Transformed empirical processes and modified Kolmogorov-Smirnov tests for multivariate distributions
Let \((X_1,...,X_n) \) be a sample of i.i.d. random variables with distribution \(F\) on \(R^d\), \(d\geq 1\). Consider \(H_0: F=F_0\) and a family of alternatives \(H_n: F=F^{(\delta/\sqrt{n})}\), where \(F^{(\tau)}\) is contiguous to \(F^{(0)}=F_0\) with density \(f^{(\tau)}\) with respect to \(F_0\), and such that there exists an \(L^2(R^d,dF_0)\) function \(k=k(x)\) satisfying \[ \| \frac{1}{\tau}(\sqrt{f^{(\tau)}}-1) -\frac{k}{2}\|_{L^2}\to 0\;\text{as} \tau \to 0. \] The aim of the paper is the construction of goodness-of-fit tests based on signed measures \(\hat{w}_x\) (\(x\in R^d\)) such that the following holds: \((a)\) When \(x\) is replaced by a random variable \(X\), the resulting measure \(\hat{w}_X(A)\) evaluated on any measurable set \(A\) is a random variable; \((b)\) The measure \(\hat{w}_n=n^{-1/2}\sum_{k=1}^{n}\hat{w}_{X_k}\) associated with the sample has some normalized limit distribution under \(H_0\); \((c)\) The asymptotic distributions of \(\hat{w}_n\) under \(H_n\) and \(H_0\) differ as much as possible. These random measures will be called a \textit{transformed empirical processes} (in short: TEP). They play, in the construction of goodness-of-fit tests, the same role as the empirical processes in classical Kolmogorov-Smirnov tests. The resulting tests are consistent against any fixt alternative, and, for each sequence of contiguous alternatives, a test in each class can be chosen so as to optimize the discrimination of those alternatives.
scientific article; zbMATH DE number 1138417
scientific article