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Prof. Xiaojun Song
Associate Professor of Business Statistics and Econometrics
Guanghua School of Management
Peking University

In this paper, we establish Neyman’s smooth tests for the equality of conditional distributions. Unlike the traditional smooth tests, always based on parametric residuals, our method requires nonparametric estimation of the conditional cumulative distribution function (CDF). The proposed smooth test statistics are asymptotically chi-square distributed under the null hypothesis that the conditional distributions of two populations are equal. The chi-square distribution for the smooth test statistic works for the general two sample sizes. We also discuss the cases where the conditioning variables follow the same or different distributions. To address the issue of random denominators in the CDF estimator, we introduce the density-weighted smooth test statistic, which imposes fewer restrictions on the asymptotic variance. The finite sample size and power properties of our proposed tests are studied, showing that the tests perform well in controlling the size once the bandwidth used in the CDF estimator is selected within a reasonable range and have decent power under the alternatives.