Universal Donsker Classes and Metric Entropy

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–Cervonenkis comdinatorial conditions defined later in this section Donsker the Vapnik-Cervonenkis 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 Cervonenkis (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].

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