Definition
It is known that under certain conditions empirical measures uniformly converge to the probability measure P (see Glivenko–Cantelli theorem). The theory of empirical processes provides the rate of this convergence.
A centered and scaled version of the empirical measure is the signed measure
It induces a map on measurable functions f given by
By the central limit theorem, converges in distribution to a normal random variable N(0, P(A)(1 − P(A))) for fixed measurable set A. Similarly, for a fixed function f, converges in distribution to a normal random variable, provided that and exist.
Definition
- is called an empirical process indexed by, a collection of measurable subsets of S.
- is called an empirical process indexed by, a collection of measurable functions from S to .
A significant result in the area of empirical processes is Donsker's theorem. It has led to a study of Donsker classes: sets of functions with the useful property that empirical processes indexed by these classes converge weakly to a certain Gaussian process. While it can be shown that Donsker classes are Glivenko–Cantelli classes, the converse is not true in general.
Read more about this topic: Empirical Process
Famous quotes containing the word definition:
“The definition of good prose is proper words in their proper places; of good verse, the most proper words in their proper places. The propriety is in either case relative. The words in prose ought to express the intended meaning, and no more; if they attract attention to themselves, it is, in general, a fault.”
—Samuel Taylor Coleridge (1772–1834)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)