Asymptotic expectancy

The asymptotic fidelity to expectations , also called asymptotic undistortion or asymptotic undistorted , is a property of a point estimator in mathematical statistics . Asymptotically unambiguous estimators are clearly those which are not unambiguous for finite samples, i.e. have a systematic bias . However, this disappears in the limit value with ever larger sample sizes.

definition

Given is a statistical model that formalizes the infinite repetition of an experiment. Furthermore, let it be a series of point estimates ${\displaystyle (X^{\mathbb {N} },{\mathcal {A}}^{\mathbb {N} },(P_{\vartheta }^{\mathbb {N} })_{\vartheta \in \Theta })}$

${\displaystyle T_{n}\colon X^{n}\to \mathbb {R} }$

given and a function to be estimated

${\displaystyle g\colon \Theta \to \mathbb {R} }$.

Then the sequence is called asymptotically unbiased if ${\displaystyle T_{n}}$

${\displaystyle \lim _{n\to \infty }\operatorname {E} _{\vartheta }(T_{n})=g(\vartheta )}$for everyone . ${\displaystyle \vartheta \in \Theta }$

It denotes the expected value with regard to the probability measure . ${\displaystyle \operatorname {E} _{\vartheta }}$ ${\displaystyle P_{\vartheta }^{\mathbb {N} }}$

example

A typical asymptotically unbiased estimator arises in the normal distribution model if the variance is estimated using the maximum likelihood method when the expected value is unknown .

The statistical model is given by

${\displaystyle (\mathbb {R} ^{\mathbb {N} },{\mathcal {B}}(\mathbb {R} ^{\mathbb {N} }),{\mathcal {N}}_{\mu ,\sigma ^{2}}^{\otimes \mathbb {N} })}$

for , the maximum likelihood estimator for a sample of size through ${\displaystyle (\mu ,\sigma ^{2})\in \mathbb {R} \times (0,\infty )}$ ${\displaystyle n}$

${\displaystyle T_{n}(X)={\frac {1}{n}}\sum _{i=1}^{n}\left(X_{i}-{\tfrac {1}{n}}\sum _{j=1}^{n}X_{j}\right)^{2}}$,

the (uncorrected) sample variance. The function to be estimated is

${\displaystyle g(\sigma )=\sigma ^{2}}$

For the sake of simplicity, denote the corresponding probability distribution of the statistical model. Then according to this calculation ${\displaystyle P(\mu ,\sigma ^{2})={\mathcal {N}}_{\mu ,\sigma ^{2}}^{\otimes \mathbb {N} }}$

${\displaystyle \operatorname {E} _{P(\mu ,\sigma ^{2})}(T_{n})={\frac {n-1}{n}}\sigma ^{2}}$.

The estimator is therefore not true to expectations. In particular, applies to the distortion

${\displaystyle \operatorname {Bias} _{T_{n}}(\sigma )={\frac {\sigma ^{2}}{n}}}$

The estimator is asymptotically true to expectations, because it is

${\displaystyle \lim _{n\to \infty }\operatorname {E} _{P(\mu ,\sigma ^{2})}(T_{n})=\sigma ^{2}=g(\sigma )}$.

More general formulations

There are even more general formulations than those given above. The prerequisites that it is always a repetition of the same experiment (infinite product model) are dropped.

Formally, a probability space is then defined for as well as a sequence of random variables on this probability space. ${\displaystyle (\Omega ,{\mathcal {A}},P_{\vartheta })}$ ${\displaystyle \vartheta \in \Theta }$ ${\displaystyle (X_{i})_{i\in \mathbb {N} }}$

A sequence of point estimators is called asymptotically fair-to-expectation for the function if ${\displaystyle T_{n}=T_{n}(X_{1},X_{2},\dots ,X_{n})}$ ${\displaystyle g}$

${\displaystyle \lim _{n\to \infty }\operatorname {E} _{\vartheta }{\bigl (}T_{n}(X_{1},X_{2},\dots ,X_{n}){\bigr )}=g(\vartheta )}$

for everyone . ${\displaystyle \vartheta \in \Theta }$

Web links

• OV Shalaevskii: Asymptotically-unbiased estimator . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .
• Hans-Otto Georgii: Stochastics . Introduction to probability theory and statistics. 4th edition. Walter de Gruyter, Berlin 2009, ISBN 978-3-11-021526-7 , doi : 10.1515 / 9783110215274 .
• 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 .

Individual evidence

1. ^ Georgii: Stochastics. 2009, p. 200.
2. ^ Rüschendorf: Mathematical Statistics. 2014, p. 351.
3. ↑ Czado, Schmidt: Mathematical Statistics. 2011, p. 105.