The Annals of Statistics

On Distributions Determined by Random Variables Distributed Over the $n$-Cube

Iain D. Currie

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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.

Article information

Ann. Statist., Volume 9, Number 4 (1981), 822-833.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 62E15: Exact distribution theory 62G30: Order statistics; empirical distribution functions

Geometric probability order statistics linear combinations internally studentized deviate


Currie, Iain D. On Distributions Determined by Random Variables Distributed Over the $n$-Cube. Ann. Statist. 9 (1981), no. 4, 822--833. doi:10.1214/aos/1176345522.

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