One-Sample V-Statistics

Let X I, … , Xn be a random sample from an unknown distribution. Given a known function h, consider estimation of the “parameter“

In order to simplify the formulas, it is assumed throughout this section that the function h is permutation symmetric in its r arguments. (A given h could always be replaced by a symmetric one.) The statistic h(XI , •.. , Xr) is an unbiased estimator for (), but it is unnatural, as it uses only the first r observations. A U-statistic with kernel h remedies this; it is defined as

where the sum is taken over the set of all unordered subsets f3 of r different integers chosen from ﹛I, … , n﹜. Because the observations are i.i.d., U is an unbiased estimator for () also. Moreover, U is permutation symmetric in Xl, … Xn, and has smaller variance than h(Xb … , Xr). In fact, if X (1) , •.• , X(n) denote the values Xl, … , Xn stripped from their order (the order statistics in the case of real-valued variables), then

Because a conditional expectation is a projection, and projecting decreases second moments, the variance of the U -statistic U is smaller than the variance of the naive estimator h(XJ, … , Xr).

In this section it is shown that the sequence ./ii(U–0) is asymptotically nonnal under the condition that Eh2(X\, … , Xr ) < 00.

12.1 Example. A U-statistic of degree r = 1 is a mean n-I1::7=\h(Xi ). The asserted asymptotic nonnality is then just the central limit theorem.