## Abstract

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.

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.

Original language | English |
---|---|

Title of host publication | Computational and Methodological Statistics and Biostatistics: Contemporary Essays in Advancement |

Editors | Andriette Bekker, Ding Chen, Johan Ferreira |

Publisher | Springer |

Pages | 211-252 |

Volume | 1 |

ISBN (Electronic) | 978-3-030-42196-0 |

ISBN (Print) | 978-3-030-42195-3 |

DOIs | |

Publication status | Published - 2020 |

### Publication series

Name | Emerging Topics in Statistics and Biostatistics |
---|---|

Publisher | Springer |

## Keywords

- asymptotic distributions
- near-exact distributions
- EGIG distribution
- GIG (Generalized Integer Gamma) distribution