A Multivariate Two-Sample Test Based on the Number of Nearest Neighbor Type Coincidences

For independent d-variate random samples X₁, ⋯, Xn₁ i.i.d. f(x), Y₁, ⋯, Yn₂ i.i.d. $g(x)$, where the densities f and g are assumed to be continuous a.e., consider the number T of all k nearest neighbor comparisons in which observations and their neighbors belong to the same sample. We show that, if $f = g$ a.e., the limiting (normal) distribution of T, as min(n₁, n₂) → ∞, n₁/(n₁ + n₂) → τ, 0 < τ < 1, does not depend on f. An omnibus procedure for testing the hypothesis H₀: f = g a.e. is obtained by rejecting H₀ for large values of T. The result applies to a general distance (generated by a norm on Rᵈ) for determining nearest neighbors, and it generalizes to the multisample situation.