Statistical Functional

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 /math> is the expected value of ''X''. The associated ''statistical function'' is the sample ''k''-th central moment,

   :$T_n=m_k=T(F_n) = \frac 1n \sum_^n (x_i - \overline x)^k.$
   

2. 
   The chi-squared goodness-of-fit statistic is a statistical function ''T''(''F''_''n''), corresponding to the statistical functional

   :$T(F) = \sum_^k \frac,$

   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

   :$T(F) = \int (F(x) - F_0(x))^2 \, w(x;F_0) \, dF_0(x),$
   where ''w''(''x''; ''F''₀) is a specified weight function and ''F''₀ 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 $w(x;F_0)=$_0(x)(1-F_0(x)) then ''T''(''F''_''n'') is the Anderson–Darling statistic.
   

Representation as a V-statistic

Suppose ''x''₁, ..., ''x''_''n'' is a sample. In typical applications the statistical function has a representation as the V-statistic
:$V_ = \frac \sum_^n \cdots \sum_^n h(x_, x_, \dots, x_),$
where ''h'' is a symmetric kernel function. SerflingSerfling (1980, Section 6.5) 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 ''x''₁, ..., ''x''_''n'', the corresponding V-statistic is defined

:$V_ = \frac \sum_^n \sum_^n h(x_i, x_j).$

Example of a V-statistic

4. 
   An example of a degree-2 V-statistic is the second central moment ''m''₂.

   If ''h''(''x'', ''y'') = (''x'' − ''y'')²/2, the corresponding V-statistic is

   :$V_ = \frac \sum_^n \sum_^n \frac(x_i - x_j)^2 = \frac \sum_^n (x_i - \bar x)^2,$
   which is the maximum likelihood estimator of variance. With the same kernel, the corresponding U-statistic is the (unbiased) sample variance:

   :$s^2= ^ \sum_ \frac(x_i - x_j)^2 = \frac \sum_^n (x_i - \bar x)^2$.
   

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. 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''(''X''₁, ..., ''X''_''k'')) = 0 for ''k'' < ''m'', and Var(''h''(''X''₁, ..., ''X''_''k'')) > 0 for ''k'' = ''m'';


2. ''n''^''m''/2''R''_''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 $E$^2(X_1,X_2)\infty, \, E, h(X_1,X_1), <\infty, and E (x,X_1)equiv 0. Then nV_2,n converges in distribution to a weighted sum of independent chi-squared variables:

:$n V_ \sum_^\infty \lambda_k Z^2_k,$

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 Lee (1990, p. 160) for the kernel function.

See also

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

Notes

References

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{{Statistics, inference, collapsed

Estimation theory
Asymptotic theory (statistics)