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 Cramer–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 typically approximated using nonsmooth empirical distribution functions (EDFs). In a recent article, Li, Li, and Racine establish the benefits of smoothing the EDF for inference, though their theoretical results are limited to the case where the covariates are observed and the distributions unobserved, while in the current setting some covariates and their distributions are unobserved (i.e., the test relies on population error terms from a location-scale model) which necessarily involves a separate theoretical approach. We demonstrate how replacing the nonsmooth distributions of unobservables with their kernel-smoothed sample counterparts can lead to substantial power improvements, and extend existing approaches to the smooth multivariate and mixed continuous and discrete data setting in the presence of unobservables. Theoretical underpinnings are provided, Monte Carlo simulations are undertaken to assess finite-sample performance, and illustrative applications are provided.

Key Words:

• Inference
• Kernel smoothing
• Kolmogorov–Smirnov

Notes

Notes

1 By “unstructured” we mean a model of the form $Y_i=\mu (X_i)+{ϵ}_i$ with $E({ϵ}_i|X_i)=0$.

2 This approach compares two kernel smoothed univariate distributions; see the function KS.test in the R package Qiu (2014) which implements this procedure using Wang, Cheng, and Yang’s (2013) plug-in bandwidth and uses the asymptotic distribution for critical values which is known to be problematic.

3 As noted by a referee, by working with kernel estimators we obtain a test that is not invariant to transformations of the regressors. A remedy could be to replace $k((x_s^c-X_s^c)/h_s)$ by nearest neighbor windows of the form $k(({\tilde{F}}_{X_s^c}(x_s^c)-{\tilde{F}}_{X_s^c}(X_s^c))/h_s)$, where ${\tilde{F}}_{X_s^c}$ is the nonsmoothed empirical distribution of $X_s^c$. It is clear that this is invariant under any monotone transformation of $X_s^c$.

4 We proceed with the Kolmogorov–Smirnov statistic by way of illustration as the Cramer–von Mises statistic requires multivariate integration for its computation while the Kolmogorov–Smirnov statistic does not. However, as will be demonstrated for the Kolmogorov–Smirnov approach, replacing the nonsmooth distribution functions in the Cramer–von Mises statistic with their kernel-smoothed counterparts would be expected to reveal similar power gains.

5 If anything, the nonsmooth version appears to be slightly over-sized for smaller n and the smooth version slightly under-sized for smaller n, but this admits a fair comparison of power curves. Next, we vary $\delta \in [0,1]$, and present power curves.

Additional information

Funding

Racine would like to gratefully acknowledge support from the Natural Sciences and Engineering Research Council of Canada (NSERC: www.nserc.ca), the Social Sciences and Humanities Research Council of Canada (SSHRC: www.sshrc.ca), and the Shared Hierarchical Academic Research Computing Network (SHARCNET: www.sharcnet.ca). I. Van Keilegom acknowledges financial support from the European Research Council (2016–2021, Horizon 2020/ERC grant agreement no. 694409).