Abstract
Remainder bounds in Fréchet differentiability of functionals for p-variation norms are found for empirical distribution functions. For \(1 \le p \le 2\) the p-variation of the empirical process \(n^{1/2} (F_n - F)\) is of order \(n^{1 - p/2} \) in probability up to a factor (log log n)p/2. For \((F,G) \mapsto \smallint FdG\) and for \((F,G) \mapsto F \circ G^{ - 1} \) this yields nearly optimal remainder bounds. Also, p-variation gives new proofs for the asymptotic distributions of the Cramér-von Mises-Rosenblatt and Watson two-sample statistics when the two sample sizes m, n go to infinity arbitrarily.
Mathematics Department, Massachusetts Institute of Technology
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Dudley, R.M. (2010). Empirical Processes and p-variation. In: Giné, E., Koltchinskii, V., Norvaisa, R. (eds) Selected Works of R.M. Dudley. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5821-1_23
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