Inference about the Ratio of Two Parameters, with Application to Whale Censusing

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Abstract
Inference for the quotient of two parameters estimated separately may be obtained by the delta method. When the distribution of linear transformations involving the numerator and the denominator is available, more exact and elementary methods may be used. Non-Bayesian and Bayesian approaches are developed. The application of the methods to estimating the stock abundance of Northeastern Atlantic minke whales, where the ratio is a raw estimate divided by a measure of observation efficiency, is explained and discussed. The Bayesian approach allows exact inference in quite general situations using only a single, rapidly implemented, one-dimensional numerical integration. A simple analytic approximation is given for the common situation where the joint posterior distribution of the numerator and denominator can be approximated by a normal distribution that gives very little probability to negative values of the denominator. The Bayesian approach also permits the incorporation of model uncertainty (or disagreement) in a natural way, and this was the basis for the conclusions of the International Whaling Commission Scientific Committee at its 1990 meeting.

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