Confidence Bounds for Positive Ratios of Normal Random Variables

Abstract

Some applications of ratios of normal random variables require both the numerator and denominator of the ratio to be positive if the ratio is to have a meaningful interpretation. In these applications, there may also be substantial likelihood that the variables will assume negative values. An example of such an application is when comparisons are made in which treatments may have either efficacious or deleterious effects on different trials. Classical theory on ratios of normal variables has focused on the distribution of the ratio and has not formally incorporated this practical consideration. When this issue has arisen, approximations have been used to address it. In this article, we provide an exact method for determining (1 − α) confidence bounds for ratios of normal variables under the constraint that the ratio is composed of positive values and connect this theory to classical work in this area. We then illustrate several practical applications of this method.

Keywords:

• Confidence intervals
• Estimation
• Ratios of normal variables

Mathematics Subject Classification:

• Primary 62F25
• Secondary 62F03

Notes

¹For the ratio to be meaningful, both the numerator and denominator should exhibit ratio properties. In particular, these quantities should be measured on an equal interval scale with an absolute zero point. This requirement can be met if both the numerator and denominator are differences in improvement on an interval scale relative to a common reference point.

²This result was sent by the authors for comment to D. B. Owen who pointed out that he had mentioned it, without an explicit proof, in Owen (1956).

³One could also consider the joint probability Pr(X > cY and Y > 0); however, we found this approach to be less numerically tractable. In addition, the resulting term does not easily reduce to a single integral, so that it is no longer straightforward to connect the current approach to the classical approach as described by Hinkley (1969).