When Moments of Ratios are Ratios of Moments

Abstract

Finding the moments of a ratio can be very difficult, but if the ratio is independent of its own denominator the solution simplifies to the moments of the numerator divided by those of the denominator. Such an independence situation occurs surprisingly frequently in important statistical problems. However, its occurrence has not always been noticed and, consequently, the device of obtaining moments of ratios as ratios of moments has often not been employed when it should have. The history of the device is one of discovery, inadequate dissemination leading to neglect and rediscovery, with the eventual users often unaware of the true originators. This paper charts that history and discusses the applications of the device, including some of recent vintage.