On the Distribution of Linear Combinations of Chi-Square Random Variables

The distribution of linear combinations of independent chi-square
random variables is intimately related with the distribution of quadratic
forms in normal random variables As such, this distribution has been
studied by many authors. However, there is still some room left for
improvement, since while some simpler approximations do not yield
sufficiently good results, other approximations which show a better
performance are sometimes too complicated to be implemented in
practical terms. In this paper the exact distribution of linear combinations
of independent chi-square random variables is obtained, for some
particular cases, in closed finite highly manageable forms, while for more
general cases near-exact approximations are obtained, which are able to
yield very manageable and well-performing approximations. Numerical
studies compare the performance of these near-exact distributions with
other existing approximations and distributions and show how sharp
are the approximations provided by these near-exact distributions. A
useful subproduct that is obtained is closed form expressions for the
distribution of quadratic forms and for some instances of ratios of
quadratic forms, useful in ANOVA and other linear or mixed-linear
models where heterocedasticity is present or assumed. Solutions for
the problem of the distribution of the statistic associated with the
Behrens–Fisher problem are then in turn obtained as a much useful
subproduct of the distribution of ratios of quadratic forms. Modules
programmed in MathematicaⓇ, MAXIMA and R for the implementation
of the distributions developed are made available at the site https://
sites.google.com/site/lincombchisquares.