Empirical measure

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In probability theory, an empirical measure is a random measure arising 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 ${\displaystyle P}$. We collect observations ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$ and compute relative frequencies. We can estimate ${\displaystyle P}$, or a related distribution function ${\displaystyle F}$ 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 ${\displaystyle X_{1},X_{2},\dots }$ be a sequence of independent identically distributed random variables with values in the state space S with probability distribution P.

Definition

The empirical measure P_n is defined for measurable subsets of S and given by

${\displaystyle P_{n}(A)={1 \over n}\sum _{i=1}^{n}I_{A}(X_{i})={\frac {1}{n}}\sum _{i=1}^{n}\delta _{X_{i}}(A)}$

where ${\displaystyle I_{A}}$ is the indicator function and ${\displaystyle \delta _{X}}$ is the Dirac measure.

Properties

• 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).
• For a fixed partition ${\displaystyle A_{i}}$ of S, random variables ${\displaystyle Y_{i}=nP_{n}(A_{i})}$ form a multinomial distribution with event probabilities ${\displaystyle P(A_{i})}$
  • The covariance matrix of this multinomial distribution is ${\displaystyle Cov(Y_{i},Y_{j})=nP(A_{i})(\delta _{ij}-P(A_{j}))}$.

Definition

${\displaystyle {\bigl (}P_{n}(c){\bigr )}_{c\in {\mathcal {C}}}}$ is the empirical measure indexed by ${\displaystyle {\mathcal {C}}}$, a collection of measurable subsets of S.

To generalize this notion further, observe that the empirical measure ${\displaystyle P_{n}}$ maps measurable functions ${\displaystyle f:S\to \mathbb {R} }$ to their empirical mean,

${\displaystyle f\mapsto P_{n}f=\int _{S}f\,dP_{n}={\frac {1}{n}}\sum _{i=1}^{n}f(X_{i})}$

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 ${\displaystyle f}$, ${\displaystyle P_{n}f}$ is a random variable with mean ${\displaystyle \mathbb {E} f}$ and variance ${\displaystyle {\frac {1}{n}}\mathbb {E} (f-\mathbb {E} f)^{2}}$.

By the strong law of large numbers, P_n(A) converges to P(A) almost surely for fixed A. Similarly ${\displaystyle P_{n}f}$ converges to ${\displaystyle \mathbb {E} f}$ almost surely for a fixed measurable function ${\displaystyle f}$. The problem of uniform convergence of P_n to P was open until Vapnik and Chervonenkis solved it in 1968.^[1]

If the class ${\displaystyle {\mathcal {C}}}$ (or ${\displaystyle {\mathcal {F}}}$) is Glivenko–Cantelli with respect to P then P_n converges to P uniformly over ${\displaystyle c\in {\mathcal {C}}}$ (or ${\displaystyle f\in {\mathcal {F}}}$). In other words, with probability 1 we have

${\displaystyle \|P_{n}-P\|_{\mathcal {C}}=\sup _{c\in {\mathcal {C}}}|P_{n}(c)-P(c)|\to 0,}$

${\displaystyle \|P_{n}-P\|_{\mathcal {F}}=\sup _{f\in {\mathcal {F}}}|P_{n}f-\mathbb {E} f|\to 0.}$

The empirical distribution function provides an example of empirical measures. For real-valued iid random variables ${\displaystyle X_{1},\dots ,X_{n}}$ it is given by

${\displaystyle F_{n}(x)=P_{n}((-\infty ,x])=P_{n}I_{(-\infty ,x]}.}$

In this case, empirical measures are indexed by a class ${\displaystyle {\mathcal {C}}=\{(-\infty ,x]:x\in \mathbb {R} \}.}$ It has been shown that ${\displaystyle {\mathcal {C}}}$ is a uniform Glivenko–Cantelli class, in particular,

${\displaystyle \sup _{F}\|F_{n}(x)-F(x)\|_{\infty }\to 0}$

with probability 1.

• Empirical risk minimization
• Poisson random measure

• Billingsley, P. (1995). Probability and Measure (Third ed.). New York: John Wiley and Sons. ISBN 0-471-80478-9.
• Donsker, M. D. (1952). "Justification and extension of Doob's heuristic approach to the Kolmogorov–Smirnov theorems". Annals of Mathematical Statistics. 23 (2): 277–281. doi:10.1214/aoms/1177729445.
• Dudley, R. M. (1978). "Central limit theorems for empirical measures". Annals of Probability. 6 (6): 899–929. doi:10.1214/aop/1176995384. JSTOR 2243028.
• Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge Studies in Advanced Mathematics. Vol. 63. Cambridge, UK: Cambridge University Press. ISBN 0-521-46102-2.
• Wolfowitz, J. (1954). "Generalization of the theorem of Glivenko–Cantelli". Annals of Mathematical Statistics. 25 (1): 131–138. doi:10.1214/aoms/1177728852. JSTOR 2236518.