V-statistic

V-statistics are a class of statistics named for Richard von Mises who developed their asymptotic distribution theory in a fundamental paper in 1947. V-statistics are closely related to U-statistics (U for 'unbiased') introduced by Wassily Hoeffding in 1948. A V-statistic is a statistical function (of a sample) defined by a particular statistical functional of a probability distribution. V-statistics are a class of statistics named for Richard von Mises who developed their asymptotic distribution theory in a fundamental paper in 1947. V-statistics are closely related to U-statistics (U for 'unbiased') introduced by Wassily Hoeffding in 1948. A V-statistic is a statistical function (of a sample) defined by a particular statistical functional of a probability distribution. Statistics that can be represented as functionals T ( F n ) {displaystyle T(F_{n})} of the empirical distribution function ( F n ) {displaystyle (F_{n})} are called statistical functionals. Differentiability of the functional T plays a key role in the von Mises approach; thus von Mises considers differentiable statistical functionals. Suppose x1, ..., xn is a sample. In typical applications the statistical function has a representation as the V-statistic where h is a symmetric kernel function. Serfling discusses how to find the kernel in practice. Vmn is called a V-statistic of degree m. A symmetric kernel of degree 2 is a function h(x, y), such that h(x, y) = h(y, x) for all x and y in the domain of h. For samples x1, ..., xn, the corresponding V-statistic is defined In examples 1–3, the asymptotic distribution of the statistic is different: in (1) it is normal, in (2) it is chi-squared, and in (3) it is a weighted sum of chi-squared variables. Von Mises' approach is a unifying theory that covers all of the cases above. Informally, the type of asymptotic distribution of a statistical function depends on the order of 'degeneracy,' which is determined by which term is the first non-vanishing term in the Taylor expansion of the functional T. In case it is the linear term, the limit distribution is normal; otherwise higher order types of distributions arise (under suitable conditions such that a central limit theorem holds). There are a hierarchy of cases parallel to asymptotic theory of U-statistics. Let A(m) be the property defined by: Case m = 1 (Non-degenerate kernel):