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

Properties

Definition

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,

In particular, the empirical measure of A is simply the empirical mean of the indicator function, P_n(A) = P_n I_A.