A Smooth Nonparametric, Multivariate, Mixed-Data Location-Scale Test

Abstract

A number of tests have been proposed for assessing the location-scale assumption that is often invoked by practitioners. Existing approaches include Kolmogorov-Smirnov and Cramérvon-Mises statistics that each involve measures of divergence between unknown joint distribution functions and products of marginal distributions. In practice, the unknown distribution functions embedded in these statistics are approximated using non-smooth empirical distribution functions. We demonstrate how replacing the non-smooth distributions with their kernel-smoothed counterparts can lead to substantial power improvements. In so doing we extend existing approaches to the smooth multivariate and mixed continuous and discrete data setting thereby extending the reach of existing approaches. Theoretical underpinnings are provided, Monte Carlo simulations are undertaken to assess finite-sample performance, and illustrative applications are provided.

Bootstrap of residual processes in regression: to smooth or not to smooth?

“…The estimator of the error distribution has been shown to be very useful for testing hypotheses regarding several features of model (1.1), like e.g. testing for the form of the regression function m(·) ), comparing regression curves (Pardo-Fernández et al (2007)), testing independence between ε and X (Einmahl and Van Keilegom (2008) and Racine and Van Keilegom (2017)), testing for symmetry of the error distribution (Koul (2002), Neumeyer and Dette (2007)), among others. The idea in each of these papers is to compare an estimator of the error distribution obtained under the null hypothesis with an estimator that is not based on the null.…”