LOCAL AND VARIABLE BANDWIDTHS AND LOCAL LINEAR-REGRESSION

Two types of non-global bandwidth, which may be called local and varia ble, have been defined in attempts to improve the performance of kerne l density estimators. In nonparametric regression, local linear fittin g has become a method of much popularity. It is natural, therefore, to consider the use of non-global bandwidths in the local linear context , and indeed local bandwidths are often used. In this paper, it is obs erved that a natural proposal in the literature for combining variable bandwidths with local linear fitting fails in the sense that the resu lting mean squared error properties are those normally associated with local rather than variable bandwidths. We are able to understand why this happens in terms of weightings that are involved. We also attempt to investigate how the bias reduction expected of well-chosen variabl e bandwidths might be achieved in conjunction with local linear fittin g.