Two-Sample Tests for Multivariate Distributions

X(1), X(2), ⋯, X(m), Y(1), Y(2), ⋯, Y(n) are independent k-variate random variables. The distribution of $X(i)$ has pdf $f(x)$, say, where x denotes a k-dimensional vector throughout this paper, and the distribution of $Y(j)$ has pdf $g(x)$, say. We assume that $f(x)$ and $g(x)$ are piecewise continuous, and that each has a finite upper bound, which it is not necessary to specify. Denote by 2Rᵢ the distance from $X(i)$ to the nearest of the points X(1), ⋯, X(i - 1), X(i + 1), ⋯, X(m), and denote by Sᵢ the number of points Y(1), ⋯, Y(n) contained in the open sphere : | x - X(i) | < Rᵢ\. Clearly, the joint distribution of Sᵢ, Sⱼ is the same as the joint distribution of Si', Sj', for any subscripts with i ≠ j, i' ≠ j'. Let r be a non-negative integer, and α any fixed positive value. $Q(r)$ denotes the Lebesgue integral ∫Eₖ 2ᵏ α f² (x) g(x) ʳ g(x) + 2ᵏα f(x) r + 1 dx, where Eₖ denotes Euclidean k-space. We will show that limm → ∞, m/n = α Pm, n S₁ = s₁, S₂ = s₂ = Q(s₁)Q(s₂), for any non-negative integers s₁,s₂, the approach being uniform in s₁,s₂. Thus, in the limit S₁, S₂ are independently distributed, with limm → ∞, m/n = α Pm, n S₁ = s₁ = Q(s₁). In [1], which discussed the univariate case, Sᵢ was defined as the number of Y's closer to $X(i)$ than to any other X to their right. In the present paper, Sᵢ is defined as the number of Y's in another neighborhood of $X(i)$. Our present definition of Sᵢ does not become for $k = 1$ the same as the definition of Sᵢ in [1]. Rather, in the univariate case, our present definition of Sᵢ is the number of Y's lying within a distance Rᵢ on either side of $X(i)$. However, if limm → ∞, m/n = α Pm, n S₁, = s₁, S₂ = s₂ is computed for the univariate case using the definition of Sᵢ given in [1], the only way in which it differs from Q(s₁)Q(s₂) is that α is replaced by α/2. Thus it seems reasonable to treat the Sᵢ as defined here as k-dimensional analogues of the Sᵢ as defined in [1], at least for large samples. An intuitive reason for α being replaced by α/2 is that in our present case, ∑ᵐi = 1 Sᵢ may be less than n, whereas in [1] this sum must always equal n. Thus in our present case, we are in a sense discarding some of the Y's, which lowers n relative to m and thus raises α by a certain factor (2, as it happens). In our present case, ∑ Sᵢ may be less than n because the Rᵢ are chosen to make the spheres around the X's non-overlapping, thus simplifying the analysis. The Rᵢ were chosen to give the largest possible non-overlapping spheres because it would seem intuitively that the larger the spheres, the more rapid the approach of the probabilities to their limiting values.