Estimating a Mixed-Exponential Response Law

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Abstract
This paper is concerned with the estimation of the parameters of a mixed-exponential response law, given some sample observations. Two alternative laws are considered, each having two parameters. The first law expresses the proportion F(t) of the population who have responded within time t in the form F(t) = (1 – e −αt )ξ. The second law states that F(t) = 1 – (1 + λt)−v. The problem has been encountered in the context of market research, in estimating the “penetration” of a market by a new product; and it is therefore described in those terms. But mathematically identical problems arise also in life testing and in fisheries research. The problem is similar to the well-known one in toxicology of fitting a response law (logistic, probit, or whatever) to dose-mortality data, with the exception that in the present case the points on the empirical response curve are not independent, the basic sampling distribution being multinomial instead of independent-binomial. In regard to statistical methodology, the paper aims to illustrate the role of the likelihood function in estimation. Whether the likelihood function has a “large-sample” appearance depends not only on the sample size but on the parametrization.

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