A nonparametric test for equality of distributions with mixed categorical and continuous data

In this paper we consider the problem of testing for equality of two density or two conditional density functions defined over mixed discrete and continuous variables. We smooth both the discrete and continuous variables, with the smoothing parameters chosen via least-squares cross-validation. The test statistics are shown to have (asymptotic) normal null distributions. However, we advocate the use of bootstrap methods in order to better approximate their null distribution in finite-sample settings and we provide asymptotic validity of the proposed bootstrap method. Simulations show that the proposed tests have better power than both conventional frequency-based tests and smoothing tests based on ad hoc smoothing parameter selection, while a demonstrative empirical application to the joint distribution of earnings and educational attainment underscores the utility of the proposed approach in mixed data settings.