Density Based Regression for Inhomogeneous Data: Application to Lottery Experiments

Abstract

This paper presents a regression procedure for inhomogeneous data characterized by varying variance, skewness and kurtosis or by an unequal amount of data over the estimation domain. The concept is based first on the estimation of the densities of an observed variable for given values of explanatory variable(s). These density functions are then used to estimate the relation between all the variables. The mean, quantile (including median) and mode re-gression estimators are proposed, with the last one appearing to be the maximum likelihood estimator in the density based approach. The paper demonstrates the advantages of the pro-posed methodology, which eliminates most of the estimation problems arising from data inhomogeneity. These include the computational inconveniences of the standard quantile and mode regression techniques. The proposed methodology, when applied to lottery experiments, makes it possible to confirm and to extend the previously presented conclusion (Kontek, 2010) that lottery valuations are only nonlinear with respect to probability when medians and means are considered. Such nonlinearity disappears once modes are considered. This means that the most likely behavior of a group is fully rational. The regression procedure presented in this paper is, however, very general and may be applied in many other cases of data inhomogeneity and not just lottery experiments.