Appendix A: Classical Statistics Is Logically Untenable
1998
We have not mentioned in this book most of the methodology of so-called classical statistics. The reason is that classical statistics is based on deductive analysis (the logic of mathematics), whereas statistical inference and decision theory are concerned with inductive analysis (probability judgments). Why is classical statistics logically untenable? D. Basu (1988), in his careful examination of the methodology due to R. A. Fisher, summed up his answer to this question as follows: “It took me the greater part of the next two decades to realize that statistics deals with the mental process of induction and is therefore essentially antimathematics. How can there be a deductive theory of statistics?” To be more concrete, we consider two favorite ideas of classical statistics, namely, unbiasedness and confidence intervals. Unbiasedness. The posterior mean is a Bayes estimator of a parameter, say, θ, with respect to squared error loss. It is also a function of the data. No Bayes estimator (based on a corresponding proper prior) can be unbiased in the sense of classical statistics; see Bickel and Blackwell (1967). An estimator (a function of data) is called unbiased in the sense of classical statistics if EF [ θ ^ | θ] =θ for each θ ∈ Θ. In this case, F is the probability distribution corresponding to sample data. Most unbiased estimators are in fact inadmissible with respect to squared error loss in the sense of classical statistics. D. Basu provides the following example. In the case of the exponential density 1 θ e− x θ , is an unbiased estimator for θ in the sense of classical statistics, where T is the total time on test (TTT) and k is the number of observed failures.