To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

From Wikipedia, the free encyclopedia

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.

YouTube Encyclopedic

  • 1/5
    Views:
    1 669
    490 070
    1 549 548
    459 913
    120 811
  • ✪ Population vs Sample and Parameter vs Statistic - Statistics #12
  • ✪ MAT 110 Basic Statistics Lesson 1 (video 1).mp4
  • ✪ Hypothesis testing and p-values | Inferential statistics | Probability and Statistics | Khan Academy
  • ✪ Choosing which statistical test to use - statistics help
  • ✪ Introduction to Statistics

Transcription

Contents

Statistical functions

Statistics that can be represented as functionals of the empirical distribution function 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 , where is the expected value of X. The associated statistical function is the sample k-th central moment,
  2. The chi-squared goodness-of-fit statistic is a statistical function T(Fn), corresponding to the statistical functional
    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
    where w(xF0) 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 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

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(xy), 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

Example of a V-statistic

  1. An example of a degree-2 V-statistic is the second central moment m2. If h(x, y) = (xy)2/2, the corresponding V-statistic is
    which is the maximum likelihood estimator of variance. With the same kernel, the corresponding U-statistic is the (unbiased) sample variance:
    .

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 and variance , where .

Case m = 2 (Degenerate kernel):

Suppose A(2) is true, and and . Then nV2,n converges in distribution to a weighted sum of independent chi-squared variables:

where are independent standard normal variables and 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. ^ a b c d von Mises (1947)
  2. ^ Lee (1990)
  3. ^ Koroljuk & Borovskich (1994)
  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. 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.
This page was last edited on 12 March 2019, at 18:44
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.