# V-statistic

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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.  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.

## Statistical functions

Statistics that can be represented as functionals $T(F_{n})$ of the empirical distribution function $(F_{n})$ 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

1. The k-th central moment is the functional$T(F)=\int (x-\mu )^{k}\,dF(x)$ , where $\mu =E[X]$ is the expected value of X. The associated statistical function is the sample k-th central moment,
$T_{n}=m_{k}=T(F_{n})={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{k}.$ 2. The chi-squared goodness-of-fit statistic is a statistical function T(Fn), corresponding to the statistical functional
$T(F)=\sum _{i=1}^{k}{\frac {(\int _{A_{i}}\,dF-p_{i})^{2}}{p_{i}}},$ 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
$T(F)=\int (F(x)-F_{0}(x))^{2}\,w(x;F_{0})\,dF_{0}(x),$ 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 $w(x;F_{0})=[F_{0}(x)(1-F_{0}(x))]^{-1}$ 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

$V_{mn}={\frac {1}{n^{m}}}\sum _{i_{1}=1}^{n}\cdots \sum _{i_{m}=1}^{n}h(x_{i_{1}},x_{i_{2}},\dots ,x_{i_{m}}),$ 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

$V_{2,n}={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}h(x_{i},x_{j}).$ ### 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
$V_{2,n}={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {1}{2}}(x_{i}-x_{j})^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2},$ which is the maximum likelihood estimator of variance. With the same kernel, the corresponding U-statistic is the (unbiased) sample variance:

$s^{2}={n \choose 2}^{-1}\sum _{i .

## 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.  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.  Let A(m) be the property defined by:

A(m):
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 $\sigma ^{2}$ and variance $(\mu _{4}-\sigma ^{4})/n$ , where $\mu _{4}=E(X-E(X))^{4}$ .

Case m = 2 (Degenerate kernel):

Suppose A(2) is true, and $E[h^{2}(X_{1},X_{2})]<\infty ,\,E|h(X_{1},X_{1})|<\infty ,$ and $E[h(x,X_{1})]\equiv 0$ . Then nV2,n converges in distribution to a weighted sum of independent chi-squared variables:

$nV_{2,n}{\stackrel {d}{\longrightarrow }}\sum _{k=1}^{\infty }\lambda _{k}Z_{k}^{2},$ where $Z_{k}$ are independent standard normal variables and $\lambda _{k}$ 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.