Cabaña, A.; Cabaña, E. M. Transformed empirical processes and modified Kolmogorov-Smirnov tests for multivariate distributions. (English) Zbl 0910.62044 Ann. Stat. 25, No. 6, 2388-2409 (1997). 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 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. Reviewer: A.Mogulskij (Novosibirsk) Cited in 1 ReviewCited in 11 Documents MSC: 62G10 Nonparametric hypothesis testing 62H15 Hypothesis testing in multivariate analysis 62G30 Order statistics; empirical distribution functions 62G20 Asymptotic properties of nonparametric inference 60G15 Gaussian processes Keywords:goodness-of-fit; power improvement; transformed empirical processes × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Caba na, A. (1993). Transformations du pont brownien empirique et tests de ty pe Kolmogorov- Smirnov. C.R. Acad. Sci. Paris Sér. I. Math. 317 315-318. · Zbl 0780.60077 [2] Caba na, A. (1996). 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