# Specification testing in semi-parametric transformation models

## TL;DR

This paper develops a goodness-of-fit test for semi-parametric transformation models, using a distance measure between nonparametric and parametric transformations, with asymptotic theory and bootstrap implementation.

## Contribution

It introduces a novel test for assessing the fit of parametric transformation classes in semi-parametric models, including asymptotic analysis and a bootstrap procedure.

## Key findings

- Asymptotic validity of the test under the null hypothesis.
- The test can distinguish between large and small deviations from the true transformation.
- A bootstrap algorithm effectively implements the proposed test.

## Abstract

In transformation regression models the response is transformed before fitting a regression model to covariates and transformed response. We assume such a model where the errors are independent from the covariates and the regression function is modeled nonparametrically. We suggest a test for goodness-of-fit of a parametric transformation class based on a distance between a nonparametric transformation estimator and the parametric class. We present asymptotic theory under the null hypothesis of validity of the semi-parametric model and under local alternatives. A bootstrap algorithm is suggested in order to apply the test. We also consider relevant hypotheses to distinguish between large and small distances of the parametric transformation class to the `true' transformation.