Generalised kernel smoothing for non-negative stationary ergodic processes

In this paper, we consider a generalised kernel smoothing estimator of the regression function with nonnegative support, using gamma probability densities as kernels, which are non-negative and have naturally varying shapes. It is based on a generalisation of Hille’s lemma and a perturbation idea that allows us to deal with the problem at the boundary. Its uniform consistency and asymptotic normality are obtained at interior and boundary points, under a stationary ergodic process assumption, without using traditional mixing conditions. The asymptotic mean squared error of the estimator is derived and the optimal value of smoothing parameter is also discussed. Graphical illustrations of the proposed estimator are provided for simulated as well as for real data. A simulation study is also carried out to compare our method with the competing local linear method