Basu's theorem

Let $(P\_\backslash theta;\; \backslash theta\; \backslash in\; \backslash Theta)$ be a family of distributions on a measurable space $(X,\; \backslash mathcal\{A\})$ and a statistic $T$ maps from $(X,\; \backslash mathcal\{A\})$ to some measurable space $(Y,\; \backslash mathcal\{B\})$. If $T$ is a boundedly complete sufficient statistic for $\backslash theta$, and $A$ is ancillary to $\backslash theta$, then conditional on $\backslash theta$, $T$ is independent of $A$. That is, $T\backslash perp\backslash !\backslash !\backslash !\backslash perp\; A|\backslash theta$.

Let $P\_\backslash theta^T$ and $P\_\backslash theta^A$ be the marginal distributions of $T$ and $A$ respectively.

Denote by $A^\{-1\}(B)$ the preimage of a set $B$ under the map $A$. For any measurable set $B\; \backslash in\; \backslash mathcal\{B\}$ we have

:$P\_\backslash theta^A(B)\; =\; P\_\backslash theta\; (A^\{-1\}(B))\; =\; \backslash int\_\{Y\}\; P\_\backslash theta(A^\{-1\}(B)\; \backslash mid\; T=t)\; \backslash \; P\_\backslash theta^T\; (dt).$

The distribution $P\_\backslash theta^A$ does not depend on $\backslash theta$ because $A$ is ancillary. Likewise, $P\_\backslash theta(\backslash cdot\; \backslash mid\; T\; =\; t)$ does not depend on $\backslash theta$ because $T$ is sufficient. Therefore

:$\backslash int\_Y\; \backslash big[\; P(A^\{-1\}(B)\; \backslash mid\; T=t)\; -\; P^A(B)\; \backslash big]\; \backslash \; P\_\backslash theta^T\; (dt)\; =\; 0.$

Note the integrand (the function inside the integral) is a function of $t$ and not $\backslash theta$. Therefore, since $T$ is boundedly complete the function

:$g(t)\; =\; P(A^\{-1\}(B)\; \backslash mid\; T=t)\; -\; P^A(B)$

is zero for $P\_\backslash theta^T$ almost all values of $t$ and thus

:$P(A^\{-1\}(B)\; \backslash mid\; T=t)\; =\; P^A(B)$

for almost all $t$. Therefore, $A$ is independent of $T$.

Let X₁, X₂, ..., X_n be independent, identically distributed normal random variables with mean μ and variance σ².

Then with respect to the parameter μ, one can show that

:$\backslash widehat\{\backslash mu\}=\backslash frac\{\backslash sum\; X\_i\}\{n\},$

the sample mean, is a complete and sufficient statistic – it is all the information one can derive to estimate μ, and no more – and

:$\backslash widehat\{\backslash sigma\}^2=\backslash frac\{\backslash sum\; \backslash left(X\_i-\backslash bar\{X\}\backslash right)^2\}\{n-1\},$

the sample variance, is an ancillary statistic – its distribution does not depend on μ.

Therefore, from Basu's theorem it follows that these statistics are independent conditional on $\backslash mu$, conditional on $\backslash sigma^2$.

This independence result can also be proven by Cochran's theorem.

Further, this property (that the sample mean and sample variance of the normal distribution are independent) characterizes the normal distribution – no other distribution has this property.{{cite journal

|doi=10.2307/2983669

|first=R.C. |last=Geary |authorlink=Roy C. Geary

|year=1936

|title=The Distribution of "Student's" Ratio for Non-Normal Samples

|journal=Supplement to the Journal of the Royal Statistical Society

|volume=3 |issue=2 |pages=178–184

|jfm=63.1090.03

|jstor=2983669

}}

{{Reflist}}

{{More footnotes|date=December 2009}}

• {{cite journal

|last=Basu |first=D. |authorlink=Debabrata Basu

|title=On Statistics Independent of a Complete Sufficient Statistic

|journal= Sankhyā

|volume=15 |issue=4 |year=1955 |pages=377–380

|mr=74745

|zbl=0068.13401

|jstor=25048259

}}

• Mukhopadhyay, Nitis (2000). Probability and Statistical Inference. Statistics: A Series of Textbooks and Monographs. 162. Florida: CRC Press USA. {{ISBN|0-8247-0379-0}}.
• {{cite journal |doi=10.2307/2685927 |last=Boos |first=Dennis D.|author2=Oliver, Jacqueline M. Hughes |date=Aug 1998 |title=Applications of Basu's Theorem |journal=The American Statistician |volume=52 |issue=3 |pages=218–221 |mr=1650407 |jstor=2685927|url=http://www.lib.ncsu.edu/resolver/1840.4/229 }}
• {{cite journal|authorlink=Malay Ghosh|title=Basu's Theorem with Applications: A Personalistic Review|

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}}

{{Statistics|state=collapsed}}

Category:Theorems in statistics

Category:Independence (probability theory)

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