Shape testing in quantile varying coefficient models with heteroscedastic error

ABSTRACT

The interest is in regression quantiles in varying coefficient models for analysing longitudinal data. The coefficients are allowed to vary with time, and the error variance (the variability function) varies with the covariates to allow for heteroscedasticity. The functional coefficients are estimated using penalized splines (P-splines), not requiring specification of the error distribution. A likelihood-ratio-type test is considered to test the shape (constancy, monotonicity and/or convexity) of the functional coefficients. Further, testing procedures based on -norm, -norm and -norm of the differences of the P-splines coefficients are considered to test for constant functional coefficients. These norm-based tests perform better than the likelihood-ratio-type test in our simulation study. An extreme value test for testing monotonicity or convexity also performs better than the likelihood-ratio-type test. The likelihood-ratio-type test is, however, useful when testing the shape of the coefficients in signal and in variability function simultaneously. A real-data example demonstrates the testing procedures.

KEYWORDS:

• Heteroscedasticity
• likelihood-ratio test
• qualitative shape testing
• quantile regression
• varying coefficient models

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors gratefully acknowledge support from the IAP Research Network P7/06 of the Belgian State (Belgian Science Policy). I. Gijbels acknowledges support from the KU Leuven Research Council (GOA/12/014). A. Verhasselt acknowledges support from the Fonds Wetenschappelijk Onderzoek (FWO) research grant 1518917N. M.A. Ibrahim and A. Verhasselt acknowledge support from the Special Research Fund (BOF) of Hasselt University [14NO6VHAA]. The infrastructure of the VSC-Flemish Supercomputer Center, funded by the Hercules Foundation and the Flemish Government – department EWI, was used for the simulations.