**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.

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] }

- 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*(*F*_{n}), corresponding to the statistical functionalwhere

*A*_{i}are the*k*cells and*p*_{i}are the specified probabilities of the cells under the null hypothesis. - The Cramér–von-Mises and Anderson–Darling goodness-of-fit statistics are based on the functional
where

*w*(*x*;*F*_{0}) is a specified weight function and*F*_{0}is a specified null distribution. If*w*is the identity function then*T*(*F*_{n}) is the well known Cramér–von-Mises goodness-of-fit statistic; if then*T*(*F*_{n}) is the Anderson–Darling 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^{ [6] } 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

- An example of a degree-2 V-statistic is the second central moment
*m*_{2}. If*h*(*x*,*y*) = (*x*−*y*)^{2}/2, the corresponding V-statistic iswhich 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.^{ [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*(*X*_{1}, ...,*X*_{k})) = 0 for*k*<*m*, and Var(*h*(*X*_{1}, ...,*X*_{k})) > 0 for*k*=*m*; *n*^{m/2}*R*_{mn}tends to zero (in probability). (*R*_{mn}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(F_{n}) is asymptotically normal.

In the variance example (4), m_{2} is asymptotically normal with mean and variance , where .

**Case m = 2** (Degenerate kernel):

Suppose *A*(2) is true, and and . Then nV_{2,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 *V*_{2,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] }

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

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- Hoeffding, W. (1948). "A class of statistics with asymptotically normal distribution".
*Annals of Mathematical Statistics*.**19**(3): 293–325. doi: 10.1214/aoms/1177730196 . JSTOR 2235637. - 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. ISBN 0-7923-2608-3. - Lee, A.J. (1990).
*U**-Statistics: theory and practice*. New York: Marcel Dekker, Inc. ISBN 0-8247-8253-4. - Neuhaus, G. (1977). "Functional limit theorems for
*U*-statistics in the degenerate case".*Journal of Multivariate Analysis*.**7**(3): 424–439. doi: 10.1016/0047-259X(77)90083-5 . - Rosenblatt, M. (1952). "Limit theorems associated with variants of the von Mises statistic".
*Annals of Mathematical Statistics*.**23**(4): 617–623. doi: 10.1214/aoms/1177729341 . JSTOR 2236587. - Serfling, R.J. (1980).
*Approximation theorems of mathematical statistics*. New York: John Wiley & Sons. ISBN 0-471-02403-1. - Taylor, R.L.; Daffer, P.Z.; Patterson, R.F. (1985).
*Limit theorems for sums of exchangeable random variables*. New Jersey: Rowman and Allanheld. - von Mises, R. (1947). "On the asymptotic distribution of differentiable statistical functions".
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