V-statistic

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.

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. 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 1n \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; 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)=[F_0(x)(1-F_0(x))]^{-1} then T(F n ) is the Anderson–Darling 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 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_{2,n} = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n h(x_i, x_j). 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_{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 . Var(h(X 1 , ..., X k )) = 0 for k 1 , ..., X k )) > 0 for k = m; 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), m 2 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)] and E[h(x,X_1)]\equiv 0 . Then nV 2,n converges in distribution to a weighted sum of independent chi-squared variables: n V_{2,n} {\stackrel d \longrightarrow} \sum_{k=1}^\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.

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