A new estimation procedure for linear combinations of exponentials
Cornell, Richard Garth
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Many experimental problems in the natural sciences result in data which can best be represented by linear combinations of exponentials of the form f(t) = ∑[with p above and k=1 below] αk e-λkt. Among such problems are those dealing with growth, decay, ion concentration, and survival and mortality. Also, in general, the solution to any problem which may be represented by linear differential equations with constant coefficients is a linear combination of exponentials. In most problems like those which have been mentioned, the parameters αk and λk have biological or physical significance. Therefore, in fitting the. function f(t) to the data it is not only necessary that the function approximate the data closely, but it is also necessary that the parameters αk and λk be accurately estimated. Furthermore, a measure of the accuracy of the estimation of the parameters is required. A new estimation procedure for linear combinations of exponentials is developed in this paper. Unlike the iterative maximum likelihood and least-squares methods for estimating the parameters for such a model, the new procedure is noniterative and can be easily applied. Also, in contrast to other non-iterative methods, error estimates are available for the parameter estimates yielded by the new procedure. In the model for the new procedure the points ti at which observations are taken are assumed to be equally spaced and the number of such points is specified to be an integral multiple of the number of parameters to be estimated. Moreover, each observation is specified to have expectation f (ti), where f is the function mentioned earlier. The coefficients αk are assumed to be non-zero and the exponents λk are assumed to be distinct and positive. Then in the derivation of new procedure, the observations are reduced to as many sums as there are parameters to be estimated. Each of these sums is equated to its expected value and the resultant equations are solved for estimators of the parameters. The estimators from the new procedure are shown to be asymptotically normally distributed as either the number of points at which observations are taken or the number of observations made at each such point approaches infinity. The asymptotic variances obtained are used to form approximate confidence limits for the αk and λk. The statistical properties of the estimators are also studied. It is found that they are consistent, but not in general unbiased or efficient. Asymptotic efficiencies are calculated tor a few sets of parameter values and a bias approximation is obtained for two special cases. The new method is also shown to be optimum relative to certain similar methods and necessary conditions for the new procedure to lead to admissible estimates are studied. In the last portion of the thesis a sampling study is reported for observations generated with a model containing only one exponential term and with errors which are normally distributed. The small sample biases and variances for the estimates computed from these observations are given and the effects of changes in the parameters in the model are investigated. Then some actual experimental data are fitted using both the new procedure and some alternative methods. The final chapter in the body of the thesis contains a critical evaluation of the new procedure relative to other estimation methods.
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