Adaptation of Karber's Method for Estimating the Exponential Parameter from Quantal Data, and Its Relationship to Birth, Death, and Branching Processes

Abstract

SUMMARY Karber estimation for the exponential parameter in a simple death or failure process requires assigning right tail proportions the average of the left-truncated exponential, while interval proportions are assigned the average of a doublytruncated exponential. In the case of equal arithmetic spacing, such assignments result in a simple estimate for which the asymptotic variance and relative efficiency are obtained, but the efficiency of the estimate can be poor. Optimally, testing should be concentrated at a single dose for which the survival probability is 20.32%. Multiple doses, arithmetically or logarithmically spaced, are justified where there is sufficient prior uncertainty. The estimate obtained is identical with the maximum likelihood estimate for serial equi-spaced observation on the same items. A maximum likelihood estimate similar in form arises in the simple birth process. General use of that estimating form is suggested for branching processes. Equi-spacing of observations permits simplified estimates where the parametric form is known, valid estimates even when the parametric form is uniknown. The principle of the Karber method of estimation (also called Spearman estimation or Spearman-Karber estimation) is rather simple, and its application to the analysis of quantal data has been effectively extended by Cornfield and Mantel [1950]. One extension that Cornfield and Mantel made was to drop the requirement of equal spacing of the dose levels. If at the successively increasing (log) dose levels X1 , X, * - , Xk, the independently observed proportions of animals responding are pi = ri/ni, and the dosage range covered is so extensive that P, = 0, p, = 1, then the KdIrber estimate of the mean of the tolerance distribution is given by