V-statistic

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Short description: Statistics named for Richard von Mises

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 [math]\displaystyle{ T(F_n) }[/math] of the empirical distribution function [math]\displaystyle{ (F_n) }[/math] 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 [math]\displaystyle{ T(F)=\int(x-\mu)^k \, dF(x) }[/math], where [math]\displaystyle{ \mu = E[X] }[/math] is the expected value of X. The associated statistical function is the sample k-th central moment,
    [math]\displaystyle{ T_n=m_k=T(F_n) = \frac 1n \sum_{i=1}^n (x_i - \overline x)^k. }[/math]
  2. The chi-squared goodness-of-fit statistic is a statistical function T(Fn), corresponding to the statistical functional
    [math]\displaystyle{ T(F) = \sum_{i=1}^k \frac{(\int_{A_i} \, dF - p_i)^2}{p_i}, }[/math]
    where Ai are the k cells and pi 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
    [math]\displaystyle{ T(F) = \int (F(x) - F_0(x))^2 \, w(x;F_0) \, dF_0(x), }[/math]
    where w(x; F0) is a specified weight function and F0 is a specified null distribution. If w is the identity function then T(Fn) is the well known Cramér–von-Mises goodness-of-fit statistic; if [math]\displaystyle{ w(x;F_0)=[F_0(x)(1-F_0(x))]^{-1} }[/math] then T(Fn) is the Anderson–Darling statistic.

Representation as a V-statistic

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

[math]\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}), }[/math]

where h is a symmetric kernel function. Serfling[6] discusses how to find the kernel in practice. Vmn 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, ..., xn, the corresponding V-statistic is defined

[math]\displaystyle{ V_{2,n} = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n h(x_i, x_j). }[/math]

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
    [math]\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, }[/math]
    which is the maximum likelihood estimator of variance. With the same kernel, the corresponding U-statistic is the (unbiased) sample variance:
    [math]\displaystyle{ s^2= {n \choose 2}^{-1} \sum_{i \lt j} \frac{1}{2}(x_i - x_j)^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar x)^2 }[/math].

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, ..., Xk)) = 0 for k < m, and Var(h(X1, ..., Xk)) > 0 for k = m;
  2. nm/2Rmn tends to zero (in probability). (Rmn 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(Fn) is asymptotically normal.

In the variance example (4), m2 is asymptotically normal with mean [math]\displaystyle{ \sigma^2 }[/math] and variance [math]\displaystyle{ (\mu_4 - \sigma^4)/n }[/math], where [math]\displaystyle{ \mu_4=E(X-E(X))^4 }[/math].

Case m = 2 (Degenerate kernel):

Suppose A(2) is true, and [math]\displaystyle{ E[h^2(X_1,X_2)]\lt \infty, \, E|h(X_1,X_1)|\lt \infty, }[/math] and [math]\displaystyle{ E[h(x,X_1)]\equiv 0 }[/math]. Then nV2,n converges in distribution to a weighted sum of independent chi-squared variables:

[math]\displaystyle{ n V_{2,n} {\stackrel d \longrightarrow} \sum_{k=1}^\infty \lambda_k Z^2_k, }[/math]

where [math]\displaystyle{ Z_k }[/math] are independent standard normal variables and [math]\displaystyle{ \lambda_k }[/math] 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 V2,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]

See also

Notes

  1. ↑ 1.0 1.1 1.2 1.3 (von Mises 1947)
  2. ↑ (Lee 1990)
  3. ↑ (Koroljuk Borovskich)
  4. ↑ (Hoeffding 1948)
  5. ↑ von Mises (1947), p. 309; Serfling (1980), p. 210.
  6. ↑ Serfling (1980, Section 6.5)
  7. ↑ Serfling (1980, Ch. 5–6); Lee (1990, Ch. 3)
  8. ↑ See Lee (1990, p. 160) for the kernel function.

References

  • Hoeffding, W. (1948). "A class of statistics with asymptotically normal distribution". Annals of Mathematical Statistics 19 (3): 293–325. doi:10.1214/aoms/1177730196. 
  • 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 ed.). Dordrecht: Kluwer Academic Publishers. ISBN 0-7923-2608-3. 
  • Lee, A.J. (1990). U-Statistics: theory and practice. New York: Marcel Dekker, Inc.. ISBN 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. doi:10.1016/0047-259X(77)90083-5. 
  • Rosenblatt, M. (1952). "Limit theorems associated with variants of the von Mises statistic". Annals of Mathematical Statistics 23 (4): 617–623. doi:10.1214/aoms/1177729341. 
  • Serfling, R.J. (1980). Approximation theorems of mathematical statistics. New York: John Wiley & Sons. ISBN 0-471-02403-1. 
  • Taylor, R.L.; Daffer, P.Z.; Patterson, R.F. (1985). Limit theorems for sums of exchangeable random variables. New Jersey: Rowman and Allanheld. 
  • von Mises, R. (1947). "On the asymptotic distribution of differentiable statistical functions". Annals of Mathematical Statistics 18 (2): 309–348. doi:10.1214/aoms/1177730385. 



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