Discrete Beta-Exponential Distribution

Consider the estimation of the regression parameters in the usual linear model. For design densities with infinite support, it has been shown by Faraldo Roca and González Manteiga 1] Faraldo Roca, P. and González Manteiga, W. 1987. “Efficiency of a new class of linear regression estimates obtained by preliminary nonparametric estimation”. In New Perspectives in Theoretical and Applied Statistics Edited by: Puri, M. L., Vilaplana, J. P. and Wertz, W. 229–242. New York: John Wiley.  Google Scholar] that it is possible to modify the classical least squares procedure and to obtain estimators for the regression parameters whose MSE's (mean squared errors) are smaller than those of the usual least squares estimators. The modification consists of presmoothing the response variables by a kernel estimator of the regression function. These authors also show that the gain in efficiency is not possible for a design density with compact support. We show that in this case local linear presmoothing does not fix this inefficiency problem, in spite of the well known fact that local linear fitting automatically corrects the bias in the endpoints of the (design density) support. We demonstrate on a theoretical basis how this inefficiency problem can be rectified in the compact design case: we prove that presmoothing with boundary kernels studied in Müller 2] Müller, H.-G. 1991. Smooth optimum kernel estimators near endpoints. Biometrika, 78: 521–530. Crossref], Web of Science ®] , Google Scholar] and Müller and Wang 3] Müller, H.-G. and Wang, J.-L. 1994. Hazard rate estimation under random censoring with varying kernels and bandwidths. Biometrics, 50: 61–76. Crossref], PubMed], Web of Science ®] , Google Scholar] leads to regression estimators which are superior over the least squares estimators. A very careful analytic treatment is needed to arrive at these asymptotic results.