V-statistic explained

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.^[1] 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)} = (1 n)/(\sum) ^{n} ^{i =1} (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
   T(F) = \sum ^{k} _{i =1} ((\int _{A _{i}} dF - p ^{2} _{i)})/(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 ^{2} _{0 (x))} 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 w(x;F _{0)=[F} _{0(x)(1-F} ^{-1} ^{0 (x))]} 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} = (1)/(n ^{m}) \sum ^{n} _{i _{1=1}} … \sum ^{n} _{i _{m=1}} h(x _{i _{1}} , x _{i _{2}} , ... , x _{i _{m}} ), where h is a symmetric kernel function. Serfling^[2] 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} = (1)/(n ^{2}) \sum ^{n} _{i =1} \sum ^{n} _{j =1} 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} = (1)/(n ^{2}) \sum ^{n} _{i =1} \sum ^{n} _{j =1} (1)/(2) (x _{i} - x ^{2} _{j)} = (1)/(n) \sum ^{n} _{i =1} (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} (1)/(2) (x _{i} - x ^{2} _{j)} = (1)/(n -1) \sum ^{n} _{i =1} (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. 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.^[3] 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} _{4= E(X - E(X))} .

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 nV_2,n converges in distribution to a weighted sum of independent chi-squared variables:n V _{2,n} {\stackrel d \longrightarrow} \sum ^{inf ty} _{k =1} λ _{k} Z ^{2} _{k,}

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.^[4]

See also

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

References

• Hoeffding . W. . 1948 . A class of statistics with asymptotically normal distribution . Annals of Mathematical Statistics . 19 . 3 . 293–325 . 2235637 . 10.1214/aoms/1177730196 . free .
• Book: Koroljuk . V.S. . Borovskich . Yu.V. . 1994 . Theory of U-statistics . English translation by P.V.Malyshev and D.V.Malyshev from the 1989 Ukrainian . Kluwer Academic Publishers . Dordrecht . 0-7923-2608-3 .
• Book: Lee, A.J. . 1990 . U-Statistics: theory and practice . Marcel Dekker, Inc. . New York . 0-8247-8253-4 .
• Neuhaus . G. . 1977 . Functional limit theorems for U-statistics in the degenerate case . Journal of Multivariate Analysis . 7 . 3 . 424–439 . 10.1016/0047-259X(77)90083-5 . free .
• Rosenblatt . M. . 1952 . Limit theorems associated with variants of the von Mises statistic . Annals of Mathematical Statistics . 23 . 4 . 617–623 . 2236587 . 10.1214/aoms/1177729341 . free .
• Book: Serfling, R.J. . 1980 . Approximation theorems of mathematical statistics . John Wiley & Sons . New York . 0-471-02403-1 .
• Book: Taylor . R.L. . Daffer . P.Z. . Patterson . R.F. . 1985 . Limit theorems for sums of exchangeable random variables . Rowman and Allanheld . New Jersey .
• von Mises . R. . 1947 . On the asymptotic distribution of differentiable statistical functions . Annals of Mathematical Statistics . 18 . 2 . 309–348 . 2235734 . 10.1214/aoms/1177730385 . free .

Notes and References

1. von Mises (1947), p. 309; Serfling (1980), p. 210.
2. Serfling (1980, Section 6.5)
3. Serfling (1980, Ch. 5–6); Lee (1990, Ch. 3)
4. See Lee (1990, p. 160) for the kernel function.