What does the Rao-Blackwell theorem imply?

The Rao–Blackwell theorem states that if g(X) is any kind of estimator of a parameter θ, then the conditional expectation of g(X) given T(X), where T is a sufficient statistic, is typically a better estimator of θ, and is never worse.

Are sufficient statistics unbiased?

Any estimator of the form U = h(T) of a complete and sufficient statistic T is the unique unbiased estimator based on T of its expectation. Hence, if T is complete and sufficient, U = h(T) is the MVUE of its expectation.

What is a Blackwell experiment?

Blackwell one experiment is more informative than another if, for every set of feasible. actions, it yields a richer menu of experiment-wise expected payoffs (i.e., expected-loss. vectors) each of which corresponds to an action taken contingent on the experimental. observations.

What is Rao Blackwellized particle filter?

Rao-Blackwellized Particle Filters (RBPF) incorporates the Rao–Blackwell theorem to improve the sampling done in a particle filter by marginalizing out some variables. Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks. Improved Techniques for Grid Mapping With Rao-Blackwellized Particle Filters.

What is efficiency of an estimator?

For an unbiased estimator, efficiency indicates how much its precision is lower than the theoretical limit of precision provided by the Cramer-Rao inequality. A measure of efficiency is the ratio of the theoretically minimal variance to the actual variance of the estimator.

How do you show that a statistic is complete?

A statistic T is called complete if Eg(T) = 0 for all θ and some function g implies that P(g(T) = 0;θ) = 1 for all θ.

What is efficiency in theory of estimation?

Since the expected value of an unbiased estimator is equal to the parameter value, . Therefore, as the. term drops out from being equal to 0. If an unbiased estimator of a parameter θ attains. for all values of the parameter, then the estimator is called efficient.

How do you find the efficiency of an estimator?

For an unbiased estimator, efficiency indicates how much its precision is lower than the theoretical limit of precision provided by the Cramer-Rao inequality. A measure of efficiency is the ratio of the theoretically minimal variance to the actual variance of the estimator. This measure falls between 0 and 1.

What does it mean for a statistic to be complete?

A complete statistic is formally defined as: Suppose a statistic T(Y) has a pdf or pmf f(t|θ). Then T(Y) is a complete statistic if Eθ[g(T(Y))] = 0 for all θ ∈ Θ implies that Pθ[g(T(Y)) = 0] = 1 for all θ ∈ Θ (Olive, 2014).

How do you show ancillary statistics?

A statistics is ancillary if its distribution does not depend on θ. More precisely, a statistic S(X) is ancillary for Θ it its distribution is the same for all θ ∈ Θ. That is, Pθ(S(X) ∈ A) is constant for θ ∈ Θ for any set A. (Xi − ¯X)2.