V-statistic

Statistics that can be represented as functionals $T(F\_n)$ of the empirical distribution function $(F\_n)$ are called statistical functionals.von Mises (1947), p. 309; Serfling (1980), p. 210. Differentiability of the functional T plays a key role in the von Mises approach; thus von Mises considers differentiable statistical functionals.

1. The k-th central moment is the functional $T(F)=\backslash int(x-\backslash mu)^k\; \backslash ,\; dF(x)$, where $\backslash mu\; =\; E[X]$ 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.

2. 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.

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; F₀) is a specified weight function and F₀ 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.

Suppose x₁, ..., x_n is a sample. In typical applications the statistical function has a representation as the V-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. 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 x₁, ..., 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).

4. An example of a degree-2 V-statistic is the second central moment m₂.

   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 < j} \frac{1}{2}(x_i - x_j)^2 =

   \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar x)^2.

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.Serfling (1980, Ch. 5–6); Lee (1990, Ch. 3) Let A(m) be the property defined by:

:A(m):

1. Var(h(X₁, ..., X_k)) = 0 for k < m, and Var(h(X₁, ..., X_k)) > 0 for k = m;
2. n^m/2R_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 $\backslash sigma^2$ and variance $(\backslash mu\_4\; -\; \backslash sigma^4)/n$, where $\backslash mu\_4=E(X-E(X))^4$.

Case m = 2 (Degenerate kernel):

Suppose A(2) is true, and and $E[h(x,X\_1)]\backslash equiv\; 0$. Then nV_2,n converges in distribution to a weighted sum of independent chi-squared variables:

:$n\; V\_\{2,n\}\; \{\backslash stackrel\; d\; \backslash longrightarrow\}\; \backslash sum\_\{k=1\}^\backslash infty\; \backslash lambda\_k\; Z^2\_k,$

where $Z\_k$ are independent standard normal variables and $\backslash lambda\_k$ 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.

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

{{Reflist}}

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{{refend}}

{{Statistics|inference|collapsed}}

Category:Estimation theory

Category:Asymptotic theory (statistics)