probability theory, an empirical measure is a random measurearising from a particular realization of a (usually finite) sequence of random variables. The precise definition is found below. 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. We collect observations and compute relative frequencies. We can estimate , or a related distribution function by means of the empirical measure or empirical distribution function, respectively. These are uniformly good estimates under certain conditions. Theorems in the area of empirical processes provide rates of this convergence.
Let be a sequence of independent identically distributed
random variables with values in the state space "S" with probability measure"P".
Definition :The "empirical measure" is defined for measurable subsets of "S" and given by:::where is the
indicator functionand is the Dirac measure.
Definition: is the "empirical measure" indexed by , a collection of measurable subsets of "S".
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, .
For a fixed measurable function "f", is a random variable with mean and variance .
By the strong
law of large numbers, converges to "P(A)" almost surelyfor fixed "A". Similarly converges to almost surely for a fixed measurable function "f". The problem of uniform convergence of to "P" was open until Vapnikand Chervonenkissolved it in 1968.
If the class (or ) is Glivenko-Cantelli with respect to "P" then converges to "P" uniformly over (or ). In other words, with probability 1 we have::
Empirical distribution function
The "empirical distribution function" provides an example of empirical measures. For real-valued
iidrandom variables it is given by
In this case, empirical measures are indexed by a class It has been shown that is a uniform
Glivenko-Cantelli class, in particular,
with probability 1.
Poisson random measure
* P. Billingsley, Probability and Measure, John Wiley and Sons, New York, third edition, 1995.
* M.D. Donsker, Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems, Annals of Mathematical Statistics, 23:277--281, 1952.
* R.M. Dudley, Central limit theorems for empirical measures, Annals of Probability, 6(6): 899â€“929, 1978.
* R.M. Dudley, Uniform Central Limit Theorems, Cambridge Studies in Advanced Mathematics, 63, Cambridge University Press, Cambridge, UK, 1999.
* J. Wolfowitz, Generalization of the theorem of Glivenko-Cantelli. Annals of Mathematical Statistics, 25, 131-138, 1954.
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