Suppose that U is a U-statistic of degree two based on N random observations drawn without replacement from a nite population. For the distribution of a standardized version of U we construct an Edgeworth expansion with remainder O(N ?1) provided that the linear part of the statistic satisses a Cram er type condition.

Motivated by applications to goodness of fit U-statistic testing, the jackknife empirical likelihood (JEL) for vector U-statistics is justified with two approaches and the Wilks theorems are proved. This extends empirical likelihood (EL) for general estimating equations (GEE’s) to U-statistics based GEE’s. The results are extended to allow for the use of estimated constraints and for the number...

The U-statistic elegantly and usefully generalizes the notion of a sample mean. Typical examples include (i) sample mean: h(x1, x2) = 12 (x1 + x2); (ii) sample variance: h(x1, x2) = 12 (x1 − x2); (iii) Gini’s mean difference: h(x1, x2) = |x1 − x2|; (iv) one-sample Wilcoxon’s statistic: h(x1, x2) = 1(x1 + x2 ≤ 0). The non-degenerate U-statistic shares many limiting properties with the sample mea...

We study the problem of independence testing given independent and identically distributed pairs taking values in a ?-finite, separable measure space. Defining natural dependence D(f) as squared L2-distance between joint density f product its marginals, we first show that there is no valid test uniformly consistent against alternatives form {f:D(f)??2}. therefore restrict attention to impose ad...

Motivated by applications to goodness of fit degenerate U-statistics testing, the Jackknife empirical likelihood approach for U-statistics is generalized to degenerate U-statistics. The proposed empirical likelihood based goodness of fit tests are asymptotically distribution free. The asymptotic theory for testing shift of location, spatial depth testing for central symmetry, Cramér-von Mises t...

We derive moment inequalities for the supremum of empirical processes of U-Statistic structure and give application to kernel type density  estimation ‎and estimation of the distribution function for functions of observations.