[Paper Review] Expansion for moments of regression quantiles with application to nonparametric testing

Enno Mammen, Ingrid Van Keilegom+1 more|arXiv (Cornell University)|Jun 26, 2013

Statistical Methods and Inference31 references3 citations

TL;DR

This paper develops a nonparametric test for parametric specifications of regression quantiles using kernel smoothing, with asymptotic theory derived via higher-order Edgeworth expansions of Nadaraya-Watson quantile estimator moments. The key contribution is a test statistic asymptotically equivalent to a weighted L²-distance between parametric and nonparametric fits, with a normal limit distribution under the null, enabling valid inference without restrictive bandwidth or dimensionality assumptions.

ABSTRACT

We discuss nonparametric tests for parametric specifications of regression quantiles. The test is based on the comparison of parametric and nonparametric fits of these quantiles. The nonparametric fit is a Nadaraya-Watson quantile smoothing estimator. An asymptotic treatment of the test statistic requires the development of new mathematical arguments. An approach that makes only use of plugging in a Bahadur expansion of the nonparametric estimator is not satisfactory. It requires too strong conditions on the dimension and the choice of the bandwidth. Our alternative mathematical approach requires the calculation of moments of Nadaraya-Watson quantile regression estimators. This calculation is done by application of higher order Edgeworth expansions.

Motivation & Objective

• To develop a nonparametric goodness-of-fit test for parametric regression quantile models.
• To overcome limitations of standard Bahadur expansion approaches that require strong bandwidth and dimensionality restrictions.
• To establish asymptotic normality of a test statistic based on the L²-distance between parametric and nonparametric quantile fits.
• To derive the asymptotic distribution of the test under the null hypothesis using higher-order Edgeworth expansions of Nadaraya-Watson quantile estimator moments.
• To provide a robust, omnibus test with power against diverse alternatives in quantile regression models.

Proposed method

• Proposes a test statistic based on the integrated squared difference between a nonparametric Nadaraya-Watson quantile estimator and a parametric fit.
• Uses kernel smoothing with a product kernel and bandwidth $ h $, assuming all bandwidths are of the same order.
• Defines the nonparametric estimator $ \widehat{r}_{\alpha}(x) $ as the minimizer of a weighted check function over residuals from the parametric fit.
• Applies higher-order Edgeworth expansions to compute moments of the Nadaraya-Watson quantile estimator, enabling asymptotic analysis.
• Decomposes the test statistic into components and proves asymptotic normality via U-statistic central limit theorems and moment calculations.
• Establishes that the test statistic $ \widehat{T}_A $ is asymptotically equivalent to a weighted $ L^2 $-distance between nonparametric and parametric fits.

Experimental results

Research questions

• RQ1 Can a nonparametric test for parametric regression quantile models be developed without relying on restrictive Bahadur expansion assumptions?
• RQ2 What is the asymptotic distribution of a kernel-based test statistic comparing parametric and nonparametric quantile fits?
• RQ3 How can higher-order Edgeworth expansions be used to derive the moments of Nadaraya-Watson quantile estimators under general conditions?
• RQ4 Does the proposed test maintain asymptotic normality under the null hypothesis when bandwidth and dimension are not restricted?
• RQ5 Can the test detect deviations from parametric specifications across a range of quantiles and covariate values?

Key findings

• The test statistic $ \widehat{T}_A $ is asymptotically equivalent to a weighted $ L^2 $-distance between the nonparametric and parametric quantile estimators.
• Under the null hypothesis, $ nh^{d/2}T_{n7} $ converges in distribution to a normal random variable with mean $ D_A $ and variance $ V_A $, where $ D_A $ is the integrated squared bias and $ V_A $ the asymptotic variance.
• The terms $ T_{n1} $ through $ T_{n6} $ are asymptotically negligible, with $ T_{n6} = o_P((nh^{d/2})^{-1}) $, ensuring the dominant contribution comes from $ T_{n7} $.
• The asymptotic normality of $ nh^{d/2}T_{n7} $ is established via U-statistic central limit theorems, with verification of moment conditions and degeneracy control.
• The test is robust to dimensionality and bandwidth choices, avoiding the strong regularity conditions required by direct Bahadur expansion methods.
• The theoretical framework supports omnibus testing with power against a broad class of alternatives, including those not detectable by mean regression-based tests.