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| In [[statistics]], '''Basu's theorem''' states that any [[completeness (statistics)|boundedly complete]] [[sufficient statistic]] is [[statistical independence|independent]] of any [[ancillary statistic]]. This is a 1955 result of [[Debabrata Basu]].<ref>Basu (1955)</ref>
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| It is often used in statistics as a tool to prove independence of two statistics, by first demonstrating one is complete sufficient and the other is ancillary, then appealing to the theorem.{{Citation needed|date=December 2009}} An example of this is to show that the sample mean and sample variance of a normal distribution are independent statistics, which is done in the Examples section below. This property (independent of sample mean and sample variance) characterizes normal distributions.
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| == Statement ==
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| Let ''P<sub>θ</sub>'' be a family of distributions on a [[measurable space]] (''X, Σ''). Then if ''T'' is a boundedly complete sufficient statistic for ''θ'', and ''A'' is ancillary to ''θ'', then ''T'' is independent of ''A''.
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| === Proof ===
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| Let ''P<sub>θ</sub><sup>T</sup>'' and ''P<sub>θ</sub><sup>A</sup>'' be the [[marginal distribution]]s of ''T'' and ''A'' respectively.
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| :<math>P_\theta^A(B) = P_\theta (A^{-1} B) = \int_{T(X)} P_\theta(A^{-1}B | T=t) \ P_\theta^T (dt) \,</math>
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| The ''P<sub>θ</sub><sup>A</sup>'' does not depend on ''θ'' because ''A'' is ancillary. Likewise, ''P<sub>θ</sub>''(·|''T = t'') does not depend on ''θ'' because ''T'' is sufficient. Therefore:
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| :<math> \int_{T(X)} \big[ P(A^{-1}B | T=t) - P^A(B) \big] \ P_\theta^T (dt) = 0 \,</math>
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| Note the integrand (the function inside the intergal) is a function of ''t'' and not ''θ''. Therefore, since ''T'' is boundedly complete:
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| :<math>P(A^{-1}B | T=t) = P^A(B) \quad \text{for all }t\,</math>
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| Therefore, ''A'' is independent of ''T''.
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| ==Example==
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| ===Independence of sample mean and sample variance of a normal distribution===
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| Let ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub> be [[Independent and identically-distributed random variables|independent, identically distributed]] [[Normal distribution|normal]] [[random variable]]s with [[mean]] ''μ'' and [[variance]] ''σ''<sup>2</sup>.
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| Then with respect to the parameter ''μ'', one can show that
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| :<math>\widehat{\mu}=\frac{\sum X_i}{n},\,</math>
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| the sample mean, is a complete sufficient statistic – it is all the information one can derive to estimate ''μ,'' and no more – and
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| :<math>\widehat{\sigma}^2=\frac{\sum \left(X_i-\bar{X}\right)^2}{n-1},\,</math>
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| the sample variance, is an ancillary statistic – its distribution does not depend on ''μ.''
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| Therefore, from Basu's theorem it follows that these statistics are independent.
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| This independence result can also be proven by [[Cochran's theorem]].
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| Further, this property (that the sample mean and sample variance of the normal distribution are independent) ''[[characterization (mathematics)|characterizes]]'' the normal distribution – no other distribution has this property.<ref>{{cite journal
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| |doi=10.2307/2983669
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| |first=R.C. |last=Geary |authorlink=Roy C. Geary
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| |year=1936
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| |title=The Distribution of the "Student's" Ratio for the Non-Normal Samples
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| |journal=Supplement to the Journal of the Royal Statistical Society
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| |volume=3 |issue=2 |pages=178–184
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| |jfm=63.1090.03
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| |jstor=2983669
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| }}</ref>
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| ==Notes==
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| {{Reflist}}
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| {{More footnotes|date=December 2009}}
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| ==References==
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| * {{cite journal
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| |last=Basu |first=D. |authorlink=Debabrata Basu
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| |title=On Statistics Independent of a Complete Sufficient Statistic
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| |journal= [[Sankhya (journal)|Sankhyā]]
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| |volume=15 |issue=4 |year=1955 |pages=377–380
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| |mr=74745
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| |zbl=0068.13401
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| |jstor=25048259
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| }}
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| * Mukhopadhyay, Nitis (2000). ''Probability and Statistical Inference''. Statistics: A Series of Textbooks and Monographs. '''162'''. Florida: CRC Press USA. ISBN 0-8247-0379-0.
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| * {{cite journal |doi=10.2307/2685927 |last=Boos |first=Dennis D.|coauthors=Oliver, Jacqueline M. Hughes |year=Aug |month=1998 |title=Applications of Basu's Theorem |journal=[[The American Statistician]] |volume=52 |issue=3 |pages=218–221 |publisher=[[American Statistical Association]] |location=Boston |mr=1650407 |jstor=2685927}}
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| * {{cite journal|authorlink=Malay Ghosh|title=Basu's Theorem with Applications: A Personalistic Review|
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| first=Malay|last=Ghosh | journal=Sankhyā: the Indian Journal of Statistics, Series A|volume=64|number=3|date=October 2002|pages=509–531 | mr=1985397 |jstor=25051412}}
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| {{Statistics|state=collapsed}}
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| [[Category:Statistical theorems]]
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| [[Category:Statistical inference]]
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| [[Category:Articles containing proofs]]
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