Expansion for moments of regression quantiles with applications to nonparametric testing

Mammen, Enno;Van Keilegom, Ingrid;Yu, Kyusang
(2019) Bernoulli : a journal of mathematical statistics and probability — Vol. 25, n° 2, p. 793-827 (2019)

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Abstract
Wediscussnonparametrictestsforparametricspecificationsofregressionquantiles.Thetestisbasedonthe 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.
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Mammen, E., Van Keilegom, I., & Yu, K. (2019). Expansion for moments of regression quantiles with applications to nonparametric testing. Bernoulli : a journal of mathematical statistics and probability, 25(2), 793-827. https://doi.org/10.3150/17-bej986 (Original work published 2019)