Asymptotic normality

The asymptotic normality is in mathematical statistics a property of Statistics . Statistics that have this property are called asymptotically normal statistics or asymptotically normally distributed statistics . Asymptotically normal statistics are characterized by the fact that their distribution in the limit value converges to the standard normal distribution (with regard to the convergence in distribution ). This enables the construction of approximate statistical methods.

definition

Given a family of probability measures indexed with an index set as well as a probability space and a sequence of random variables on this probability space. ${\displaystyle \Theta }$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle (\Omega ,{\mathcal {A}},P_{\vartheta })}$ ${\displaystyle (X_{i})_{i\in \mathbb {N} }}$

Then a sequence of statistics is called asymptotically normally distributed or asymptotically normal if there are consequences ${\displaystyle T_{n}=T_{n}(X_{1},X_{2},\dots ,X_{n})}$

${\displaystyle (a_{n}(\vartheta ))_{n\in \mathbb {N} }}$ and ${\displaystyle (b_{n}(\vartheta ))_{n\in \mathbb {N} }}$

there so that

${\displaystyle {\frac {T_{n}(X_{1},X_{2},\dots ,X_{n})-a_{n}(\vartheta )}{b_{n}(\vartheta )}}{\stackrel {n\to \infty }{\implies }}{\mathcal {N}}_{0,1}}$ in distribution to all . ${\displaystyle \vartheta \in \Theta }$

The normalized and rescaled distributions thus converge to the standard normal distribution .

use

Asymptotically normally distributed statistics are an aid in asymptotic statistics. If the distribution of a statistic is unknown, but is asymptotically normally distributed, one can go through it ${\displaystyle T_{N}}$

${\displaystyle P(T_{N}(X)\leq t)\approx \Phi \left({\frac {t-a_{N}(\vartheta )}{b_{N}(\vartheta )}}\right)}$

approach. Here the distribution function denotes the standard normal distribution. The sample size should be large enough to keep the approximation error small. This possibility of approximation allows exact statistical methods, which are tailored to Gaussian distributed statistics ( Gaussian test etc.), to be transferred as an approximate method to asymptotically normally distributed statistics. ${\displaystyle \Phi }$ ${\displaystyle N}$

literature

• Claudia Czado, Thorsten Schmidt: Mathematical Statistics . Springer-Verlag, Berlin Heidelberg 2011, ISBN 978-3-642-17260-1 , doi : 10.1007 / 978-3-642-17261-8 .