Chapman-Robbins inequality

The Chapman-Robbins inequality is a mathematical statement in estimation theory , a branch of mathematical statistics . For an unbiased estimator, it provides a lower bound for the variance of the estimator and thus also an estimate of its quality. With additional regularity requirements, the Chapman-Robbins inequality also provides a point-wise version of the Cramér-Rao inequality .

The inequality is named after Douglas George Chapman and Herbert Robbins .

formulation

Framework

A statistical model is given . Be strong and be of dominating , that is all there is a density function ${\displaystyle (X,{\mathcal {A}},(P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \vartheta _{0}\in \Theta }$ ${\displaystyle (P_{\vartheta })_{\vartheta \in \Theta }}$ ${\displaystyle P_{\vartheta _{0}}}$ ${\displaystyle \vartheta \in \Theta }$

${\displaystyle f_{\vartheta }:={\frac {\mathrm {d} P_{\vartheta }}{\mathrm {d} P_{\vartheta _{0}}}}}$

from regarding . ${\displaystyle P_{\vartheta }}$ ${\displaystyle P_{\vartheta _{0}}}$

Furthermore, let the set of all square-integrable functions (see Lp-space ) and the set of all unbiased estimators for the parameter function . ${\displaystyle L^{2}(P_{\vartheta _{0}}):=L^{2}(X,{\mathcal {A}},P_{\vartheta _{0}})}$ ${\displaystyle P_{\vartheta _{0}}}$ ${\displaystyle D_{g}}$ ${\displaystyle g}$

Then

${\displaystyle D_{g}(\vartheta _{0}):=D_{g}\cap L^{2}(P_{\vartheta _{0}})}$

the set of all unbiased estimators for with finite variance with respect to and ${\displaystyle g}$ ${\displaystyle P_{\vartheta _{0}}}$

${\displaystyle {\mathcal {F}}_{\vartheta _{0}}:=\{f_{\vartheta }\,|\,f_{\vartheta }\in L^{2}(P_{\vartheta _{0}})\}}$

the set of all density functions with finite variance with respect to . ${\displaystyle P_{\vartheta _{0}}}$

statement

It applies to everyone : ${\displaystyle T\in D_{g}(\vartheta _{0})}$

${\displaystyle \operatorname {Var} _{\vartheta _{0}}(T)\geq \sup _{f_{\vartheta }\in {\mathcal {F}}_{\vartheta _{0}}}{\frac {\left(g(\vartheta )-g(\vartheta _{0})\right)^{2}}{\operatorname {Var} _{\vartheta _{0}}(f_{\vartheta })}}}$

Transition to the Cramér-Rao inequality

The Chapman-Robbins inequality yields a pointwise version of the Cramér-Rao inequality under the following conditions :

• The derivation   in   exists   for all of them . ${\displaystyle x\in X}$ ${\displaystyle {\frac {\partial }{\partial \vartheta }}f_{\vartheta }(x)}$ ${\displaystyle \vartheta _{0}}$
• The quotient     converges for in versus   . ${\displaystyle {\frac {f_{\vartheta }-1}{\vartheta -\vartheta _{0}}}}$ ${\displaystyle \vartheta \to \vartheta _{0}}$ ${\displaystyle L^{2}(P_{\vartheta _{0}})}$ ${\displaystyle {\frac {\partial }{\partial \vartheta }}f_{\vartheta }|_{\vartheta =\vartheta _{0}}}$

• The parameter function   is differentiable in. ${\displaystyle g\colon \Theta \rightarrow \mathbb {R} }$ ${\displaystyle \vartheta _{0}}$

It follows from these assumptions

${\displaystyle \lim _{\vartheta \to \vartheta _{0}}{\frac {g(\vartheta )-g(\vartheta _{0})}{\vartheta -\vartheta _{0}}}=g'(\vartheta _{0})}$

such as

${\displaystyle \lim _{\vartheta \to \vartheta _{0}}{\frac {\operatorname {Var} _{\vartheta _{0}}(f_{\vartheta })}{(\vartheta -\vartheta _{0})^{2}}}=I(\vartheta _{0})}$,

where the Fisher information is in the point . ${\displaystyle I(\vartheta _{0})}$ ${\displaystyle \vartheta _{0}}$

From the Chapman-Robbins inequality it follows that

${\displaystyle \operatorname {Var} _{\vartheta _{0}}(T)\geq {\frac {(g'(\vartheta _{0}))^{2}}{I(\vartheta _{0})}}}$,

the Cramér-Rao inequality in the point . ${\displaystyle \vartheta _{0}}$

literature

• Douglas G. Chapman & Herbert Robbins: Minimum Variance Estimation Without Regularity Assumptions . In: Annals of Mathematical Statistics . tape 22 , no. 4 , 1951, pp. 581-586 , doi : 10.1214 / aoms / 1177729548 , JSTOR : 2236927 .
• Ludger Rüschendorf: Mathematical Statistics . Springer Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-41996-6 , doi : 10.1007 / 978-3-642-41997-3 .