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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. 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 theorem is named after [[Calyampudi Radhakrishna Rao|C.R. Rao]] and [[David Blackwell]]. The process of transforming an estimator using the Rao–Blackwell theorem can be referred to as '''Rao–Blackwellization'''. The transformed [[estimator]] is called the '''Rao–Blackwell estimator'''.<ref name=LS1/><ref name=LS2/><ref name=LS3/> ==Definitions== *An [[estimator]] δ(''X'') is an [[observable variable|''observable'' random variable]] (i.e. a [[statistic]]) used for estimating some ''[[unobservable]]'' quantity. 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. The average height of those 40—the "sample average"—may be used as an estimator of the unobservable "population average". *A [[sufficiency (statistics)|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. 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. In the most frequently cited examples, the "unobservable" quantities are parameters that parametrize a known family of [[probability distribution]]s according to which the data are distributed. ::In other words, a [[sufficiency (statistics)|sufficient statistic]] ''T(X)'' for a parameter θ is a [[statistic]] such that the conditional probability of the data ''X'', given ''T''(''X''), does not depend on the parameter θ. *A '''Rao–Blackwell estimator''' δ<sub>1</sub>(''X'') of an unobservable quantity θ is the [[Conditional expectation|conditional expected value]] E(δ(''X'') | ''T''(''X'')) of some estimator δ(''X'') given a sufficient statistic ''T''(''X''). Call δ(''X'') the '''"original estimator"''' and δ<sub>1</sub>(''X'') the '''"improved estimator"'''. It is important that the improved estimator be ''observable'', i.e. that it does not depend on θ. Generally, the conditional expected value of one function of these data given another function of these data ''does'' depend on θ, but the very definition of sufficiency given above entails that this one does not. *The ''[[mean squared error]]'' of an estimator is the expected value of the square of its deviation from the unobservable quantity being estimated of θ. ==The theorem== ===Mean-squared-error version=== One case of Rao–Blackwell theorem states: :The mean squared error of the Rao–Blackwell estimator does not exceed that of the original estimator. In other words, :<math>\operatorname{E}((\delta_1(X)-\theta)^2)\leq \operatorname{E}((\delta(X)-\theta)^2).</math> 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(''Y''<sup>2</sup>) cannot be less than [E(''Y'')]<sup>2</sup>. That inequality is a case of [[Jensen's inequality]], although it may also be shown to follow instantly from the frequently mentioned fact that :<math> 0 \leq \operatorname{Var}(Y) = \operatorname{E}((Y-\operatorname{E}(Y))^2) = \operatorname{E}(Y^2)-(\operatorname{E}(Y))^2.</math> More precisely, the mean square error of the Rao-Blackwell estimator has the following decomposition<ref name=LiaoBerg2018>{{cite journal|title= Sharpening Jensen's Inequality |author1=J. G. Liao |author2=A. Berg | date = 22 June 2018 | journal = The American Statistician | volume = 73 | issue = 3 | pages = 278–281 | doi = 10.1080/00031305.2017.1419145| arxiv = 1707.08644 | s2cid = 88515366 }}</ref> :<math> \operatorname{E}[(\delta_1(X)-\theta)^2]=\operatorname{E}[(\delta(X)-\theta)^2]-\operatorname{E}[\operatorname{Var}(\delta(X)\mid T(X))]</math> Since <math>\operatorname{E}[\operatorname{Var}(\delta(X)\mid T(X))]\ge 0</math>, the Rao-Blackwell theorem immediately follows. ===Convex loss generalization=== The more general version of the Rao–Blackwell theorem speaks of the "expected loss" or [[risk function]]: :<math>\operatorname{E}(L(\delta_1(X)))\leq \operatorname{E}(L(\delta(X)))</math> where the "loss function" ''L'' may be any [[convex function]]. If the loss function is twice-differentiable, as in the case for mean-squared-error, then we have the sharper inequality<ref name=LiaoBerg2018/> :<math>\operatorname{E}(L(\delta(X)))-\operatorname{E}(L(\delta_1(X)))\ge \frac{1}{2}\operatorname{E}_T\left[\inf_x L''(x)\operatorname{Var}(\delta(X)\mid T)\right].</math> ==Properties== The improved estimator is [[bias of an estimator|unbiased]] if and only if the original estimator is unbiased, as may be seen at once by using the [[law of total expectation]]. The theorem holds regardless of whether biased or unbiased estimators are used. The theorem seems very weak: it says only that the Rao–Blackwell estimator is no worse than the original estimator. In practice, however, the improvement is often enormous.<ref name=LS4/> ==Example== Phone calls arrive at a switchboard according to a [[Poisson process]] at an average rate of λ per minute. This rate is not observable, but the numbers ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> of phone calls that arrived during ''n'' successive one-minute periods are observed. It is desired to estimate the probability ''e''<sup>&minus;λ</sup> that the next one-minute period passes with no phone calls. An ''extremely'' crude estimator of the desired probability is :<math>\delta_0=\left\{\begin{matrix}1 & \text{if}\ X_1=0, \\ 0 & \text{otherwise,}\end{matrix}\right.</math> i.e., it estimates this probability to be 1 if no phone calls arrived in the first minute and zero otherwise. Despite the apparent limitations of this estimator, the result given by its Rao–Blackwellization is a very good estimator. The sum :<math> S_n = \sum_{i=1}^n X_{i} = X_1+\cdots+X_n</math> can be readily shown to be a sufficient statistic for λ, i.e., the ''conditional'' distribution of the data ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub>, depends on λ only through this sum. Therefore, we find the Rao–Blackwell estimator :<math>\delta_1=\operatorname{E}(\delta_0\mid S_n=s_n).</math> After doing some algebra we have :<math>\begin{align} \delta_1 &= \operatorname{E} \left (\mathbf{1}_{\{X_1=0\}} \Bigg| \sum_{i=1}^n X_{i} = s_n \right ) \\ &= P \left (X_{1}=0 \Bigg| \sum_{i=1}^n X_{i} = s_n \right ) \\ &= P \left (X_{1}=0, \sum_{i=2}^n X_{i} = s_n \right ) \times P \left (\sum_{i=1}^n X_{i} = s_n \right )^{-1} \\ &= e^{-\lambda}\frac{\left((n-1)\lambda\right)^{s_n}e^{-(n-1)\lambda}}{s_n!} \times \left (\frac{(n\lambda)^{s_n}e^{-n\lambda}}{s_n!} \right )^{-1} \\ &= \frac{\left((n-1)\lambda\right)^{s_n}e^{-n\lambda}}{s_n!} \times \frac{s_n!}{(n\lambda)^{s_n}e^{-n\lambda}} \\ &= \left(1-\frac{1}{n}\right)^{s_n} \end{align}</math> Since the average number of calls arriving during the first ''n'' minutes is ''n''λ, one might not be surprised if this estimator has a fairly high probability (if ''n'' is big) of being close to :<math>\left(1-{1 \over n}\right)^{n\lambda}\approx e^{-\lambda}.</math> So δ<sub>1</sub> is clearly a very much improved estimator of that last quantity. In fact, since ''S''<sub>''n''</sub> is [[completeness (statistics)|complete]] and δ<sub>0</sub> is unbiased, δ<sub>1</sub> is the unique minimum variance unbiased estimator by the [[Lehmann–Scheffé theorem]]. ==Idempotence== Rao–Blackwellization is an [[idempotent]] operation. Using it to improve the already improved estimator does not obtain a further improvement, but merely returns as its output the same improved estimator. ==Completeness and Lehmann&ndash;Scheffé minimum variance== If the conditioning statistic is both [[Completeness (statistics)|complete]] and [[Sufficient statistic|sufficient]], and the starting estimator is unbiased, then the Rao&ndash;Blackwell estimator is the unique "[[minimum-variance unbiased estimator|best unbiased estimator]]": see [[Lehmann–Scheffé theorem]]. An example of an improvable Rao–Blackwell improvement, when using a minimal sufficient statistic that is '''not complete''', was provided by Galili and Meilijson in 2016.<ref>{{cite journal|title= An Example of an Improvable Rao–Blackwell Improvement, Inefficient Maximum Likelihood Estimator, and Unbiased Generalized Bayes Estimator |author1=Tal Galili |author2=Isaac Meilijson | date = 31 Mar 2016 | journal = The American Statistician | volume = 70 | issue = 1 | pages = 108–113 | doi=10.1080/00031305.2015.1100683| pmc = 4960505 | pmid=27499547}}</ref> Let <math>X_1, \ldots, X_n</math> be a random sample from a scale-uniform distribution <math>X \sim U \left( (1-k) \theta, (1+k) \theta \right),</math> with unknown mean <math>E[X]=\theta</math> and known design parameter <math>k \in (0,1)</math>. In the search for "best" possible unbiased estimators for <math>\theta,</math> it is natural to consider <math>X_1</math> as an initial (crude) unbiased estimator for <math>\theta</math> and then try to improve it. Since <math>X_1</math> is not a function of <math>T = \left( X_{(1)}, X_{(n)} \right)</math>, the minimal sufficient statistic for <math>\theta</math> (where <math>X_{(1)} = \min( X_i )</math> and <math>X_{(n)} = \max( X_i )</math>), it may be improved using the Rao–Blackwell theorem as follows: :<math>\hat{\theta}_{RB}=E_{\theta} \left [X_1|X_{(1)}, X_{(n)} \right ]=\frac{X_{(1)}+X_{(n)}}{2}.</math> However, the following unbiased estimator can be shown to have lower variance: :<math>\hat{\theta}_{LV} = \frac{1}{2 \left (k^2 \frac{n-1}{n+1}+1\right )} \left[ (1-k){{X}_{(1)}}+(1+k){{X}_{(n)}} \right].</math> And in fact, it could be even further improved when using the following estimator: :<math>\hat{\theta}_{BAYES} =\frac{n+1}{n} \left[ 1-\frac{\frac{\left( \frac{{{X}_{(1)}}}{1-k} \right)}{\left( \frac{{{X}_{(n)}}}{1+k} \right)}-1}{{{\left[ \frac{\left( \frac{{{X}_{(1)}}}{1-k} \right)}{\left( \frac{{{X}_{(n)}}}{1+k} \right)} \right]}^{n+1}}-1} \right] \frac{X_{(n)}}{1+k}</math> The model is a [[Scale parameter|scale model]]. Optimal [[Equivariant Estimator|equivariant estimators]] can then be derived for [[Loss function|loss functions]] that are [[Invariant estimator|invariant]].<ref>{{Cite journal|last=Taraldsen|first=Gunnar|date=2020|title=Micha Mandel (2020), "The Scaled Uniform Model Revisited," The American Statistician, 74:1, 98–100: Comment|url=https://doi.org/10.1080/00031305.2020.1769727|journal=The American Statistician|volume=74|issue=3|pages=315|doi=10.1080/00031305.2020.1769727|s2cid=219493070 |issn=0003-1305}}</ref> ==See also== * [[Basu's theorem]] &mdash; Another result on complete sufficient and [[ancillary statistic]]s ==References== {{reflist|refs= * <ref name=LS1>{{Cite journal |last=Blackwell |first=D. |author-link=David Blackwell |title=Conditional expectation and unbiased sequential estimation |journal=[[Annals of Mathematical Statistics]] |volume=18 |issue=1 |pages=105–110 |year=1947 |doi=10.1214/aoms/1177730497 |mr=19903 | zbl = 0033.07603 |doi-access=free }} </ref> <ref name=LS2>{{Cite journal |last=Kolmogorov |first=A. N. |author-link=Andrey Kolmogorov |title=Unbiased estimates |journal=Izvestiya Akad. Nauk SSSR. Ser. Mat. |year=1950 |volume=14 |pages=303–326 |mr=36479 }} </ref> <ref name=LS3>{{cite journal |last=Rao |first=C. Radhakrishna |title=Information and accuracy attainable in the estimation of statistical parameters |journal=Bulletin of the Calcutta Mathematical Society |volume=37 |issue=3 |year=1945 |pages=81–91 }} </ref> <ref name=LS4>{{cite web |url=https://statmodeling.stat.columbia.edu/2020/01/29/rao-blackwellization-and-discrete-parameters-in-stan/ |title=Rao-Blackwellization and discrete parameters in Stan |last=Carpenter |first=Bob |date=January 20, 2020 |website=Statistical Modeling, Causal Inference, and Social Science |access-date=September 13, 2021 |quote="The Rao-Blackwell theorem states that the marginalization approach has variance less than or equal to the direct approach. In practice, this difference can be enormous."}} </ref> }} ==External links== * {{SpringerEOM |title=Rao–Blackwell–Kolmogorov theorem |id=R/r077550 |first=M.S. |last=Nikulin}} {{Statistics|inference|collapsed}} {{DEFAULTSORT:Rao-Blackwell Theorem}} [[Category:Theorems in statistics]] [[Category:Estimation theory]] </textarea><div class="templatesUsed"><div class="mw-templatesUsedExplanation"><p><span id="templatesused">Pages transcluded onto the current version of this page<span class="posteditwindowhelplinks"> (<a href="/wiki/Help:Transclusion" title="Help:Transclusion">help</a>)</span>:</span> </p></div><ul> <li><a href="/wiki/Template:Ambox" title="Template:Ambox">Template:Ambox</a> (<a href="/w/index.php?title=Template:Ambox&action=edit" title="Template:Ambox">view source</a>) (template editor protected)</li><li><a href="/wiki/Template:Citation" title="Template:Citation">Template:Citation</a> (<a href="/w/index.php?title=Template:Citation&action=edit" title="Template:Citation">view 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