We consider a problem of checking whether the coefficient of the scale and location function is a constant. Both the scale and location functions are modeled as single-index models. Two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals of the difference of the empirical residual processes are proposed. The asymptotic distribution of the estimator for single-index parameter is derived, and the empirical distribution function of residuals is shown to converge to a Gaussian process. Moreover, the proposed test statistics can be able to detect local alternatives that converge to zero at a parametric convergence rate. A bootstrap procedure is further proposed to calculate critical values. Simulation studies and a real data analysis are conducted to demonstrate the performance of the proposed methods.
Jun Zhang. Cuizhen Niu. Gaorong Li. "Exploring the constant coefficient of a single-index variation." Braz. J. Probab. Stat. 33 (1) 57 - 86, February 2019. https://doi.org/10.1214/17-BJPS377