V-statistics are a class of statistics named for Richard von Mises who developed their asymptotic distribution theory in a fundamental paper in 1947.[1] V-statistics are closely related to U-statistics[2][3] (U for "unbiased") introduced by Wassily Hoeffding in 1948.[4] 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 of the empirical distribution function are called statistical functionals.[5]Differentiability of the functional T plays a key role in the von Mises approach; thus von Mises considers differentiable statistical functionals.[1]
Examples of statistical functions
The k-th central moment is the functional, where is the expected value of X. The associated statistical function is the sample k-th central moment,
The chi-squared goodness-of-fit statistic is a statistical function T(Fn), corresponding to the statistical functional
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.
Representation as a V-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[6] 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
Example of a V-statistic
An example of a degree-2 V-statistic is the second central momentm2.
If h(x, y) = (x − y)2/2, the corresponding V-statistic is
which is the maximum likelihood estimator of variance. With the same kernel, the corresponding U-statistic is the (unbiased) sample variance:
.
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.[1] 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.[7] Let A(m) be the property defined by:
A(m):
Var(h(X1, ..., Xk)) = 0 for k < m, and Var(h(X1, ..., Xk)) > 0 for k = m;
nm/2Rmn tends to zero (in probability). (Rmn is the remainder term in the Taylor series for T.)
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[1] (Example 3) is an example of a degenerate kernel V-statistic.[8]
Koroljuk, V.S.; Borovskich, Yu.V. (1994). Theory of U-statistics (English translation by P.V.Malyshev and D.V.Malyshev from the 1989 Ukrainian ed.). Dordrecht: Kluwer Academic Publishers. ISBN0-7923-2608-3.
Lee, A.J. (1990). U-Statistics: theory and practice. New York: Marcel Dekker, Inc. ISBN0-8247-8253-4.