The distribution function of a random variable of the form $\sum^n_{i = 1} a_i Y_1 Y_2 \cdots Y_i$ where $a_i > 0$ and $0 \leq Y_i \leq 1$ is considered. A geometric argument is used to obtain the distribution function as a repeated integral. The result is used first to obtain the distribution function of a linear combination of variables defined over the simplex $X_i \geq 0, \sum^n_{i = 1} X_i \leq 1$. As a second application the distribution of certain quadratic forms over the simplex is obtained. This result yields as a special case the distribution of the internally studentized extreme deviate; the cases of normal and exponential samples are considered in detail and the required distributions obtained.
Publié le : 1981-07-14
Classification:
Geometric probability,
order statistics,
linear combinations,
internally studentized deviate,
60D05,
62E15,
62G30
@article{1176345522,
author = {Currie, Iain D.},
title = {On Distributions Determined by Random Variables Distributed Over the $n$-Cube},
journal = {Ann. Statist.},
volume = {9},
number = {1},
year = {1981},
pages = { 822-833},
language = {en},
url = {http://dml.mathdoc.fr/item/1176345522}
}
Currie, Iain D. On Distributions Determined by Random Variables Distributed Over the $n$-Cube. Ann. Statist., Tome 9 (1981) no. 1, pp. 822-833. http://gdmltest.u-ga.fr/item/1176345522/