V-statistic

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. 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 ${\displaystyle T(F)=\int (x-\mu {)}^{k}dF(x)}$, 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

   ${\displaystyle T(F)=\int (F(x)-F_0(x){)}^{2}w(x;F_0)d{F}_0(x),}$

   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 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

1. 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

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

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

   ${\displaystyle s^2={(\genfrac{}{}{0ex}{}{n}{2})}^{-1}\sum _{i<j}\frac{1}{2}(x_i-x_j{)}^2=\frac{1}{n-1}\sum _{i=1}^n(x_i-\overline{x}{)}^2}$.