On a Class of Admissible Partitions

Abstract

In many multi-decision problems the existence of admissible decision functions (for definitions we refer to [5]) depends upon the existence of corresponding partitions of the sample space into regions of specified shape. Usually the requirements for such statistical partitions differ from those relating to less restricted partitions of a set $S$ according to a vector of finite measures on the measurable subsets of $S$, usually referred to under the general title of the "ham sandwich problem" (see, e.g., [3], where further references may be found). Nevertheless, as indicated here, the solutions to a wide class of division problems rest very heavily on the fundamental result of Lyapunov and generalizations of this (see, e.g., [4]). Several multi-decision problems (Section 4 below) relating to the mean $\mu$ of a $k$-variate normal distribution $N(\mu, \Sigma)$ reduce to the problem of locating (hence called "topothetical," cf. [2]) the parameter point $\mu$ into one of $k + 1$ convex $k$-dimensional polyhedral cones $\omega_1, \omega_2, \cdots, \omega_{k+1}$ (hereafter referred to as "cones") with common vertex $\mu_0$ which form a partition of the parameter space $E_k$ of $\mu$ (see (1) below). Let us identify the sample space $E_k$ of an observation $X$ from $N(\mu, \Sigma)$ with the parameter space $E_k$. It was shown in [2] that the family $R_\omega$ of all translations $R(\tau) = (R_1(\tau), \cdots, R_{k+1}(\tau))$ of the system $\omega = (\omega_1, \cdots, \omega_{k+1})$ (see Definition 2) defines a class of admissible procedures, henceforth referred to as partitions; the decision $d_i$ that $\mu \varepsilon \omega_i$ is taken when the actual observation $x \varepsilon R_i$. Furthermore, there exists a unique partition $R(\tau_0) \varepsilon R_\omega$ which is minimax. The minimax character of $R(\tau_0)$ amounts to the following proposition: There exists a unique, in $R_\omega$, partition of $E_k$ into $k + 1$ cones with the same probability content under the normal $k$-variate distribution (Corollary 5.1 of [2]). The distribution may be assumed spherical normal (unit variance in any direction) without any loss of generality, since a nonsingular linear transformation $T$ such that $T\Sigma T' = I$ preserves the shape of the partition $\omega$. The purpose of this note is to extend the above result to the case of arbitrary probability contents for such conical regions (Theorem 1), and at the same time show how the corresponding partitions are related to classes of admissible partitions for a family of classification and topothetical problems relating to the normal mean $\mu$. Several problems which have been extensively studied in the statistical literature emerge as special cases of our general topothetical problem (Section 4).