Cochran's theorem

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

Suppose U₁, ..., U_n are i.i.d. standard normally distributed random variables, and an identity of the form

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

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

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

We first show that the matrices B⁽ⁱ⁾ can be simultaneously diagonalized and that their non-zero eigenvalues are all equal to +1. We then use the vector basis that diagonalize them to simplify their characteristic function and show their independence and distribution.

Each of the matrices B⁽ⁱ⁾ has rank r_i and thus r_i non-zero eigenvalues. For each i, the sum ${\displaystyle C^{(i)}\equiv \sum _{j\neq i}B^{(j)}}$ has at most rank ${\displaystyle \sum _{j\neq i}r_{j}=N-r_{i}}$. Since ${\displaystyle B^{(i)}+C^{(i)}=I_{N\times N}}$, it follows that C⁽ⁱ⁾ has exactly rank N − r_i.

...
Wikipedia