Note on Multivariate Goodness-of-fit Tests

Abstract

Let $X_1, X_2, \cdots, X_n$ be $m$-dimensional statistically independent random vectors with common distribution function $F$. It is frequently desirable to test the hypothesis that $F$ is a member of some class of distribution functions $\mathscr{H}_0$. For the scalar case, $(m = 1)$, much research has been done; see for example [1], [2], [3]. For $m > 1$ comparatively little has been accomplished, and a useful extension of the techniques used for $m = 1$ awaits the solution of certain problems in stochastic processes with a vector parameter; see, for example, [4]. In this paper consistent tests are developed for any given class $\mathscr{H}_0$. These tests can be constructed to have size $\alpha$ and prescribed power $1 - \beta$ against alternatives whose probability assignment to at least one of a certain given class of sets $\{B(v)\}$ differs from that of each member of $\mathscr{H}_0$ by at least a prescribed value $K$. The range of such alternatives is seen in Section 3 to be rather wide, so that at least in theory, the suggested tests would seem to be rather useful. The tests are constructed by mapping the set of all $m$-dimensional distribution functions in a one to one measurable manner into a subset of the set of one-dimensional distribution functions. Such mappings are, of course, reasonably well known; see Halmos [7], p. 153. The purpose of this note is to show how such mappings offer sufficient flexibility for the construction of a class of tests which are useful for most ordinary purposes. Simpson [6] suggested tests based on mapping bivariate distributions into univariate distributions. However no mention was made there of consistency, or of power in terms of the type of alternative here mentioned.