In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, an empirical measure is a
random measure In probability theory, a random measure is a measure-valued random element. Random measures are for example used in the theory of random processes, where they form many important point processes such as Poisson point processes and Cox processes. ...
arising from a particular realization of a (usually finite) sequence of
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
s. The precise definition is found below. Empirical measures are relevant to
mathematical statistics
Mathematical statistics is the application of probability theory, a branch of mathematics, to statistics, as opposed to techniques for collecting statistical data. Specific mathematical techniques which are used for this include mathematical ...
.
The motivation for studying empirical measures is that it is often impossible to know the true underlying
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more g ...
. 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 process
In probability theory, an empirical process is a stochastic process that describes the proportion of objects in a system in a given state.
For a process in a discrete state space a population continuous time Markov chain or Markov population model ...
es provide rates of this convergence.
Definition
Let
be a sequence of
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group)