Distribution-free testing in linear and parametric regression (Q825054)

The paper under review (pertaining to distribution-free testing in linear and parametric regression) considers the problem of testing the hypothesis that the regression function belongs to a specified parametric family of functions (which depend on a finite-dimensional parameter). The author's objective is to describe a new asymptotically distribution-free method for testing such hypotheses. More specifically, his objective is to construct asymptotically distribution-free version of the regression empirical process, so that functionals from this process, employed as test statistics, will be asymptotically distribution-free. The core of the method consists of the application of unitary operators described in [\textit{E. Khmaladze}, Ann. Stat. 41, No. 6, 2979--2993 (2013; Zbl 1294.62095); Bernoulli 22, No. 1, 563--588 (2016; Zbl 1345.60094); Trans. A. Razmadze Math. Inst. 174, No. 2, 155--173 (2020; Zbl 1458.62189)] and studied in [\textit{L. A. Roberts}, Stat. Probab. Lett. 150, 47--53 (2019; Zbl 1459.62069); \textit{T. T. M. Nguyen}, Metrika 80, No. 2, 153--170 (2017; Zbl 1394.62074)]. First, the method is illustrated with simple linear regression model. In Section 2 the general form of one-dimensional linear regression is considered. Then in Section 3 general parametric regression is studied. In Section 4 the case of finite-dimensional vector covariates is analyzed. Finally, Section 5 is devoted to power considerations.