Cochran’s theorem

Let X be multivariate normally distributed as 𝑵𝒑(𝟎,𝑰) such that


where each

  1. 1.

    Qi is a quadratic formMathworldPlanetmath

  2. 2.

    Qi=𝐗T𝐁i𝐗, where 𝐁i is a p by p square matrixMathworldPlanetmath

  3. 3.
  4. 4.


Then any two of the following imply the third:

  1. 1.


  2. 2.

    each Qi has a chi square distribution ( with ri of freedom, χ2(ri)

  3. 3.

    Qi’s are mutually independent

As an example, suppose X12χ2(m1) and X22χ2(m2). Furthermore, assume X12X22 and m1>m2, then


This corollary is known as Fisher’s theorem.

Title Cochran’s theorem
Canonical name CochransTheorem
Date of creation 2013-03-22 14:33:01
Last modified on 2013-03-22 14:33:01
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Theorem
Classification msc 62J10
Classification msc 62H10
Classification msc 62E10
Defines Fisher’s theorem