Application of the empirical characteristic function to compare and estimate densities by pooling information

Summary  Independent measurements are taken from distinct populations which may differ in mean, variance and in shape, for instance in the number of modes and the heaviness of the tails. Our goal is to characterize differences between these different populations. To avoid pre-judging the nature of the heterogeneity, for instance by assuming a parametric form, and to reduce the loss of information by calculating summary statistics, the observations are transformed to the empirical characteristic function (ECF). An eigen decomposition is applied to the ECFs to represent the populations as points in a low dimensional space and the choice of optimal dimension is made by minimising a mean square error. Interpretation of these plots is naturally provided by the corresponding density estimate obtained by inverting the ECF projected on the reduced dimension space. Some simulated examples indicate the promise of the technique and an application to the growth of Mirabilis plants is given.