A Test of Fit for Multivariate Distributions

Abstract

Suppose $X$ is a chance variable taking values in $k$-dimensional Euclidean space. That is, $X = (Y_1, \cdots, Y_k),$ where $Y_i$ is a univariate chance variable. The joint distribution of $(Y_1, \cdots, Y_k)$ has density $f(y_1, \cdots, y_k),$ say. We shall call a function $h(y_1, \cdots, y_k)$ "piecewise continuous" if it is everywhere bounded, and $k$-dimensional Euclidean space can be broken into a finite number of Borel-measurable subregions, such that in the interior of each subregion $h(y_1, \cdots, y_k)$ is continuous, and the set of all boundary points of all subregions has Lebesgue measure zero. We assume that $f(y_1, \cdots, y_k)$ is piecewise continuous. Let $h(y_1, \cdots, y_k)$ be some given nonnegative piecewise continuous function, and let $X_1, \cdots, X_n$ be independent chance variables, each with the density $f(y_1, \cdots, y_k).$ Choose a nonnegative number $t,$ and for each $i$, construct a $k$-dimensional sphere with center at $X_i = (Y_{i1}, \cdots, Y_{ik})$ and of $k$-dimensional volume $$\frac{th(Y_{i1}, \cdots, Y_{ik})}{n}.$$ Such a sphere will be called "of type $s$" if it contains exactly $s$ of the $(n - 1)$ points $X_1, \cdots, X_{i-1}, X_{i+1}, \cdots, X_n.$ Let $R_n(t; s)$ denote the proportion of the $n$ spheres which are of type $s$. For typographical simplicity, we denote the vector $(y_1, \cdots, y_k)$ by $y.$ Let $S(t; s)$ denote the multiple integral $$(t^s/s!) \int^\infty_{-\infty} \cdots \int^\infty_{-\infty} h^s(y)f^{s+1}(y) \exp \{-th(y)f(y)\} dy_1 \cdots dy_k.$$ It is shown that $R_n(t; s)$ converges stochastically to $S(t; s)$ as $n$ increases. This result is then used to construct a test of the hypothesis that the unknown density $f(y)$ is equal to a given density $g(y).$