V statistic

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 ) = ∫ ( x − μ ) k d F ( x ) , where μ = 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 ) = 1 n ∑ i = 1 n ( x i − x ¯ ) k .
2. The chi-squared goodness-of-fit statistic is a statistical function T(F_n), corresponding to the statistical functional T ( F ) = ∑ i = 1 k ( ∫ A i d F − 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 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

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

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 V 2 , n = 1 n 2 ∑ i = 1 n ∑ j = 1 n 1 2 ( x i − x j ) 2 = 1 n ∑ i = 1 n ( x i − 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 2 ) − 1 ∑ i < j 1 2 ( x i − x j ) 2 = 1 n − 1 ∑ i = 1 n ( x i − 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:

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 σ 2 and variance ( μ 4 − σ 4 ) / n , where μ 4 = E ( X − E ( X ) ) 4 .

Case m = 2 (Degenerate kernel):

Suppose A(2) is true, and E [ h 2 ( X 1 , X 2 ) ] < ∞ , E | h ( X 1 , X 1 ) | < ∞ , and E [ h ( x , X 1 ) ] ≡ 0 . Then nV_2,n converges in distribution to a weighted sum of independent chi-squared variables:

where Z k are independent standard normal variables and λ 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.