The Helmert transformation is defined by the matrix so that the element a ij in row i,

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The Helmert transformation is defined by the matrix 

A = r-1/2 2-1/2 -1/2 -2-1/2 r-1/2 -1/2 r-1/2  -1/2 0 0 0 : 0 6-1/2 6-1/2 -2 x 6-1/2 0 0 : 0 12-1/2 12-1/2

so that the element aij in row i, column j is

dij 1--1/2 {j(j-1)}-1/2 -(j-1)/2j-1/2 0 (j = 1) (i < j) (i > j > 1) (i=j > 1).

It is also useful to write aj for the ( column) vector which consists of the jth column of the matrix A. Show that if the variates Xi are independently N(θi,1), then the variates Wj = ajT /(X - µ) = ∑ai j (Xi - µj) are independently normally distributed with unit variance and such that EW= 0 for j > 1 and  w*w=\w =(x  ;) = (x  )^( - ). - - j

By taking aiJ ∝ θ- µj for i > j, ajj = 0 for i jj such tha ∑aij = 0, extend this result to the general case and show that E W∝ y = ∑(0i - µi)2. Deduce that the distribution of a non-central chi-squared variate depends only of r and y.

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