Universal Donsker Classes and Metric Entropy

Abstract

When $\mathfrak{F}$ is a universal Donsker class, then for independent, indetically distributed (i.i.d) observation $\mathbf{X}_1,\ldots,\mathbf{X}_n$ with an unknown law P, for any $\mathfrak{f}_i$in $\mathfrak{F},$ $i=1,\ldots,m,\quad n^{-1/2}\left\{ \mathfrak{f}_1\left(\mathbf{X}_1\right)+\ldots+\mathfrak{f}_i\left(\mathbf{X}_n\right)\right\}_{1\leq i\leq m}$ is asymptotically normal with mean Vector $n^{1/2}\left\{\int\mathfrak{f}_i\left(\mathbf{X}_n\right)d\mathbf{P}\left(x\right)\right\}_{1_\leq i\leq m}$ and covariance matrix $\int\mathfrak{f}_i\mathfrak{f}_j d\mathbf{P}-\int\mathfrak{f}_id\mathbf{P}\int\mathfrak{f}_jd\mathbf{P},$ uniformly for ${\mathfrak{f}_i}\in \mathfrak{F}.$ Then, for certain Statistics formed frome the $\mathfrak{f}_i\left(\mathbf{X}_k\right),$ even where $\mathfrak{f}_i$ may be chosen depending on the $\mathbf{X}_k$ there will be asymptotic distribution as $n \rightarrow \infty.$ For example, for $\mathbf{X}^2$ statistics, where $f_i$ are indicators of disjoint intervals, depending suitably on $\mathbf{X}_1,\ldots,\mathbf{X}_n$, whose union is the real line, $\mathbf{X}^2$ quadratic forms have limiting distributions [Roy (1956) and Watson (1958)] which may, however, not be $\mathbf{X}^2$ distributions and may depend on P [Chernoff and Lehmann (1954)]. Universal Donsker classes of sets are, up to mild measurability conditions, just classes satisfying the Vapnik–Červonenkis comdinatorial conditions defined later in this section Donsker the Vapnik-Červonenkis combinatorial conditions defined later in this section [Durst and Dudley (1981) and Dudley (1984) Chapter 11]. The use of such classes allows a variety of extensions of the Roy–Watson results to general (multidimensional) sample spaces [Pollard (1979) and Moore and Subblebine (1981)]. Vapnik and Červonenkis (1974) indicated application of their families of sets to classification (pattern recognition) problems. More recently, the classes have been applied to tree-structured classifiacation [Breiman, Friedman, Olshen and Stone (1984), Chapter 12].

Received August 1985; revised August 1986.

This research was partially supported by National Science Foundation Grant DMS-8506638.

AMS 1980 subject claasifications. Primary 60F17, 60F05; secondary 60G17, 60G20.