Two-step Smoothing Estimation of the Time-variant Parameter with Application to Temperature Data

In this article, we develop two nonparametric smoothing estimators for parameter of a time-variant parametric model. This parameter can be from any parametric family or from any parametric or semi-parametric regression model. Estimation is based on a two-step procedure, in which we first get the raw estimate of the parameter at a set of disjoint time points and then compute the final estimator at any time by smoothing the raw estimators. We will call these estimators two-step local polynomial smoothing estimator and two-step kernel smoothing estimator. We derive these two two-step smoothing estimators by modeling raw estimates of the time-variant parameter from any regression model or probability model and then establish a mathematical relationship between these two estimators. Our two-step estimation method is applied to temperature data from Dhaka, the capital city of Bangladesh. Extensive simulation studies under different cross-sectional and longitudinal frameworks have been conducted to check the finite sample MSE of our estimators. Narrower bootstrap confidence bands and smaller MSEs from application and simulation results show the superiority of the local polynomial smoothing estimator over the kernel smoothing estimator