Empirical measure

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.

Definition

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

${\displaystyle P_n(A)=\frac{1}{n}\sum _{i=1}^n{I}_A(X_i)=\frac{1}{n}\sum _{i=1}^n{\delta }_{X_i}(A)}$

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 ${\displaystyle Y_i=n{P}_n(A_i)}$ 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

${\displaystyle (P_n(c){)}_{c\in \mathcal{C}}}$ 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,

${\displaystyle f\mapsto P_{n}f={\int }_{S}fd{P}_n=\frac{1}{n}\sum _{i=1}^{n}f(X_i)}$

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.

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

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

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