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 ${\displaystyle T(F_{n})}$ of the empirical distribution function ${\displaystyle (F_{n})}$ 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 ${\displaystyle T(F)=\int (x-\mu )^{k}\,dF(x)}$, where ${\displaystyle \mu =E[X]}$ is the expected value of X. The associated statistical function is the sample k-th central moment,

   ${\displaystyle T_{n}=m_{k}=T(F_{n})={\frac {1}{n))\sum _{i=1}^{n}(x_{i}-{\overline {x)))^{k}.}$

2. The chi-squared goodness-of-fit statistic is a statistical function T(F_n), corresponding to the statistical functional

   ${\displaystyle T(F)=\sum _{i=1}^{k}{\frac {(\int _{A_{i))\,dF-p_{i})^{2)){p_{i))},}$

   where A_i are the k cells and p_i 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

   ${\displaystyle T(F)=\int (F(x)-F_{0}(x))^{2}\,w(x;F_{0})\,dF_{0}(x),}$

   where w(x; F₀) is a specified weight function and F₀ is a specified null distribution. If w is the identity function then T(F_n) is the well known Cramér–von-Mises goodness-of-fit statistic; if ${\displaystyle w(x;F_{0})=[F_{0}(x)(1-F_{0}(x))]^{-1))$ then T(F_n) is the Anderson–Darling statistic.

Representation as a V-statistic

Suppose x₁, ..., x_n is a sample. In typical applications the statistical function has a representation as the V-statistic

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

where h is a symmetric kernel function. Serfling^[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 x₁, ..., x_n, the corresponding V-statistic is defined

${\displaystyle V_{2,n}={\frac {1}{n^{2))}\sum _{i=1}^{n}\sum _{j=1}^{n}h(x_{i},x_{j}).}$

Example of a V-statistic

4. 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

   ${\displaystyle V_{2,n}={\frac {1}{n^{2))}\sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {1}{2))(x_{i}-x_{j})^{2}={\frac {1}{n))\sum _{i=1}^{n}(x_{i}-{\bar {x)))^{2},}$

   which is the maximum likelihood estimator of variance. With the same kernel, the corresponding U-statistic is the (unbiased) sample variance:

   ${\displaystyle s^{2}={n \choose 2}^{-1}\sum _{i<j}{\frac {1}{2))(x_{i}-x_{j})^{2}={\frac {1}{n-1))\sum _{i=1}^{n}(x_{i}-{\bar {x)))^{2))$.

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):

1. Var(h(X₁, ..., X_k)) = 0 for k < m, and Var(h(X₁, ..., X_k)) > 0 for k = m;
2. n^m/2R_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), m₂ is asymptotically normal with mean ${\displaystyle \sigma ^{2))$ and variance ${\displaystyle (\mu _{4}-\sigma ^{4})/n}$, where ${\displaystyle \mu _{4}=E(X-E(X))^{4))$.

Case m = 2 (Degenerate kernel):

Suppose A(2) is true, and ${\displaystyle E[h^{2}(X_{1},X_{2})]<\infty ,\,E|h(X_{1},X_{1})|<\infty ,}$ and ${\displaystyle E[h(x,X_{1})]\equiv 0}$. Then nV_2,n converges in distribution to a weighted sum of independent chi-squared variables:

${\displaystyle nV_{2,n}{\stackrel {d}{\longrightarrow ))\sum _{k=1}^{\infty }\lambda _{k}Z_{k}^{2},}$

where ${\displaystyle Z_{k))$ are independent standard normal variables and ${\displaystyle \lambda _{k))$ 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 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]