## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 19, Number 2 (1948), 228-237.

### Inversion Formulae for the Distribution of Ratios

#### Abstract

The use of the repeated Cauchy principal value affords greater facility in the application of inversion formulae involving characteristic functions. Formula (2) below is especially useful in obtaining the inversion formula (1) for the distribution of the ratio of linear combinations of random variables which may be correlated. Formulae (1), (10), (12) generalize the special cases considered by Cramer [2], Curtiss [4], Geary [6], and are free of some restrictions they impose. The results are further generalized in section 6, where inversion formulae are given for the joint distribution of several ratios. In section 7, the joint distribution of several ratios of quadratic forms in random variables $X_1, X_2,\cdots,X_n$ having a multivariate normal distribution is considered.

#### Article information

**Source**

Ann. Math. Statist., Volume 19, Number 2 (1948), 228-237.

**Dates**

First available in Project Euclid: 28 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177730247

**Digital Object Identifier**

doi:10.1214/aoms/1177730247

**Mathematical Reviews number (MathSciNet)**

MR25005

**Zentralblatt MATH identifier**

0032.03403

**JSTOR**

links.jstor.org

#### Citation

Gurland, John. Inversion Formulae for the Distribution of Ratios. Ann. Math. Statist. 19 (1948), no. 2, 228--237. doi:10.1214/aoms/1177730247. https://projecteuclid.org/euclid.aoms/1177730247