NONPARAMETRIC-ESTIMATION OF JOINT DISCRETE-CONTINUOUS PROBABILITY DENSITIES WITH APPLICATIONS

This investigation considers nonparametric estimation of the joint probability density function of a random vector (X, Y) where X is discrete and Y is continuous. Using both the kernel density estimation (for the continuous co-ordinate) and a discrete analog (for the discrete co-ordinate), we define a class of 'kernel' type joint estimates. Basic properties of this estimate are studied. In addition, applications of the results to non-parametric regression when the regressor is a discrete random variable as well as to the case of mixed regressors, and to a discrete version of 'index coefficient' (cf. Powell et al., 1989, Econometrica, 57, 1403-1430) are presented. Optimal choices of the smoothing parameters are also discussed.