On Semiparametric Mode Regression Estimation By ALI GANNOUN Chaire de Statistique Appliqu´ ee and CEDRIC CNAM 292, Rue Saint-Martin 75141 Paris, France JEROME SARACCO Institut de Math´ ematiques de Bordeaux, UMR CNRS 5251, Universit´ e Bordeaux 1 351 cours de la lib´ eration 33405 TALENCE Cedex and KEMING YU Department of Mathematical Sciences Brunel University, Uxbridge, UB8, 3PH, United Kingdom Abstract It has been found that, for a variety of probability distributions, there is a surprising linear relation between mode, mean and median. In this paper, the relation between mode, mean and median regression functions is assumed to follow a simple parametric model. We pro- pose a semiparametric conditional mode (mode regression) estimation for an unknown (unimodal) conditional distribution function in the context of regression model, so that any m-step-ahead mean and me- dian forecasts can then be substituted into the resultant model to deliver m-step-ahead mode prediction. In the semiparametric model, Least Squared Estimator (LSEs) for the model parameters and the simultaneous estimation of the unknown mean and median regres- sion functions by the local linear kernel method are combined to infer about the parametric and nonparametric components of the proposed model. The asymptotic normality of these estimators is derived, and the asymptotic distribution of the parameter estimates is also given and is shown to follow usual parametric rates in spite of the pres- ence of the nonparametric component in the model. These results are applied to obtain a data-based test for the dependence of mode regression over mean and median regression under a regression model. Keywords. Asymptotic normality, hypothesis testing, local linear kernel estimate, mode, prediction, rate of convergence, semiparametric regression 1