Consider a semiparametric transformation model of the form Λ θ (Y ) =
m(X ) + ε, where Y is a univariate dependent variable, X is a d-dimensional covari-
ate, and ε is independent of X and has mean zero. We assume that {Λ θ : θ ∈ Θ} is
a parametric family of strictly increasing functions, while m is an unknown regres-
sion function. The goal of the paper is to develop tests for the null hypothesis that
m(·) 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 ε estimated under the null hypothesis and the dis-
tribution of ε without making use of this null hypothesis. The estimated distributions
are based on a profile likelihood estimator of θ and a local polynomial estimator of
m(·). 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.

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• Goodness-of-fit tests in semiparametric transformation models