The Berry-Esseen Theorem for $U$-Statistics

An asymptotic test for a geometric process against a lattice distribution with monotone hazard

“…A first solution for this problem could consist in using the classical chi-square (X 2 , for short) test-statistic, or one of its modifications, such as smooth tests (Best, Rayner;1989), or modified chi-square tests (see, for instance, Bolshev, Mirvaliev;1978). However, in the present case this solution is not completely satisfactory, at least form a theoretical point of view.…”

The Edgeworth Expansion for U-Statistics of Degree Two

“…The speed of convergence to normality was investigated by Bickel (1974), Chan and Wierman (1977), Callaert and Janssen (1978) and Helmers and van Zwet (1982) who showed in increasing generality that (1.8) X where <I> denotes the standard normal distribution function (d.f.). If (1.2) and (1.7) are satisfied, so that asymptotic normality is ensured, then Ejg(X 1 )j 3 < oo suffices to establish (1.8).…”

“…In view of (1.6), (1.22) and the fact that Ei11NI 2 +1l = O(N 2 +1l) [cf. Callaert and Janssen (1978)] we may rewrite (2.8) as…”

A Berry-Esseen Bound for Symmetric Statistics

“…Bickel (1974) established the Berry-Esseen bound (1.8) under the additional assumption that h is bounded. Chan and Wierman (1977) and Callaert and Janssen (1978) successively reduced this assumption first to Eh 4 (X 1 , X 2 ) < oo and then to E[h(X 1 , X 2 W < oo . Helmers and Van Zwet (1982) showed that E[g(X 1 W < oo suffices.…”