# Empirical measure

In

probability theory , an**empirical measure**is arandom measure arising from a particular realization of a (usually finite) sequence ofrandom variable s. The precise definition is found below. Empirical measures are relevant tomathematical statistics .The motivation for studying empirical measures 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 computerelative frequencies . We can estimate $P$, or a related distribution function $F$ by means of the empirical measure or empirical distribution function, respectively. These are uniformly good estimates under certain conditions. Theorems in the area ofempirical process es provide rates of this convergence.**Definition**Let $X\_1,\; X\_2,\; dots$ be a sequence of independent identically distributed

random variable s with values in the state space "S" withprobability measure "P".**Definition**:The "empirical measure" $P\_n$ is defined for measurable subsets of "S" and given by::$P\_n(A)\; =\; \{1\; over\; n\}\; sum\_\{i=1\}^n\; I\_A(X\_i)=frac\{1\}\{n\}sum\_\{i=1\}^n\; delta\_\{X\_i\}(A)$:where $I\_A$ is theindicator function and $delta\_X$ is theDirac measure .For a fixed measurable set "A", $nP\_n(A)$ is a binomial random variable with mean "nP(A)" and variance "nP(A)(1-P(A))". In particular, $P\_n(A)$ is an unbiased estimator of "P(A)".

**Definition**:$igl(P\_n(c)igr)\_\{cinmathcal\{C$ is the "empirical measure" indexed by $mathcal\{C\}$, a collection of measurable subsets of "S".To generalize this notion further, observe that the empirical measure $P\_n$ maps

measurable function s $f:S\; o\; mathbb\{R\}$ to their "empirical mean ",:$fmapsto\; P\_n\; f=int\_S\; fdP\_n=frac\{1\}\{n\}sum\_\{i=1\}^n\; f(X\_i)$

In particular, the empirical measure of "A" is simply the empirical mean of the indicator function, $P\_n(A)=P\_n\; I\_A$.

For a fixed measurable function "f", $P\_nf$ is a random variable with mean $mathbb\{E\}f$ and variance $frac\{1\}\{n\}mathbb\{E\}(f\; -mathbb\{E\}\; f)^2$.

By the strong

law of large numbers , $P\_n(A)$ converges to "P(A)"almost surely for fixed "A". Similarly $P\_nf$ converges to $mathbb\{E\}\; f$ almost surely for a fixed measurable function "f". The problem of uniform convergence of $P\_n$ to "P" was open untilVapnik andChervonenkis solved it in 1968.If the class $mathcal\{C\}$ (or $mathcal\{F\}$) is Glivenko-Cantelli with respect to "P" then $P\_n$ converges to "P" uniformly over $cinmathcal\{C\}$ (or $fin\; mathcal\{F\}$). In other words, with probability 1 we have:$|P\_n-P|\_mathcal\{C\}=sup\_\{cinmathcal\{C|P\_n(c)-P(c)|\; o\; 0,$:$|P\_n-P|\_mathcal\{F\}=sup\_\{finmathcal\{F|P\_nf-mathbb\{E\}f|\; o\; 0.$

**Empirical distribution function**The "empirical distribution function" provides an example of empirical measures. For real-valued

iid random variables $X\_1,dots,X\_n$ it is given by:$F\_n(x)=P\_n((-infty,x]\; )=P\_nI\_\{(-infty,x]\; \}.$

In this case, empirical measures are indexed by a class $mathcal\{C\}=\{(-infty,x]\; :xinmathbb\{R\}\}.$ It has been shown that $mathcal\{C\}$ is a uniform

Glivenko-Cantelli class , in particular,:$sup\_F|F\_n(x)-F(x)|\_infty\; o\; 0$

with probability 1.

**ee also***

Empirical process

*Poisson random measure **References*** 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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