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Goodness-of-fit tests in semiparametric transformation models. (English) Zbl 1342.62058
Summary: Consider a semiparametric transformation model of the form $$\Lambda_{\theta}(Y)= m(X) + \varepsilon$$, where $$Y$$ is a univariate dependent variable, $$X$$ is a $$d$$-dimensional covariate, and $$\varepsilon$$ is independent of $$X$$ and has mean zero. We assume that $$\{\Lambda_{\theta} : \theta\in\Theta\}$$ is a parametric family of strictly increasing functions, while $$m$$ is an unknown regression function. The goal of the paper is to develop tests for the null hypothesis that $$m(\cdot)$$ belongs to a certain parametric family of regression functions. We propose a Kolmogorov-Smirnov and a Cramér-von Mises type test statistic, which measure the distance between the distribution of $$\varepsilon$$ estimated under the null hypothesis and the distribution of $$\varepsilon$$ without making use of this null hypothesis. The estimated distributions are based on a profile likelihood estimator of $$\theta$$ and a local polynomial estimator of $$m(\cdot)$$. The limiting distributions of these two test statistics are established under the null hypothesis and under a local alternative. We use a bootstrap procedure to approximate the critical values of the test statistics under the null hypothesis. Finally, a simulation study is carried out to illustrate the performance of our testing procedures, and we apply our tests to data on the scattering of sunlight in the atmosphere.

##### MSC:
 62G08 Nonparametric regression and quantile regression 62G09 Nonparametric statistical resampling methods 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference
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