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.
Statistical functions.
Statistics that can be represented as functionals formula_1 of the empirical distribution function formula_2 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".
Representation as a V-statistic.
Suppose "x"1, ..., "x""n" 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. "V""mn" 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 "x"1, ..., "x""n", the corresponding V-statistic is defined
Asymptotic distribution.
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):
If "A"(1) is true, the statistic is a sample mean and the Central Limit Theorem implies that T(Fn) is asymptotically normal.
In the variance example (4), m2 is asymptotically normal with mean formula_5 and variance formula_6, where formula_7.
Case "m" = 2 (Degenerate kernel):
Suppose "A"(2) is true, and formula_8 and formula_9. Then nV2,n converges in distribution to a weighted sum of independent chi-squared variables:
where formula_11 are independent standard normal variables and formula_12 are constants that depend on the distribution "F" and the functional "T". In this case the asymptotic distribution is called a "quadratic form of centered Gaussian random variables". The statistic "V"2,"n" is called a "degenerate kernel V-statistic". The V-statistic associated with the Cramer–von Mises functional (Example 3) is an example of a degenerate kernel V-statistic.