On the consistency of Prony’s method

Modification’s of Prony’s classical technique for estimating rate constants in exponential fitting problems have many contemporary applications. Here the consistency of Prony’s method and of related algorithms based on maximum likelihood is discussed as the number of observations n → ∞ by considering the simplest possible models for fitting sums of exponentials to observed data. Two sampling regimes are relevant, corresponding to transient problems and problems of frequency estimation; and these are associated with rather different kinds of behaviour. The general pattern is that the stronger results are obtained for the frequency estimation problem. However, the algorithms considered are all scaling dependent and consistency is not automatic. A new feature emerges which is the importance of an appropriate choice of scale in order to ensure consistency of the estimates in certain cases. The tentative conclusion is that algorithms referred to as ORA (Objective function Reweighting Algorithm) are superior to their exact maximum likelihood counterparts referred to as GRA (Gradient condition Reweighting Algorithm), especially in the frequency estimation problem. This conclusion does not extend to fitting other families of functions such as rational functions.