Comparison of geometric and arithmetic means for bandwidth selection in Nadaraya-Watson kernel regression estimator

Nadaraya-Watson kernel regression estimator (NWKRE) is a typical kernel regression estimator which is a kernel-based and non-parametric regression method to estimate the conditional expectation of a random variable and the non-linear mapping from input to output. For NWKRE, the selection of bandwidth, i.e., smoothing parameter h, plays a very important role in the fitting performance. In order to enhance the performance of NWKRE, an adaptive Nadaraya-Watson kernel regression estimator is proposed, ANWKRE for short. There are two main strategies to determine the adaptive or local bandwidth factor λ: geometric mean and arithmetic mean based determination methods, respectively. In this paper, we firstly investigate the mathematical properties of geometric mean and arithmetic mean in the framework of regression analysis. Then, some experimental comparisons are conducted to demonstrate our theoretical results. The experimental results find that the arithmetic mean based ANWKRE can obtain a smoother regression estimation for unknown function.