## The Annals of Mathematical Statistics

### A Property of the Multivariate $t$ Distribution

Olive Jean Dunn

#### Abstract

The Student $t$ distribution has the property that the distribution evaluated from $-u$ to $+u$ is an increasing function of $\nu$, the degrees of freedom (this also applies to the distribution evaluated from $- \infty$ to $+ u$). (An elementary proof of this property, due to M. Ray Mickey, is based on consideration of the ratio between the Student density functions for two consecutive values of the degrees of freedom, and makes use of the inequality [5], $(1/4\nu) - (1/360\nu^3) \leqq \log a_\nu \leqq - (1/4\nu) + (2/45\nu^3)$, where $a_\nu$ is the constant in the Student $t$ density with $\nu$ degrees of freedom.) It is pointed out in this note that this monotonicity does not generalize to $k$ dimensions in a usual multivariate $t$ distribution; for $k$ sufficiently large, the direction of the monotonicity becomes reversed.

#### Article information

Source
Ann. Math. Statist., Volume 36, Number 2 (1965), 712-714.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177700183

Digital Object Identifier
doi:10.1214/aoms/1177700183

Mathematical Reviews number (MathSciNet)
MR172372

Zentralblatt MATH identifier
0125.37603

JSTOR
Dunn, Olive Jean. A Property of the Multivariate $t$ Distribution. Ann. Math. Statist. 36 (1965), no. 2, 712--714. doi:10.1214/aoms/1177700183. https://projecteuclid.org/euclid.aoms/1177700183