Small-Sample Tests for the Equality of Two Normal Cumulative Probabilities, Coefficients of Variation, and Sharpe Ratios

Abstract

This article considers the problem of testing the equality of the cumulative probabilities of two independent normal distributions. This is equivalent to the problem of testing the equality of the coefficients of variations from two separate populations, which in turn is equivalent to the problem of testing the equality of two Sharpe ratios in financial analyses when it is assumed that the data are normally distributed. This is a problem that has received considerable attention in the statistical and financial research literature; although, while previous approaches have relied upon large sample asymptotic results, the objective of this article is to develop a test procedure that guarantees the nominal confidence level for small sample sizes. Examples are provided to demonstrate the new test procedure, and software is available for its implementation from the authors.

AMS Subject Classification:

• 62F99

Keywords:

• Cumulative distribution function
• Coefficient of variation
• Sharpe ratio
• Normal distribution
• Two-sample comparison
• Noncentral t-distribution
• Test procedure
• Acceptance set

Acknowledgments

We sincerely thank each of the reviewers for their helpful and insightful comments and suggestions that have resulted in a much improved version of this article.