Regular statistical model

A regular statistical model or, for short, a regular model is a special statistical model in which certain additional assumptions apply. These additional assumptions provide the existence of far-reaching properties such as the existence of the score function and thus also the Fisher information . Some authors also call these additional assumptions Cramér-Rao regularity conditions , since they are often used in the context of the Cramér-Rao inequality . Not all authors use the same additional assumptions for the statistical model. This article gives an overview of the regularity requirements and the possible conclusions.

definition

A statistical model is given for which the following applies: ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$

1. It is a one-parameter model. So it is . ${\displaystyle \Theta \subset \mathbb {R} }$
2. Every probability measure has a density function with respect to a σ-finite measure , that is, is a dominated distribution class . In most cases, the density function is a probability density function, i.e. it is dominated by the Lebesgue measure (continuous case), or a counting density , i.e. is dominated by the counting measure (discrete case). ${\displaystyle P_{\vartheta }}$ ${\displaystyle f(x,\vartheta )}$ ${\displaystyle \mu }$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$

The statistical model is then called a regular statistical model if:

• ${\displaystyle \Theta }$is an open interval. This is necessary to guarantee the well-defined differentiation . Some authors only demand that there should be an open set. ${\displaystyle \vartheta }$ ${\displaystyle \Theta }$
• The density function is truly greater than 0. This requirement is necessary in order to be able to logarithmize when defining the score function. Some authors demand real positivity only on a quantity that does not depend on the parameter . This definition allows more leeway in defining the basic set. ${\displaystyle f(x,\vartheta )}$ ${\displaystyle X\times \Theta }$ ${\displaystyle A}$ ${\displaystyle \vartheta }$
• The score function

${\displaystyle S_{\vartheta }(x)={\frac {\partial }{\partial \vartheta }}\ln f(x,\vartheta )}$

exists and is finite. Some authors demand more strongly that the density function should be continuously differentiable. It is only important here that the score function exists and is finite in order to define the Fisher information based on it . ${\displaystyle \vartheta }$

• It is

${\displaystyle 0<I(\vartheta ):=\operatorname {Var} _{\vartheta }(S_{\vartheta })<\infty }$.

So the Fisher information should be really positive and finite. This guarantees the well-definition of the Cramér-Rao inequality , where the Fisher information is in the denominator. ${\displaystyle I(\vartheta )}$

• The exchange relation applies

${\displaystyle \int {\frac {\partial }{\partial \vartheta }}f(x,\vartheta )\mathrm {d} \mu (x)={\frac {\partial }{\partial \vartheta }}\int f(x,\vartheta )\mathrm {d} \mu (x)}$.

It follows that the score function is centered, so it is . This simplifies the Fisher information to . ${\displaystyle \operatorname {E} _{\vartheta }(S_{\vartheta })=0}$ ${\displaystyle I(\vartheta )=\operatorname {E} _{\vartheta }(S_{\vartheta }^{2})}$

• Some authors claim that the stronger commutation relation

${\displaystyle \int {\frac {\partial }{\partial \vartheta }}T(x)f(x,\vartheta )\mathrm {d} \mu (x)={\frac {\partial }{\partial \vartheta }}\int T(x)f(x,\vartheta )\mathrm {d} \mu (x)}$.

holds for all with finite variance. This contains the upper commutation relation as a special case. It is necessary for the proof of the Cramér-Rao inequality. If it is not required within the framework of the regularity of the statistical model, it is required as a separate property in the formulation of the Cramér-Rao inequality (see, for example, regular unbiased estimators ). ${\displaystyle T}$

use

The main task of a regular statistical model is to provide the framework conditions for the proof of the Cramér-Rao inequality . This has far-reaching consequences:

• It provides a lower bound for the variance of estimators and thus criteria for their quality and for uniformly best unbiased estimators .
• It provides the definition of a Cramér-Rao efficient estimator and thus a (non-asymptotic) efficiency concept for estimators .
• The exponential family can be motivated from it.

literature

• 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 .
• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .
• 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. ↑ Czado Schmidt: Mathematical Statistics. 2011, p. 115.
2. ^ Rüschendorf: Mathematical Statistics. 2014, pp. 159–160.
3. ^ Georgii: Stochastics. 2009, pp. 210-211.