Tail behaviour for suprema of empirical processes
Let F be a continuous distribution function on the unit cube \(I^ k\) of \({\mathbb{R}}^ k\) with uniform univariate marginals, and let \(W_ F\) be the ''pinned Brownian sheet based on F'' (i.e. the centered Gaussian process with covariance \(EW_ F(x)W_ F(y)=F(x\wedge y)-F(x)F(y)\), \(x,y\in I^ k\). The normalized and centered empirical process based on F converges in law to \(W_ F\), and for this fact to be useful in testing hypotheses one needs estimates for the distribution of \(\sup_{t}W_ F(t).\) It is shown in this paper that for all \(\lambda\) and appropriate c, C, \[ c(F,k)\lambda^{2(k-1)}e^{-2\lambda^ 2}\leq P\{\sup_{t}W_ F(t)>\lambda \}\leq C(k)\lambda^{\quad 2(k-1)}c^{-2\lambda^ 2}, \] with the constants identified in the case \(k=2\). This type of results originated in \textit{J. Kiefer} and \textit{J. Wolfowitz}, [Trans. Am. Math. Soc. 87, 173-186 (1958; Zbl 0088.113)] and \textit{J. Kiefer} [Pac. J. Math. 11, 649-660 (1961; Zbl 0119.349)], and there is some recent literature (Goodman, Park and Skough, Cabaña and Wschebor) on the case of uniform F. The authors apply their results to obtain sharp upper and lower class results for suprema of empirical processes and mention that their unpublished manuscript (1985) contains applications to multivariate Kolmogorov-Smirnov tests.
scientific article; zbMATH DE number 3960785
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