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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

  1. 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,
  2. 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.

  3. The Cramér–von-Mises and Anderson–Darling goodness-of-fit statistics are based on the functional

    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

  1. An example of a degree-2 V-statistic is the second central moment m2. If h(x, y) = (xy)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:

  1. Var(h(X1, ..., Xk)) = 0 for k < m, and Var(h(X1, ..., Xk)) > 0 for k = m;
  2. nm/2Rmn tends to zero (in probability). (Rmn is the remainder term in the Taylor series for T.)

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 [1] (Example 3) is an example of a degenerate kernel V-statistic. [8]

See also


  1. 1 2 3 4 von Mises (1947)
  2. Lee (1990)
  3. Koroljuk & Borovskich (1994)
  4. Hoeffding (1948)
  5. von Mises (1947), p. 309; Serfling (1980), p. 210.
  6. Serfling (1980, Section 6.5)
  7. Serfling (1980, Ch. 5–6); Lee (1990, Ch. 3)
  8. See Lee (1990, p. 160) for the kernel function.

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