Evaluating unbiased estimators by their variance clearly corresponds to evaluating estimators using a squared error loss function. In practice, however, the improvement is often enormous. If the loss function is twice-differentiable, as in the case for mean-squared-error, then we have the sharper inequality[4]The improved estimator is unbiased if and only if the original estimator is unbiased, as may be seen at once by using the law of total expectation. , it estimates this probability to be 1 if no phone calls arrived in the first minute and zero otherwise. The transformed estimator is called the Rao–Blackwell estimator.
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e. Assume the distribution of X depends on a parameter θ. [1][2][3]One case of Rao–Blackwell theorem states:In other words,The essential tools of the proof besides the definition above are the law of total expectation and the fact that for any random variable Y, E(Y2) cannot be less than [E(Y)]2. 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. Then W(X) is unbiased and σ U 2 ≥ σ W 2, where σ W 2 is the variance of W. Rao and David Blackwell links the notions of sufficient statistics and unbiased estimation.
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LetThat is, W(X) is the conditional expected value of U(X) given S(X). However, the following unbiased estimator can be shown to have lower variance:And in fact, it could be even further improved when using the following estimator:. A well known extension of the RB. That inequality is a case of Jensen’s inequality, although it may also be shown to follow instantly from the frequently mentioned fact thatMore precisely, the mean square error of the Rao-Blackwell estimator has the following decomposition[4]The more general version of the Rao–Blackwell theorem speaks of the “expected loss” or risk function:where the “loss function” L may be any convex function. Let X, a random vector represent the data.
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, the conditional distribution of the data X1, . 1007/978-3-642-04898-2_479Published: 02 December 2014
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04897-5
Online ISBN: 978-3-642-04898-2eBook Packages: Mathematics and StatisticsReference Module Computer Science and EngineeringIn statistics, the Rao–Blackwell check my blog 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. , Xn, depends on λ only through this sum. .
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Using it to improve the already improved estimator does not obtain a further improvement, but merely returns as its output the same improved estimator. An extremely crude estimator of the desired probability isi. A statistic S(X) is said to be sufficient if the conditional distribution of X given S does not depend on θ. The RB Theorem, which is constructive says the following:Let U(X) be any unbiased estimator of g(θ) and let σ U 2 be the variance of U.
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Therefore, we find the Rao–Blackwell estimatorAfter doing some algebra we haveSince the average number of calls arriving during the first n minutes is nλ, one might not be surprised if this estimator has a fairly high probability (if n is big) of being close toSo δ1 is clearly a very much improved estimator of that last quantity. In fact, since Sn is complete and δ0 is unbiased, δ1 is the unique minimum variance unbiased estimator by the Lehmann–Scheffé theorem. , Xn of phone calls that arrived during n successive one-minute periods are observed. .