Concentration and goodness-of-fit in higher dimensions: (asymptotically) distribution-free methods

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Abstract
A novel approach for constructing goodness-of-fit techniques in arbitrary finite dimensions is presented. Testing problems are considered as well as the construction of diagnostic plots. The approach is based on some new notions of mass concentration, and in fact, our basic testing problems are formulated as problems of ‘‘goodness-of-concentration.’’ It is this connection to concentration of measure that makes the approach conceptually simple. The presented test statistics are continuous functionals of certain processes which behave like the standard one-dimensional uniform empirical process. Hence, the test statistics behave like classical test statistics for goodness-of-fit. In particular, for simple hypotheses they are asymptotically distribution free with well-known asymptotic distribution. The simple technical idea behind the approach may be called a generalized quantile transformation, where the role of one-dimensional quantiles in classical situations is taken over by so-called minimum volume sets.