V-statistic

{{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.<ref name=VM>{{harvtxt|von Mises|1947}}</ref> V-statistics are closely related to U-statistics<ref>{{harvtxt|Lee|1990}}</ref><ref>{{harvtxt|Koroljuk|Borovskich|1994}}</ref> (U for "unbiased") introduced by Wassily Hoeffding in 1948.<ref>{{harvtxt|Hoeffding|1948}}</ref> 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>T(F_n)</math> of the empirical distribution function <math>(F_n)</math> are called ''statistical functionals''.<ref>von Mises (1947), p. 309; Serfling (1980), p. 210.</ref> Differentiability of the functional ''T'' plays a key role in the von Mises approach; thus von Mises considers ''differentiable statistical functionals''.<ref name=VM/>

=== Examples of statistical functions ===

<ol> <li> The ''k''-th central moment is the ''functional'' <math>T(F)=\int(x-\mu)^k \, dF(x)</math>, where <math>\mu = E[X]</math> is the expected value of ''X''. The associated ''statistical function'' is the sample ''k''-th central moment,

:<math> T_n=m_k=T(F_n) = \frac 1n \sum_{i=1}^n (x_i - \overline x)^k. </math> </li>

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

:<math> T(F) = \sum_{i=1}^k \frac{(\int_{A_i} \, dF - p_i)^2}{p_i}, </math>

where ''A''_''i'' are the ''k'' cells and ''p''_''i'' are the specified probabilities of the cells under the null hypothesis. </li>

<li> The Cramér–von-Mises and Anderson–Darling goodness-of-fit statistics are based on the functional

:<math> T(F) = \int (F(x) - F_0(x))^2 \, w(x;F_0) \, dF_0(x), </math> where ''w''(''x'';&nbsp;''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 <math>w(x;F_0)=[F_0(x)(1-F_0(x))]^{-1}</math> then ''T''(''F''_''n'') is the Anderson–Darling statistic. </li> </ol>

=== 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 :<math> 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<ref name=Serfling.a>Serfling (1980, Section 6.5)</ref> discusses how to find the kernel in practice. ''V''_''mn'' is called a V-statistic of degree&nbsp;''m''.

A symmetric kernel of degree 2 is a function ''h''(''x'',&nbsp;''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

:<math> 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 ===

<ol start="4"> <li> An example of a degree-2 V-statistic is the second central moment ''m''₂.

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

:<math> 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>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</math>. </li> </ol>

== 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.<ref name=VM/> 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&nbsp;''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.<ref>Serfling (1980, Ch. 5–6); Lee (1990, Ch. 3)</ref> Let ''A''(''m'') be the property defined by: :''A''(''m''): <ol style="list-style-type:lower-roman"> <li> Var(''h''(''X''₁, ..., ''X''_''k'')) = 0 for ''k'' < ''m'', and Var(''h''(''X''₁, ..., ''X''_''k'')) > 0 for ''k'' = ''m''; </li> <li> ''n''^''m''/2''R''_''mn'' tends to zero (in probability). (''R''_''mn'' is the remainder term in the Taylor series for ''T''.)</li> </ol>

'''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 <math>\sigma^2</math> and variance <math>(\mu_4 - \sigma^4)/n</math>, where <math>\mu_4=E(X-E(X))^4</math>.

'''Case ''m'' = 2''' (Degenerate kernel):

Suppose ''A''(2) is true, and <math>E[h^2(X_1,X_2)]<\infty, \, E|h(X_1,X_1)|<\infty, </math> and <math> E[h(x,X_1)]\equiv 0</math>. Then nV_2,n converges in distribution to a weighted sum of independent chi-squared variables:

:<math> n V_{2,n} {\stackrel d \longrightarrow} \sum_{k=1}^\infty \lambda_k Z^2_k,</math>

where <math>Z_k</math> are independent standard normal variables and <math>\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 ''V''_2,''n'' is called a ''degenerate kernel V-statistic''. The V-statistic associated with the Cramer–von Mises functional<ref name=VM/> (Example 3) is an example of a degenerate kernel V-statistic.<ref>See Lee (1990, p. 160) for the kernel function.</ref>

== See also == * U-statistic * Asymptotic distribution * Asymptotic theory (statistics)

== Notes == {{Reflist}}

== References == {{refbegin}} * {{cite journal | last = Hoeffding | first = W. | year = 1948 | title = A class of statistics with asymptotically normal distribution | journal = Annals of Mathematical Statistics | volume = 19 | issue = 3 | pages = 293–325 | jstor = 2235637 | doi=10.1214/aoms/1177730196 | doi-access = free }} * {{cite book | last1 = Koroljuk | first1 = V.S. | last2 = Borovskich | first2 = Yu.V. | year = 1994 | title = Theory of ''U''-statistics | edition = English translation by P.V.Malyshev and D.V.Malyshev from the 1989 Ukrainian | publisher = Kluwer Academic Publishers | location = Dordrecht | isbn = 0-7923-2608-3 }} * {{cite book | last = Lee | first = A.J. | year = 1990 | title = ''U''-Statistics: theory and practice | publisher = Marcel Dekker, Inc. | location = New York | isbn = 0-8247-8253-4 }} * {{cite journal | last = Neuhaus | first = G. | year = 1977 | title = Functional limit theorems for ''U''-statistics in the degenerate case | journal = Journal of Multivariate Analysis | volume = 7 | issue = 3 | pages = 424–439 | doi = 10.1016/0047-259X(77)90083-5 | doi-access = free }} * {{cite journal | last = Rosenblatt | first = M. | year = 1952 | title = Limit theorems associated with variants of the von Mises statistic | journal = Annals of Mathematical Statistics | volume = 23 | issue = 4 | pages = 617–623 | jstor = 2236587 | doi=10.1214/aoms/1177729341 | doi-access = free }} * {{cite book | last = Serfling | first = R.J. | year = 1980 | title = Approximation theorems of mathematical statistics | publisher = John Wiley & Sons | location = New York | isbn = 0-471-02403-1 }} * {{cite book | last1 = Taylor | first1 = R.L. | last2 = Daffer | first2 = P.Z. | last3 = Patterson | first3 = R.F. | year = 1985 | title = Limit theorems for sums of exchangeable random variables | publisher = Rowman and Allanheld | location = New Jersey }} * {{cite journal | last = von Mises | first = R. | year = 1947 | title = On the asymptotic distribution of differentiable statistical functions | journal = Annals of Mathematical Statistics | volume = 18 | issue = 2 | pages = 309–348 | jstor = 2235734 | doi=10.1214/aoms/1177730385 | doi-access = free }} {{refend}}

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Category:Estimation theory Category:Asymptotic theory (statistics)