Van Trees Inequality

The Van Trees inequality is a central inequality from Bayesian estimation theory , a branch of mathematical statistics . Similar to the Cramér-Rao inequality from frequentist statistics, it provides an estimate of the variance for point estimators and thus a possibility to compare different estimators with one another. In contrast to the Cramér-Rao inequality, the inequality dispenses with the assumption of fairness to expectations , but is therefore somewhat weaker for fair-expectation estimators. For large sample sizes, however, the Van-Trees limit differs only slightly from the Cramér-Rao limit.

The inequality is named after Harry L. van Trees , who first established the inequality in 1968.

The inequality

Framework

The one-parameter statistical model with dominant measure is given . We denote the density of respect . ${\displaystyle ({\mathcal {X}},{\mathcal {A}},{\mathcal {(}}P_{\vartheta })_{\vartheta \in \Theta })}$ ${\displaystyle \mu }$ ${\displaystyle f(x|\vartheta )}$ ${\displaystyle P_{\vartheta }}$ ${\displaystyle \mu }$

Above the parameter space there is also a probability measure with a density related to the Lebesgue measure . This means that our model is a Bayesian statistical model. ${\displaystyle (\Theta ,{\mathcal {B}}(\Theta ))}$ ${\displaystyle \pi }$ ${\displaystyle \lambda (\vartheta )}$

The following regularity conditions still apply:

• ${\displaystyle \lambda (\vartheta )}$and are both ( almost certainly) absolutely continuous functions . ${\displaystyle f(x|\cdot )}$ ${\displaystyle \mu }$
• ${\displaystyle \Theta }$is a closed interval in ${\displaystyle \mathbb {R} }$
• The function converges to at the edges of the definition interval . ${\displaystyle \lambda }$ ${\displaystyle \Theta }$ ${\displaystyle 0}$

formulation

Let be an estimator for the parameter and a random variable that is how distributed. We also assume that is true. ${\displaystyle T=T(X)}$ ${\displaystyle \vartheta }$ ${\displaystyle X}$ ${\displaystyle P_{\vartheta }}$ ${\displaystyle \operatorname {E} _{\vartheta }[{\frac {\partial }{\partial \vartheta }}\ln f(X,\vartheta )]=0}$

Be further

${\displaystyle I(\vartheta ):=\operatorname {E} _{\vartheta }[S_{\vartheta }^{2}]=\operatorname {E} _{\vartheta }[({\frac {\partial }{\partial \vartheta }}\ln f(X,\vartheta ))^{2}]}$

${\displaystyle I(\lambda ):=\operatorname {E} [({\frac {\partial }{\partial \vartheta }}\ln \lambda (\vartheta ))^{2}]}$

the Fisher information for or for a parameter in . It is the (ordinary) expectation value regarding the probability measure and the expected value with respect to the joint probability of measurement and a -distributed random variable . ${\displaystyle \vartheta }$ ${\displaystyle \lambda }$ ${\displaystyle \operatorname {E} _{\vartheta }}$ ${\displaystyle P_{\vartheta }}$ ${\displaystyle \operatorname {E} }$ ${\displaystyle X}$ ${\displaystyle \pi }$ ${\displaystyle Y}$

The van Trees inequality now states:

${\displaystyle \operatorname {E} [(T(X)-\vartheta )^{2}]\geq {\frac {1}{\operatorname {E} [I(\vartheta )]+I(\lambda )}}}$

Applications

The inequality can be used to show that there are no super-efficient estimators in one- or two-parameter models . Here, a super-efficient estimator is meant to be an estimator (not true to expectations) that falls below the Cramér-Rao inequality.

literature

• Richard D. Gill, Boris Y. Levit: Applications of the van Trees inequality: a Bayesian Cramér-Rao bound. In: Bernoulli. 1, no. 1-2, 1995, pp. 59-79. (projecteuclid.org)