abhranildas/gx2

Matlab toolbox to compute the statistics, pdf, cdf, inverse cdf and random numbers of the generalized chi-square distribution.

gx2's Issues

Error in the derivation of the mean of the normal variable?

In the derivation of the gx2 distribution from a quadratic form, the mean of the normal variable is calculated as:

gx2/gx2_params_norm_quad.m

Line 60 in 06edc5c

m=q0-dot(w,lambda);

This agrees with the formula given in the paper (https://arxiv.org/abs/2012.14331):
$m = q(\mathbf{\mu}) - \mathbf{w}\cdot\mathbf{\lambda}$

But it seems to me that there is an error, and it should be:
$m = q(\mathbf{\mu}) - \mathbf{k}\cdot\mathbf{\lambda}$
(i.e. the noncentrality parameters should be weighed by the degrees of freedom of the Chi-squared distributions, $\mathbf{k}$).

That mean is derived from a previous formula in the paper, where the normal variable is defined as:
$\sum{b_{i^\prime} y_{i^\prime}} + \tilde{q_0} - \sum{(\frac{b_i}{2D_i})^2} $

The mean of that variable is meant to be the the value of that variable when $y_{i^\prime} = 0$, i.e.:
$\tilde{q_0} - \sum{(\frac{b_i}{2D_i})^2} $

Now, $\tilde{q}_0$ is defined as $q(\mathbf{\mu})$, which accounts for the first term; and the elements in the sum of the second term correspond to the definition of $\lambda_j$, for the unique values of the eigenvalues $D_i$.

Therefore, if I have followed that derivation correctly, each value of $\lambda_j$ should be weighed by the number of eigenvalues that are equal to $\lambda_j$ (i.e. the degrees of freedom $k_i$), and not the values of the eigenvalues ($w_j$).