A consistent specification test for functional linear quantile regression models

This paper is focused on the specification test of functional linear quantile regression models. A nonparametric test statistic is proposed based on the orthogonality of residual and its conditional expectation. It is proved with mild assumptions that the proposed statistic follows asymptotically the standard normal distribution under the null hypothesis, but tends to infinity under alternative hypothesis. The asymptotic power of the test is also presented for some local alternative hypotheses. The test is easy to implement, and is shown by simulations powerful even for small sample sizes. A real data example with the Capital Bikeshare data is presented for illustration.