than comparing the mean squared errors of estimators.
Minimax strategy. According to the minimax strategy, we choose the estimator for which the largest possible value of the mean squared error is the smallest. This strategy may be regarded as the most pessimistic and risk-averse approach. In our example, T is preferred to W by this strategy. We formally define
DEFINITION 7.2.В Let 0 be an estimator of 0. It is a minimax estimator if, for any other estimator 0, we have
max E(B – 0)2 < max £(0 – 0)2. в e
We see in Figure 7.5 that W does well for the values of p around У2> whereas T does well for the values of p near 0 or 1. This suggests that we can perhaps combine the two estimators and produce an estimator which is better than either in some sense. One possible way to combine the two estimators is to define
The mean squared error of Z is computed to be (7.2.5) £(Z ~ pf = 2P2 ~2p + 1
and is graphed as the dashed curve in Figure 7.5. When we compare the three estimators T, W, and Z, we see that Z is chosen both by the subjective strategy with the uniform prior density for p and by the minimax strategy. In Chapter 8 we shall learn that Z is a Bayes estimator.