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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 $T(F_{n})$ of the empirical distribution function $(F_{n})$ 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

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,
$T_{n}=m_{k}=T(F_{n})={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\overline {x}})^{k}.$
The chi-squared goodness-of-fit statistic is a statistical function T(F_n), corresponding to the statistical functional
$T(F)=\sum _{i=1}^{k}{\frac {(\int _{A_{i}}\,dF-p_{i})^{2}}{p_{i}}},$
where A_i are the k cells and p_i are the specified probabilities of the cells under the null hypothesis.
The Cramér–von-Mises and Anderson–Darling goodness-of-fit statistics are based on the functional
$T(F)=\int (F(x)-F_{0}(x))^{2}\,w(x;F_{0})\,dF_{0}(x),$
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 $w(x;F_{0})=[F_{0}(x)(1-F_{0}(x))]^{-1}$ 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^[6] 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

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
$V_{2,n}={\frac {1}{n^{2}}}\sum _{i=1}^{n}\sum _{j=1}^{n}{\frac {1}{2}}(x_{i}-x_{j})^{2}={\frac {1}{n}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2},$
which is the maximum likelihood estimator of variance. With the same kernel, the corresponding U-statistic is the (unbiased) sample variance:
$s^{2}={n \choose 2}^{-1}\sum _{i<j}{\frac {1}{2}}(x_{i}-x_{j})^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(x_{i}-{\bar {x}})^{2}$ .

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.^[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):

Var(h(X₁, ..., X_k)) = 0 for k < m, and Var(h(X₁, ..., X_k)) > 0 for k = m;
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 $E[h^{2}(X_{1},X_{2})]<\infty ,\,E|h(X_{1},X_{1})|<\infty ,$ and $E[h(x,X_{1})]\equiv 0$ . Then nV_2,n converges in distribution to a weighted sum of independent chi-squared variables:

nV_{2,n}{\stackrel {d}{\longrightarrow }}\sum _{k=1}^{\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^[1] (Example 3) is an example of a degenerate kernel V-statistic.^[8]

Notes

^ ^a ^b ^c ^d von Mises (1947)
^ Lee (1990)
^ Koroljuk & Borovskich (1994)
^ Hoeffding (1948)
^ von Mises (1947), p. 309; Serfling (1980), p. 210.
^ Serfling (1980, Section 6.5)
^ Serfling (1980, Ch. 5–6); Lee (1990, Ch. 3)
^ 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. JSTOR 2235637.
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. JSTOR 2236587.
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. JSTOR 2235734.

[VM-1] von Mises (1947)

[2] Lee (1990)

[3] Koroljuk & Borovskich (1994)

[4] Hoeffding (1948)

[5] von Mises (1947), p. 309; Serfling (1980), p. 210.

[Serfling.a-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.

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