Empirical Distribution (Random Measure)

The empirical distribution is a random measure in stochastics , a branch of mathematics . It forms a discrete probability distribution , the exact structure of which depends on several random variables . Only at the transition to realizations of the random variables is the probability distribution clearly determined and then the empirical distribution of a sample . The empirical distribution plays an important role in the area between probability theory and statistics . For example, the expected value of the empirical distribution can be used under certain circumstances as an estimate function for the expected value of the underlying random variable.

definition

Real random variables are given . ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$

Then is called

${\displaystyle L:={\frac {1}{n}}\sum _{i=1}^{n}\delta _{X_{i}}}$

the empirical distribution of . If all random variables have the same distribution, sometimes only the sample size and one random variable are given. ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$ ${\displaystyle n}$

The empirical distribution is also defined somewhat more generally for random variables with values ​​in Polish spaces .

Demarcation

The discrete probability distribution is also used as an empirical distribution

${\displaystyle P={\frac {1}{n}}\sum _{i=1}^{n}\delta _{x_{i}}}$

marked with . This is referred to here as the empirical distribution of the sample . The empirical distribution of the random variables and the empirical distribution of the sample are closely related. The empirical distribution of the sample arises when one moves from the (indefinite) random variable to the realization of the random variable. ${\displaystyle x=(x_{1},x_{2},\dots ,x_{n})\in \mathbb {R} ^{n}}$ ${\displaystyle x}$ ${\displaystyle X_{i}}$ ${\displaystyle x_{i}=X(\omega )}$

properties

Expected value

The expected value of the empirical distribution is the sample mean , that is

${\displaystyle \operatorname {E} (L)={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}$

Median

The median of the empirical distribution is the sample median , that is

${\displaystyle m_{L}={\begin{cases}X_{({\tfrac {n+1}{2}})}&{\text{ falls }}n{\text{ ungerade }}\\{\tfrac {1}{2}}(X_{({\tfrac {n}{2}})}+X_{({\tfrac {n+1}{2}})})&{\text{ falls }}n{\text{ gerade }}\end{cases}}}$.

Here the i-th order statistic denotes . ${\displaystyle X_{(i)}}$

Variance

The variance of the empirical distribution is the (uncorrected) sample variance , so

${\displaystyle \operatorname {Var} (L)={\frac {1}{n}}\sum _{i=1}^{n}X_{i}^{2}-\left({\frac {1}{n}}\sum _{i=1}^{n}X_{i}\right)^{2}}$

Moments

The kth moment of the empirical distribution is given by

${\displaystyle m_{k}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}^{k}}$ .

The moments of the empirical distribution are also known as the sampling moment .

use

If the random variables are independently and identically distributed , the key figures of the empirical distribution can serve as an estimation function for the corresponding key figures of the random variables . The sample mean is the expected value of the empirical distribution and can be used as an estimator for the expected value of the random variable . ${\displaystyle X_{1},X_{2},\dots ,X_{n}}$ ${\displaystyle X_{i}}$ ${\displaystyle X_{i}}$

Web links

• AV Prokhorov: Empirical distribution . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ↑ ^{a b} Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 245 , doi : 10.1007 / 978-3-642-36018-3 .
2. ^ Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , p. 243 , doi : 10.1515 / 9783110215274 .
3. ↑ Klaus D. Schmidt: Measure and probability . 2nd, revised edition. Springer-Verlag, Heidelberg Dordrecht London New York 2011, ISBN 978-3-642-21025-9 , pp. 357 , doi : 10.1007 / 978-3-642-21026-6 .
4. ↑ Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , p. 75 , doi : 10.1007 / 978-3-642-17261-8 .