The asymptotic properties of a general functional of the Gasser–Muller estimator are investigated in the Sobolev space. The convergence rate, consistency, and central limit theorem are established. | Turkish Journal of Mathematics Research Article Turk J Math (2014) 38: 1090 – 1101 ¨ ITAK ˙ c TUB ⃝ doi: Functionals of Gasser–Muller estimators 1 Petre BABILUA1,∗, Elizbar NADARAYA1 , Grigol SOKHADZE1,2 Department of Mathematics, Faculty of Exact and Natural Sciences, I. Javakhishvili Tbilisi State University, Tbilisi, Georgia 2 I. Vekua Institute of Applied Mathematics of I. Javakhishvili Tbilisi State University, Tbilisi, Georgia Received: • Accepted: • Published Online: • Printed: Abstract: The asymptotic properties of a general functional of the Gasser–Muller estimator are investigated in the Sobolev space. The convergence rate, consistency, and central limit theorem are established. Key words: Nonparametric regression, Gasser–Muller estimator, functionals 1. Introduction Many researchers show interest in the study of functionals of probability distribution densities or functionals of regression functions. They consider mostly functionals of the integral type. For instance, integral functionals of a probability density function and its derivatives were studied in [9, 2, 8], whereas the same problems were investigated for a regression function in [3, 6]. A special mention should be made of [5], a work by Goldstein and Messer where the general type functional of a probability density function and the functional of the Nadaraya– Watson regression function were considered. Additionally, in [5], the problem of optimality was studied for a plug-in estimator in a functional space. In the present paper we consider a general functional of the Gasser–Muller regression function. We are concerned with the consistency issues and the conditions under which the central limit theorem is fulfilled. We determine convergence orders and deal with some related problems. An analogous topic was studied in [1] for integral functionals. Let us consider a regression model .
