Call δ( X) the "original estimator" and δ 1( X) the "improved estimator". A Rao–Blackwell estimator δ 1( X) of an unobservable quantity θ is the conditional expected value E(δ( X) | T( X)) of some estimator δ( X) given a sufficient statistic T( X).In other words, a sufficient statistic T(X) for a parameter θ is a statistic such that the conditional distribution of the data X, given T( X), does not depend on the parameter θ. In the most frequently cited examples, the "unobservable" quantities are parameters that parametrize a known family of probability distributions according to which the data are distributed. It is defined as an observable random variable such that the conditional probability distribution of all observable data X given T( X) does not depend on the unobservable parameter θ, such as the mean or standard deviation of the whole population from which the data X was taken. A sufficient statistic T( X) is a statistic calculated from data X to estimate some parameter θ for which no other statistic which can be calculated from data X provides any additional information about θ.
The average height of those 40-the "sample average"-may be used as an estimator of the unobservable "population average". For example, one may be unable to observe the average height of all male students at the University of X, but one may observe the heights of a random sample of 40 of them. a statistic) used for estimating some unobservable quantity. An estimator δ( X) is an observable random variable (i.e.6 Completeness and Lehmann–Scheffé minimum variance.The transformed estimator is called the Rao–Blackwell estimator. The process of transforming an estimator using the Rao–Blackwell theorem is sometimes called Rao–Blackwellization. The theorem is named after Calyampudi Radhakrishna Rao and David Blackwell. Sometimes one can very easily construct a very crude estimator g( X), and then evaluate that conditional expected value to get an estimator that is in various senses optimal. 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. In statistics, the Rao–Blackwell theorem, sometimes referred to as the Rao–Blackwell–Kolmogorov theorem, is a result which characterizes the transformation of an arbitrarily crude estimator into an estimator that is optimal by the mean-squared-error criterion or any of a variety of similar criteria. JSTOR ( May 2014) ( Learn how and when to remove this template message).Unsourced material may be challenged and removed.įind sources: "Rao–Blackwell theorem" – news Please help improve this article by adding citations to reliable sources. This article needs additional citations for verification.
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