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.
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
The Cramér–von-Mises and Anderson–Darling goodness-of-fit statistics are based on the functional
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 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 moment m2.
If h(x, y) = (x − y)2/2, the corresponding V-statistic is
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).
- 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.)
Case m = 1 (Non-degenerate kernel):
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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- von Mises, R. (1947). "On the asymptotic distribution of differentiable statistical functions". Annals of Mathematical Statistics. 18 (2): 309–348. doi:10.1214/aoms/1177730385. JSTOR 2235734.