Empirical process

The study of empirical processes is a branch of mathematical statistics and a sub-area of probability theory.

The motivation for studying empirical processes is that it is often impossible to know the true underlying probability measure $$P$$. We collect observations $$X_1, X_2, \dots, X_n$$  and compute relative frequencies. We can estimate $$P$$, or a related distribution function $$F$$ by means of the empirical measure or empirical distribution function, respectively. Theorems in the area of empirical processes confirm that these are uniformly good estimates or determine accuracy of the estimation.

Suppose $$X$$ is a sample space of observations. $$X$$ can be quite general; for example: the real line, some Euclidean space, a space of functions, a Riemannian manifold, or whatever might be of interest. Let $$X_1, X_2, \dots, X_n$$ be independent identically distributed (iid) random variables (rv's), with probability measure $$P$$ on $$X$$. For a measurable set $$A$$, the empirical measure $$P_n$$ is defined as


 * $$P_n(A) = {1 \over n} \operatorname{card}\{\, j \in \{\,1,\dots,n\,\} : X_j \in  A\,\}.$$

If $$C$$ is a collection of subsets of $$X$$, then the collection


 * $$\{P_n(c): c \in C\}$$

is the empirical measure indexed by $$C$$. The empirical process $$B_n$$ is defined as
 * $$B_n = \sqrt n(P_n-P).$$

and
 * $$\{B_n(c): c \in C\}$$

is the empirical process indexed by $$C.$$

A special case is the empirical process $$G_n$$ associated with empirical distribution functions $$F_n$$.


 * $$G_n(x) = \sqrt n(F_n(x)-F(x)),$$

where $$X_1, X_2, \dots, X_n$$ are real-valued random variables with distribution function $$F$$ and $$F_n$$ is defined by


 * $$F_n(x) = {1 \over n} \operatorname{card}\{\,j \in \{\,1,\dots,n\,\} : X_j \leq  x \,\}.$$

In this case,


 * $$C = \{(-\infty, x): x \in R\}.$$

Major results for this special case include Kolmogorov-Smirnov statistics, the Glivenko-Cantelli theorem and Donsker's theorem. Moreover, the empirical distribution function $$F_n$$ of a finite sequence of realizations of a random variable is the very essence of statistical inference.

Glivenko-Cantelli theorem
By the strong law of large numbers, we know that


 * $$F_n(x) {\longrightarrow} _{a.s.} F(x) . $$

However, Glivenko and Cantelli strengthened this result.

The Glivenko-Cantelli theorem (1933):


 * $$\|F_n - F\|_\infty = \sup_{x\in R} |F_n(x) - F(x)| {\longrightarrow} _{a.s.} 0. $$

Another way to state this is as follows: the sample paths of $$F_n$$ get uniformly closer to $$F$$ as $$n$$ increases; hence $$F_n$$, which we observe, is almost surely a good approximation for $$F$$, which becomes better as we collect more observations.

Donsker's theorem
By the classical central limit theorem, it follows that


 * $$G_n(x){\longrightarrow}_{dist} G(x),$$

that is, $$G_n(x)$$ converges in distribution to a Gaussian (normal) random variable $$G(x)$$ with mean 0 and variance  $$F(x)[1-F(x)].$$   Donsker (1952) showed that the sample paths of $$G_n(x)$$, as functions on the real line $$R$$, converge in distribution to a stochastic process $$G$$ in the space $$l$$∞ of all bounded functions $$f:R{\rightarrow}R$$. The function space $$l$$∞ is used in this context to remind us that we are concerned with distributional convergence in terms of sample paths. The limit process $$G$$ is a Gaussian process with zero mean and covariance given by


 * cov[G(s), G(t)] = E[G(s)G(t)] = F[min(s, t)] &minus; F(s)F(t).

The process $$G(x)$$ can be written as $$B(F(x))$$ where $$B$$ is a standard Brownian bridge on the unit interval.

If the observations $$X_1, X_2, \dots, X_n$$ are in a more general sample space $$X$$, we seek generalizations of the Glivenko-Cantelli theorem and Donsker's theorem. Also, we seek other theorems to determine rates of convergence and accuracy of estimation.

The classical empirical distribution function for real-valued random variables is a special case of the general theory with $$X$$ = $$R$$ and the class of sets $$C = \{(\infty, x]: x \in R\}$$.