I. DIVERGENCES AND DIVERGENCE STATISTICS MANY of the divergence measures used in statistics are of the f -divergence type introduced independently by I. Csiszár [1], T. Morimoto [2], and Ali and Silvey [3]. Such divergence measures have been studied in great detail in [4]. Often one is interested inequalities for one f -divergence in terms of another f -divergence. Such inequalities are for ins...

There are several theorical results about order statistics and copulas in the literature that have  been mentioned also by Nelsen cite{p20}. The present study after reviewing some of these results, relies on  simulation technique to investigate the mentioned results about order statistics and copulas. The study concentrates on two well known Archimedean Gumbel and Frank families in the case tha...

We introduce f -divergence, a concept from information theory and statistics, for convex bodies in R. We prove that f -divergences are SL(n) invariant valuations and we establish an affine isoperimetric inequality for these quantities. We show that generalized affine surface area and in particular the Lp affine surface area from the Lp Brunn Minkowski theory are special cases of f -divergences.

A U-statistic of order k with kernel f :X →Rd over a Poisson process is defined in [25] as ∑ x1,...,xk∈η 6= f (x1, . . . ,xk) under appropriate integrability assumptions on f . U-statistics play an important role in stochastic geometry since many interesting functionals can be written as Ustatistics, like intrinsic volumes of intersection processes, characteristics of random geometric graphs, v...

Let Yi = f(xi) + Ei (1 ≤ i ≤ n) with given covariates x1 < x2 < · · · < xn, an unknown regression function f and independent random errors Ei with median zero. It is shown how to apply several linear rank test statistics simultaneously in order to test monotonicity of f in various regions and to identify its local extrema.