Cochran's theorem

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In statistics, Cochran's theorem, devised by William G. Cochran,1 is a theorem used to justify results relating to the probability distributions of statistics that are used in the analysis of variance.2

Statement

Suppose U1, ..., Un are independent standard normally distributed random variables, and an identity of the form


\sum_{i=1}^n U_i^2=Q_1+\cdots + Q_k

can be written, where each Qi is a sum of squares of linear combinations of the Us. Further suppose that


r_1+\cdots +r_k=n

where ri is the rank of Qi. Cochran's theorem states that the Qi are independent, and each Qi has a chi-squared distribution with ri degrees of freedom.1 Here the rank of Qi should be interpreted as meaning the rank of the matrix B(i), with elements Bj,k(i), in the representation of Qi as a quadratic form:

Q_i=\sum_{j=1}^n\sum_{k=1}^n U_j B_{j,k}^{(i)} U_k .

Less formally, it is the number of linear combinations included in the sum of squares defining Qi, provided that these linear combinations are linearly independent.

Examples

Sample mean and sample variance

If X1, ..., Xn are independent normally distributed random variables with mean μ and standard deviation σ then

U_i = \frac{X_i-\mu}{\sigma}

is standard normal for each i. It is possible to write


\sum_{i=1}^n U_i^2=\sum_{i=1}^n\left(\frac{X_i-\overline{X}}{\sigma}\right)^2
+ n\left(\frac{\overline{X}-\mu}{\sigma}\right)^2

(here \overline{X} is the sample mean). To see this identity, multiply throughout by \sigma^2 and note that


\sum(X_i-\mu)^2=
\sum(X_i-\overline{X}+\overline{X}-\mu)^2

and expand to give


\sum(X_i-\mu)^2=
\sum(X_i-\overline{X})^2+\sum(\overline{X}-\mu)^2+
2\sum(X_i-\overline{X})(\overline{X}-\mu).

The third term is zero because it is equal to a constant times

\sum(\overline{X}-X_i)=0,

and the second term has just n identical terms added together. Thus


\sum(X_i-\mu)^2=
\sum(X_i-\overline{X})^2+n(\overline{X}-\mu)^2 ,

and hence


\sum\left(\frac{X_i-\mu}{\sigma}\right)^2=
\sum\left(\frac{X_i-\overline{X}}{\sigma}\right)^2
+n\left(\frac{\overline{X}-\mu}{\sigma}\right)^2
=Q_1+Q_2.

Now the rank of Q2 is just 1 (it is the square of just one linear combination of the standard normal variables). The rank of Q1 can be shown to be n − 1, and thus the conditions for Cochran's theorem are met.

Cochran's theorem then states that Q1 and Q2 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.3

Distributions

The result for the distributions is written symbolically as


\sum\left(X_i-\overline{X}\right)^2  \sim \sigma^2 \chi^2_{n-1}.

n(\overline{X}-\mu)^2\sim \sigma^2 \chi^2_1,

Both these random variables are proportional to the true but unknown variance σ2. Thus their ratio is does not depend on σ2 and, because they are statistically independent, the distribution of their ratio is given by


\frac{n\left(\overline{X}-\mu\right)^2}
{\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}}
   \sim F_{1,n-1}

where F1,n − 1 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.

Estimation of variance

To estimate the variance σ2, one estimator that is sometimes used is the maximum likelihood estimator of the variance of a normal distribution


\widehat{\sigma}^2=
\frac{1}{n}\sum\left(
X_i-\overline{X}\right)^2.

Cochran's theorem shows that


\frac{n\widehat{\sigma}^2}{\sigma^2}\sim\chi^2_{n-1}

and the properties of the chi-squared distribution show that the expected value of \widehat{\sigma}^2 is σ2(n − 1)/n.

Alternative formulation

The following version is often seen when considering linear regression.citation needed Suppose that Y\sim N_n(0,\sigma^2I_n) is a standard multivariate normal random vector (here I_n denotes the n-by-n identity matrix), and if A_1,\ldots,A_k are all n-by-n symmetric matrices with \sum_{i=1}^kA_i=I_n. Then, on defining r_i=Rank(A_i), any one of the following conditions implies the other two:

See also

References

  1. ^ a b Cochran, W. G. (April 1934). "The distribution of quadratic forms in a normal system, with applications to the analysis of covariance". Mathematical Proceedings of the Cambridge Philosophical Society 30 (2): 178–191. doi:10.1017/S0305004100016595. 
  2. ^ Bapat, R. B. (2000). Linear Algebra and Linear Models (Second ed.). Springer. ISBN 978-0-387-98871-9. 
  3. ^ Geary, R.C. (1936). "The Distribution of the "Student's" Ratio for the Non-Normal Samples". Supplement to the Journal of the Royal Statistical Society 3 (2): 178–184. doi:10.2307/2983669. JFM 63.1090.03. JSTOR 2983669. 







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