On the Asymptotic Behavior of Decision Procedures

Abstract

In this paper, the asymptotic behavior of decision procedures will be studied for a particular class of multiple decision problems. The study will throw some light on the desirability of the minimax decision procedure when the number of observations is large, and it will be seen that decision procedures frequently exist which are superior to the minimax decision procedure for large samples. The fact that the minimax decision procedure may be desirable in certain problems for small samples but undesirable for large samples was revealed by Hodges and Lehmann [1] in connection with estimation problems. Robbins [2] suggested the term "asymptotically subminimax" for the type of superior procedures which may then exist. A definition of this term which will be useful for an investigation of the asymptotic behavior of decision procedures, will be given in Section 1. A major part of this paper will be concerned with certain sequences of decision procedures called asymptotically admissible which have desirable properties similar to those of admissible decision procedures for the case of some fixed sample size. These asymptotically admissible decision procedures include a subclass of the asymptotically subminimax procedures, and the sequences of minimax procedures for those problems for which asymptotically subminimax procedures do not exist. The problems to be considered are those in which a random variable, $X$, is known to have a distribution function belonging to the distribution space, $\Omega = \{F_i(x)\}, i = 1, 2, \cdots, k$, and it is desired to select the true distribution function based on a sample of $n$ independent observations of $X$. It will be assumed that all $F_i(x)$ are absolutely continuous distribution functions having density functions, $f_i(x)$, and that for every constant $K$, the set of points for which $f_i(x)/f_j(x) = K (i \neq j)$ is a set of probability measure zero under every $F$ and all possible $i$ and $j$. A simple loss function, $W(F_i, d_j)$, where $d_j$ is the decision to select $F_j$, will be used with $W(F_i, d_j) = 1$ if an incorrect decision is made (i.e. $i \neq j)$, and $W(F_i, d_j) = 0$ if a correct decision is made (i.e. $i = j$). For such a loss function, the expected loss is simply the probability of making an incorrect decision. Section 1 will be concerned with asymptotically minimax sequences of decision procedures and Section 2 will be concerned with asymptotic admissibility. It will be seen that the asymptotic behavior of the minimax decision procedure depends on the limits of the components in the sequence of least favorable a priori distributions. Theorem 2.2 gives a sufficient condition, in terms of these limit values, for the minimax procedure to be asymptotically admissible. In Section 3, a detailed study will be made of the class of problems where $\Omega$ consists of $k$ univariate normal distributions having the same variance but different means. Let the means be denoted by $\theta_i (i = 1, 2, \cdots, k)$ with $\theta_1 < \theta_2 < \cdots < \theta_k$, and let $\min_i(\theta_{i+1} - \theta_i) = \gamma$. Then Theorems 3.1 and 3.6 will show that the minimax procedure is asymptotically admissible when the means can be put into sets, each set containing the same number, $\bar{n} \geqq 2$, of consecutive means, with a difference of $\gamma$ between any two consecutive means of a set, and a difference greater than $\gamma$ between any two means not belonging to the same set. Theorems 3.2 and 3.3 will show that in all other cases the minimax procedure is asymptotically inadmissible, and asymptotically subminimax procedures will be constructed for all these cases. Although a complete study of the asymptotic admissibility of asymptotically subminimax procedures will not be made in this paper, Theorem 3.7 will show that a certain asymptotically subminimax procedure is asymptotically admissible for all the cases covered by Theorem 3.3 and for some of the cases covered by Theorem 3.2. On the other hand, for those cases covered by Theorem 3.2 with an $\Omega$ consisting of only 3 means, it will be shown that every asymptotically subminimax procedure is asymptotically inadmissible.