A Method for Fitting Linear Combinations of Exponentials

Abstract

Many situationis in biology, epidemiology, economics and the physical sciences yield data which can best be described by linear combinations of exponelntials. This is particularly true in biological radiation, growth and tracer studies. In this paper it will be shown that the method of partial totals may be used to fit linear combinations of any number of exponentials to data taken at equally spaced intervals. The method has been applied to special cases of the general model before, for instance, by Stevens [1951] and Croxton and Cowden [1960]. As in the particular case considered by Stevens, it will be shown that the method, though not generally efficient, is consistent. The method is often useful in computing preliminary estimates for iterative maximum likelihood solutions. Alternatively, if computing facilities are not available for obtaining maximum likelihood estimates, the method provides a systematic, consistent and relatively simple estimation procedure. Also, since for this estimation procedure it is not necessary to assume that all observations have the same distribution, it can be utilized sometimes when other methods are inappropriate. The estimation procedure will be developed in detail and illustrated for two simple models with only one exponential term. Then the general estimation procedure will be presented analogously and concisely and will be illustrated for a model with two exponential terms. The consistency and the large sample variances of the estimators will also be considered briefly. The simple models are emphasized because they are the ones most often encountered in applications. 2. ESTIMATION PROCEDURE FOR ONE EXPONENTIAL TERM