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| In [[statistics]], '''Cochran's theorem''', devised by [[William G. Cochran]],<ref name="Cochran">{{cite journal|last=Cochran|first=W. G.|authorlink=William Gemmell Cochran|title=The distribution of quadratic forms in a normal system, with applications to the analysis of covariance|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|date=April 1934|volume=30|issue=2|pages=178–191|doi=10.1017/S0305004100016595}}</ref> is a [[theorem]] used to justify results relating to the [[probability distribution]]s of statistics that are used in the [[analysis of variance]].<ref>{{cite book |author= Bapat, R. B.|title=Linear Algebra and Linear Models|edition=Second|publisher= Springer |year=2000|isbn=978-0-387-98871-9}}</ref>
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| == Statement == | |
| Suppose ''U''<sub>1</sub>, ..., ''U''<sub>''n''</sub> are [[statistical independence|independent]] standard [[normal distribution|normally distributed]] [[random variable]]s, and an identity of the form
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| :<math>
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| \sum_{i=1}^n U_i^2=Q_1+\cdots + Q_k
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| </math>
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| can be written, where each ''Q''<sub>''i''</sub> is a sum of squares of linear combinations of the ''U''s. Further suppose that
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| :<math>
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| r_1+\cdots +r_k=n
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| </math>
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| where ''r''<sub>''i''</sub> is the [[rank (linear algebra)|rank]] of ''Q''<sub>''i''</sub>. Cochran's theorem states that the ''Q''<sub>''i''</sub> are independent, and each ''Q''<sub>''i''</sub> has a [[chi-squared distribution]] with ''r''<sub>''i''</sub> [[degrees of freedom (statistics)|degrees of freedom]].<ref name="Cochran"/> Here the rank of ''Q''<sub>''i''</sub> should be interpreted as meaning the rank of the matrix ''B''<sup>(''i'')</sup>, with elements ''B''<sub>''j,k''</sub><sup>(''i'')</sup>, in the representation of ''Q''<sub>''i''</sub> as a [[quadratic form]]:
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| :<math>Q_i=\sum_{j=1}^n\sum_{k=1}^n U_j B_{j,k}^{(i)} U_k .</math>
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| Less formally, it is the number of linear combinations included in the sum of squares defining ''Q''<sub>''i''</sub>, provided that these linear combinations are linearly independent.
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| <!--
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| Cochran's theorem is the converse of [[Fisher's theorem]]. -->
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| == Examples ==
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| === Sample mean and sample variance ===
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| If ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are independent normally distributed random variables with mean μ and standard deviation σ
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| then
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| :<math>U_i = \frac{X_i-\mu}{\sigma}</math>
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| is [[standard normal]] for each ''i''. It is possible to write | |
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| :<math>
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| \sum_{i=1}^n U_i^2=\sum_{i=1}^n\left(\frac{X_i-\overline{X}}{\sigma}\right)^2
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| + n\left(\frac{\overline{X}-\mu}{\sigma}\right)^2
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| </math>
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| (here <math>\overline{X}</math> is the [[Arithmetic mean|sample mean]]). To see this identity, multiply throughout by <math>\sigma^2</math> and note that
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| :<math>
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| \sum(X_i-\mu)^2=
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| \sum(X_i-\overline{X}+\overline{X}-\mu)^2
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| </math>
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| and expand to give
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| :<math>
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| \sum(X_i-\mu)^2=
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| \sum(X_i-\overline{X})^2+\sum(\overline{X}-\mu)^2+
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| 2\sum(X_i-\overline{X})(\overline{X}-\mu).
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| </math> | |
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| The third term is zero because it is equal to a constant times
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| :<math>\sum(\overline{X}-X_i)=0,</math>
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| and the second term has just ''n'' identical terms added together. Thus
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| :<math>
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| \sum(X_i-\mu)^2=
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| \sum(X_i-\overline{X})^2+n(\overline{X}-\mu)^2 ,
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| </math>
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| and hence
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| :<math> | |
| \sum\left(\frac{X_i-\mu}{\sigma}\right)^2=
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| \sum\left(\frac{X_i-\overline{X}}{\sigma}\right)^2
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| +n\left(\frac{\overline{X}-\mu}{\sigma}\right)^2
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| =Q_1+Q_2.
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| </math>
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| Now the rank of ''Q''<sub>2</sub> is just 1 (it is the square of just one linear combination of the standard normal variables). The rank of ''Q''<sub>1</sub> can be shown to be ''n'' − 1, and thus the conditions for Cochran's theorem are met.
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| Cochran's theorem then states that ''Q''<sub>1</sub> and ''Q''<sub>2</sub> are independent, with chi-squared distributions with ''n'' − 1 and 1 degree of freedom respectively. This shows that the sample mean and [[sample variance]] are independent. This can also be shown by [[Basu's theorem]], and in fact this property ''characterizes'' the normal distribution – for no other distribution are the sample mean and sample variance independent.<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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| ===Distributions===
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| The result for the distributions is written symbolically as
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| :<math>
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| n(\overline{X}-\mu)^2\sim \sigma^2 \chi^2_1,
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| </math>
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| :<math>
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| \sum\left(X_i-\overline{X}\right)^2 \sim \sigma^2 \chi^2_{n-1}.
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| </math>
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| Both these random variables are proportional to the true but unknown variance σ<sup>2</sup>. Thus their ratio is does not depend on σ<sup>2</sup> and, because they are statistically independent, the distribution of their ratio is given by
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| :<math>
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| \frac{n\left(\overline{X}-\mu\right)^2}
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| {\frac{1}{n-1}\sum\left(X_i-\overline{X}\right)^2}\sim \frac{\chi^2_1}{\frac{1}{n-1}\chi^2_{n-1}}
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| \sim F_{1,n-1}
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| </math>
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| where ''F''<sub>1,''n'' − 1</sub> is the [[F-distribution]] with 1 and ''n'' − 1 degrees of freedom (see also [[Student's t-distribution]]). The final step here is effectively the definition of a random variable having the F-distribution.
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| === Estimation of variance ===
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| To estimate the variance σ<sup>2</sup>, one estimator that is sometimes used is the [[maximum likelihood]] estimator of the variance of a normal distribution
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| :<math>
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| \widehat{\sigma}^2=
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| \frac{1}{n}\sum\left(
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| X_i-\overline{X}\right)^2. </math>
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| Cochran's theorem shows that
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| :<math> | |
| \frac{n\widehat{\sigma}^2}{\sigma^2}\sim\chi^2_{n-1}
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| </math>
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| and the properties of the chi-squared distribution show that the expected value of <math>\widehat{\sigma}^2</math> is σ<sup>2</sup>(''n'' − 1)/''n''.
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| ==Alternative formulation==
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| The following version is often seen when considering linear regression.{{Citation needed|date=July 2011}} Suppose that <math>Y\sim N_n(0,\sigma^2I_n)</math> is a standard [[Multivariate normal distribution|multivariate normal]] [[random vector]] (here <math>I_n</math> denotes the n-by-n [[identity matrix]]), and if <math>A_1,\ldots,A_k</math> are all n-by-n [[symmetric matrices]] with <math>\sum_{i=1}^kA_i=I_n</math>. Then, on defining <math>r_i=Rank(A_i)</math>, any one of the following conditions implies the other two:
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| * <math>\sum_{i=1}^kr_i=n ,</math>
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| * <math>Y^TA_iY\sim\sigma^2\chi^2_{r_i}</math> (thus the <math>A_i</math> are [[positive semidefinite]])
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| * <math>Y^TA_iY</math> is independent of <math>Y^TA_jY</math> for <math>i\neq j .</math>
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| == See also ==
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| * [[Cramér's theorem]], on decomposing normal distribution
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| * [[Infinite divisibility (probability)]]
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| {{refimprove|date=July 2011}}
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| | |
| ==References==
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| <references/>
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| {{Experimental design|state=expanded}}
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| {{DEFAULTSORT:Cochran's Theorem}}
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| [[Category:Statistical theorems]]
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| [[Category:Characterization of probability distributions]]
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