Fisher's theorem

When studying probability distributions involving  distributions, it often happens than one encounters the following situation:

    * One is handling n independent standard normal variables X_i ~N(0, 1),

    * And calculations involve a quadratic form in the X_i with the following particular form:

Q(X₁, X₂ , ..., X_n) = (_iX_i ²)  - Y₁² - Y₂² -...- Y_p²             p < n  

where the Y_i are orthogonal linear forms in the X_i:

Y_i = _j c_ij.X_j

the orthogonality conditions being:

_k c_ik c_jk = 1       if     i = j

_k c_ik c_jk = 0       if     i  j

Then Fisher's theorem asserts that:  

In intuitive terms, the second part of Fisher's theorem is a consequence of the fact that the quadratic form Q(X₁, X₂ , ..., X_n):

    * seems to depend on n independent standard normal variables (the X_i),

    * but in fact, depends only on n - p independent standard normal variables, hence the_{n - p} distribution.

We outline here a demonstration of Fisher's theorem. 

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This result can easily be extended to normal variables with variance ².

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Here are two important consequences of Fisher's theorem.

1) Let  X~N(µ,  ²).  

    * If   denotes the sample mean,

    * and S² denotes the sample variance

then  and S² are independent variables.

2) Let X~N(µ,  ²). For an n observation sample, denote

S² = 1/n - 1._i(x - )²

the sample variance.

Then:

(n - 1)S²/ ² ~_{n - 1}

We use these two results when we elaborate the formal definition of the t distribution.

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