A Multidimensional Goodness-of-Fit Test Based on Interpoint Distances

Abstract Distributional assumptions can be examined with multidimensional goodness-of-fit tests. We propose a conceptually simple test with an appealing logic and accessible asymptotic properties, which is generalizable to a variety of problems and appears to work well against diverse alternatives. To test whether a k-dimensional random sample X1, …, X n follows the distribution G, consider a triangle formed by two randomly selected data points X i and X j and a variable Y ∼ G. Our statistic estimates the likelihood that the side formed by the line from X i to X j is the smallest, the middle, or the largest side of the triangle.

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