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Testing the adequacy of semiparametric transformation models

Abstract

We consider a semiparametric model whereby the response variable following a transformation can be expressed by means of a regression model. In this model, the form of the transformation is specified analytically (up to an unknown transformation parameter), while the regression function is completely unknown. We develop testing procedures for the null hypothesis that this semiparametric model adequately describes the data at hand. In doing so, the test statistic is formulated on the basis of Fourier-type conditional expectations, an idea first put forward by Bierens (J Econom 20:105–134, 1982). The asymptotic distribution of the test statistic is obtained under the null as well as under alternative hypotheses. Since the limit null distribution is nonstandard, a bootstrap version is utilized in order to actually carry out the test procedure. Monte Carlo results are included that illustrate the finite-sample properties of the new method.

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Acknowledgements

Simos Meintanis acknowledges support by the Special Account for Research Grants \((\hbox {E}\mathrm{{\Lambda }}\hbox {KE})\) (Research Grant 11699) of the National and Kapodistrian University of Athens. M. Hušková acknowledges support from Grant GACR 15-096635S. James Allison thanks the National Research Foundation of South Africa for financial support. The authors would also like to thank the referees for their constructive comments that led to an improvement of the paper.

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Correspondence to S. G. Meintanis.

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S. G. Meintanis: On sabbatical leave from the University of Athens.

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Allison, J.S., Hušková, M. & Meintanis, S.G. Testing the adequacy of semiparametric transformation models. TEST 27, 70–94 (2018). https://doi.org/10.1007/s11749-017-0544-4

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Keywords

  • Transformation model
  • Goodness-of-fit test
  • Nonparametric regression
  • Bootstrap test

Mathematics Subject Classification

  • 62G08
  • 62G09
  • 62G10