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 of the empirical distribution function 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.
where Ai are the k cells and pi are the specified probabilities of the cells under the null hypothesis.
where w(x; F0) is a specified weight function and F0 is a specified null distribution. If w is the identity function then T(Fn) is the well known Cramér–von-Mises goodness-of-fit statistic; if then T(Fn) is the Anderson–Darling statistic.
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 m.discusses how to find the kernel in practice. Vmn is called a V-statistic of degree
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
which is the maximum likelihood estimator of variance. With the same kernel, the corresponding U-statistic is the (unbiased) sample variance:
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. 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).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
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 and variance , where .
Case m = 2 (Degenerate kernel):
Suppose A(2) is true, and and . Then nV2,n converges in distribution to a weighted sum of independent chi-squared variables:
where are independent standard normal variables and 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 V2,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.
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