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 .
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.
The following regularity conditions still apply:
- and are both ( almost certainly) absolutely continuous functions .
- is a closed interval in
- The function converges to at the edges of the definition interval .
formulation
Let be an estimator for the parameter and a random variable that is how distributed. We also assume that is true.
Be further
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 .
The van Trees inequality now states:
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)