Abstract
Consider the semiparametric transformation model \(\Lambda _{\theta _o}(Y)=m(X)+\varepsilon \), where \(\theta _o\) is an unknown finite dimensional parameter, the functions \(\Lambda _{\theta _o}\) and \(m\) are smooth, \(\varepsilon \) is independent of \(X\), and \({\mathbb {E}}(\varepsilon )=0\). We propose a kernel-type estimator of the density of the error \(\varepsilon \), and prove its asymptotic normality. The estimated errors, which lie at the basis of this estimator, are obtained from a profile likelihood estimator of \(\theta _o\) and a nonparametric kernel estimator of \(m\). The practical performance of the proposed density estimator is evaluated in a simulation study.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmad, I., Li, Q. (1997). Testing symmetry of an unknown density function by kernel method. Journal of Nonparametric Statistics, 7, 279–293.
Akritas, M. G., Van Keilegom, I. (2001). Non-parametric estimation of the residual distribution. Scandinavian Journal of Statistics, 28, 549–567.
Amemiya, T. (1985). Advanced econometrics. Cambridge: Harvard University Press.
Bickel, P. J., Doksum, K. (1981). An analysis of transformations revisited. Journal of the American Statistical Association, 76, 296–311.
Billingsley, P. (1968). Convergence of probability measures. New York: Wiley.
Box, G. E. P., Cox, D. R. (1964). An analysis of transformations. Journal of the Royal Statistical Society-Series B, 26, 211–252.
Carroll, R. J., Ruppert, D. (1988). Transformation and weighting in regression. New York: Chapman and Hall.
Chen, G., Lockhart, R. A., Stephens, A. (2002). Box-Cox transformations in linear models: Large sample theory and tests of normality. Canadian Journal of Statistics, 30, 177–234 (with discussion).
Cheng, F. (2005). Asymptotic distributions of error density and distribution function estimators in nonparametric regression. Journal of Statistical Planning and Inference, 128, 327–349.
Cheng, F., Sun, S. (2008). A goodness-of-fit test of the errors in nonlinear autoregressive time series models. Statistics and Probability Letters, 78, 50–59.
Dette, H., Kusi-Appiah, S., Neumeyer, N. (2002). Testing symmetry in nonparametric regression models. Journal of Nonparametric Statistics, 14, 477–494.
Efromovich, S. (2005). Estimation of the density of the regression errors. Annals of Statistics, 33, 2194–2227.
Einmahl, U., Mason, D. M. (2005). Uniform in bandwidth consistency of kernel-type function estimators. Annals of Statistics, 33, 1380–1403.
Escanciano, J. C., Jacho-Chavez, D. (2012). \(\sqrt{n}\)-uniformly consistent density estimation in nonparametric regression. Journal of Econometrics, 167, 305–316.
Fitzenberger, B., Wilke, R. A., Zhang, X. (2010). Implementing box-cox quantile regression. Econometric Reviews, 29, 158–181.
Horowitz, J. L. (1998). Semiparametric methods in economics. New York: Springer.
Linton, O., Sperlich, S., Van Keilegom, I. (2008). Estimation of a semiparametric transformation model. Annals of Statistics, 36, 686–718.
Müller, U. U., Schick, A., Wefelmeyer, W. (2004). Estimating linear functionals of the error distribution in nonparametric regression. Journal of Statistical Planning and Inference, 119, 75–93.
Nadaraya, E. A. (1964). On estimating regression. Theory of Probability and its Applications, 9, 141–142.
Neumeyer, N., Dette, H. (2007). Testing for symmetric error distribution in nonparametric regression models. Statistica Sinica, 17, 775–795.
Neumeyer, N., Van Keilegom, I. (2010). Estimating the error distribution in nonparametric multiple regression with applications to model testing. Journal of Multivariate Analysis, 101, 1067–1078.
Pinsker, M. S. (1980). Optimal filtering of a square integrable signal in Gaussian white noise. Problems of Information Transmission, 16, 52–68.
Sakia, R. M. (1992). The Box-Cox transformation technique: A review. The Statistician, 41, 169–178.
Samb, R. (2011). Nonparametric estimation of the density of regression errors. Comptes Rendus de l’Académie des Sciences-Paris, Série I, 349, 1281–1285.
Shin, Y. (2008). Semiparametric estimation of the Box-Cox transformation model. Econometrics Journal, 11, 517–537.
Vanhems, A., Van Keilegom, I. (2011). Semiparametric transformation model with endogeneity: A control function approach. Journal of Econometrics (under revision).
Watson, G. S. (1964). Smooth regression analysis. Sankhy \(\overline{a}\)-Series A, 26, 359–372.
Zellner, A., Revankar, N. S. (1969). Generalized production functions. Reviews of Economic Studies, 36, 241–250.
Author information
Authors and Affiliations
Corresponding author
Additional information
B. Colling and C. Heuchenne supported by IAP research network P7/06 of the Belgian Government (Belgian Science Policy), and by the contract ‘Projet d’Actions de Recherche Concertées’ (ARC) 11/16-039 of the ‘Communauté française de Belgique’, granted by the ‘Académie Universitaire Louvain’. R. Samb supported by IAP research network P7/06 of the Belgian Government (Belgian Science Policy). I. Van Keilegom supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement No. 203650, by IAP research network P7/06 of the Belgian Government (Belgian Science Policy), and by the contract ‘Projet d’Actions de Recherche Concertées’ (ARC) 11/16-039 of the ‘Communauté française de Belgique’, granted by the ‘Académie Universitaire Louvain’.
About this article
Cite this article
Colling, B., Heuchenne, C., Samb, R. et al. Estimation of the error density in a semiparametric transformation model. Ann Inst Stat Math 67, 1–18 (2015). https://doi.org/10.1007/s10463-013-0441-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-013-0441-x