In probability theory, an empirical measure is a random measure arising from a particular realization of a (usually finite) sequence of random variables.
Empirical measures are relevant to mathematical statistics.
The motivation for studying empirical measures is that it is often impossible to know the true underlying probability measure
, or a related distribution function
These are uniformly good estimates under certain conditions.
Theorems in the area of empirical processes provide rates of this convergence.
be a sequence of independent identically distributed random variables with values in the state space S with probability distribution P. Definition Properties Definition To generalize this notion further, observe that the empirical measure
maps measurable functions
to their empirical mean, In particular, the empirical measure of A is simply the empirical mean of the indicator function, Pn(A) = Pn IA.
For a fixed measurable function
is a random variable with mean
By the strong law of large numbers, Pn(A) converges to P(A) almost surely for fixed A.
almost surely for a fixed measurable function
The problem of uniform convergence of Pn to P was open until Vapnik and Chervonenkis solved it in 1968.
) is Glivenko–Cantelli with respect to P then Pn converges to P uniformly over
In other words, with probability 1 we have The empirical distribution function provides an example of empirical measures.
For real-valued iid random variables
it is given by In this case, empirical measures are indexed by a class