Parametric deconvolution for a common heteroscedastic case

Abstract

There exists an extensive statistics literature dealing with non-parametric deconvolution, the estimation of the underlying population probability density when sample values are subject to measurement errors. In parametric deconvolution, on the other hand, the data are known to be from a specific distribution. In this case the parameters of the distribution can be estimated by e.g. maximum likelihood. In realistic cases the measurement errors may be heteroscedastic and there may be unknown parameters associated with the distribution. The specific realistic case is investigated in which the measurement error standard deviation is proportional to the true sample values. In this case it is shown that the method of moment’s estimation is particularly simple. Estimation by maximum likelihood is computationally very expensive, since numerical integration needs to be performed for each data point, for each evaluation of the likelihood function. Method of moment’s estimation sometimes fails to give physically meaningful estimates. The origin of this problem lies in the large sampling variations of the third moment. Possible remedies are considered. Due to the fact that a convolution integral needed to be calculated for each data point, and that this has to be repeated for each iteration towards the solution, maximum likelihood computing cost is very high. New preliminary work suggests that saddle point approximations could sometimes be used for the convolution integrals. This allows much larger datasets to be dealt with. Application of the theory is illustrated with simulation and real data.