Jurin's law

2013-06-23T13:09:26Z

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'''V-statistics''' are a class of statistics named for [[Richard von Mises]] who developed their [[asymptotic theory|asymptotic distribution theory]] in a fundamental paper in 1947.<ref name=VM>{{harvtxt|von Mises|1947}}</ref> V-statistics are closely related to [[U-statistic]]s<ref>{{harvtxt|Lee|1990}}</ref><ref>{{harvtxt|Koroljuk|Borovskich|1994}}</ref> (U for “[[Bias of an estimator|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 functions''.<ref>von Mises (1947), p. 309; Serfling (1980), p. 210.</ref> [[Differentiable function|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 [[Pearson's chi-squared test|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 criterion|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''; ''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 criterion|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 ''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

:<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'' − ''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#Population variance and sample variance|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 distribution|normal]], in (2) it is [[Chi-squared distribution|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 series|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-statistic]]s.<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(Fn) is [[Asymptotic normality|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 nV2,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
| ref = harv
}}
* {{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
| ref = harv
}}
* {{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
| ref = harv
}}
* {{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
| ref = harv
}}
* {{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
| ref = harv
}}
* {{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
| ref = harv
}}
* {{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
| ref = harv
}}
* {{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
| ref = harv
}}
{{refend}}

[[Category:Estimation theory]]
[[Category:Asymptotic statistical theory]]

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Jurin's law