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Independence tests in semiparametric transformation models. (English) Zbl 1397.62149
Summary: Consider an observed response $$Y$$ which, following a certain transformation $${\mathcal Y}_\vartheta:={\mathcal T}_\vartheta(Y)$$, can be expressed by a homoskedastic nonparametric regression model reference a vector $$X$$ of regressors. If this transformation model is indeed valid then conditionally on $$X$$, the values of $${\mathcal Y}_\vartheta$$ may be viewed as being just location shifts of the regression error, for some value of the transformation parameter $$\vartheta$$. We propose tests for the validity of this model, and establish the limiting distribution of the test statistics under the null hypothesis and under alternatives. Since the null distribution is complicated we also suggest a certain resampling procedure in order to approximate the critical values of the tests, and subsequently use this type of resampling in a Monte Carlo study of the finite-sample properties of the new tests. In estimating the model we rely on the methods proposed by N. Neumeyer et al. [Stat. Sin. 26, No. 3, 925–954 (2016; Zbl 1360.62189)] for the aforementioned transformation model. Our tests however deviate from the tests suggested by N. Neumeyer [loc. cit.] in that we employ an analogue of the test suggested by Z. Hlávka et al. [J. Multivariate Anal. 102, No. 4, 816–827 (2011; Zbl 1327.62258)] involving characteristic functions, rather than distribution functions.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62G10 Nonparametric hypothesis testing 62G09 Nonparametric statistical resampling methods
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