V-statistic

Template:Short description 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,

   $T_n=m_k=T(F_n)=\frac{1}{n}\sum _{i=1}^n(x_i-\bar{x}{)}^k.$

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

   $T(F)=\sum _{i=1}^k\frac{(\underset{A_i}{\int }dF-p_i{)}^2}{p_i},$

   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

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

   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 $w(x;F_0)=[F_0(x)(1-F_0(x)){]}^{-1}$ 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

$V_{mn}=\frac{1}{n^m}\sum _{i_1=1}^n\cdots \sum _{i_m=1}^{n}h(x_{i_1},x_{i_2},\dots ,x_{i_m}),$

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

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

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

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

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

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

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, ..., 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 ${\sigma }^2$ 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 $E[h^2(X_1,X_2)]<\mathrm{\infty },E|h(X_1,X_1)|<\mathrm{\infty },$ and $E[h(x,X_1)]\equiv 0$. Then nV_2,n converges in distribution to a weighted sum of independent chi-squared variables:

$n{V}_{2,n}\stackrel{d}{⟶}\sum _{k=1}^{\mathrm{\infty }}{\lambda }_k{Z}_k^2,$

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 V_2,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.

See also

• U-statistic
• Asymptotic distribution
• Asymptotic theory (statistics)

References

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