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 Cramér-von-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 counter-parts 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. I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S ( I S B INGRID VAN KEILEGOMORSTAT, KU Leuven, Naamsestraat 69, B-3000 Leuven, Belgium.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 Cramér-von-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.

Estimation in copula models with two-piece skewed margins using the inference for margins method

“…This is obtained by replacing F Z j ,n in the classical tests with a smoothed version. As shown in [34], these type of tests have better power and are overall better when data show a more discrete behaviour. The smooth tests are close to the classical F Z j ,n -based test as shown in [41] and asymptotically equivalent in terms of ef-ficiency (see [4]).…”

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(•) (Van Keilegom et al (2008)), 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.…”