On the Local Polynomial Estimators of the Counting Process Intensity Function and its Derivatives

We consider the properties of the local polynomial estimators of a counting process intensity function and its derivatives. By expressing the local polynomial estimators in a kernel smoothing form via effective kernels, we show that the bias and variance of the estimators at boundary points are of the same magnitude as at interior points and therefore the local polynomial estimators in the context of intensity estimation also enjoy the automatic boundary correction property as they do in other contexts such as regression. The asymptotically optimal bandwidths and optimal kernel functions are obtained through the asymptotic expressions of the mean square error of the estimators. For practical purpose, we suggest an effective and easy-to-calculate data-driven bandwidth selector. Simulation studies are carried out to assess the performance of the local polynomial estimators and the proposed bandwidth selector. The estimators and the bandwidth selector are applied to estimate the rate of aftershocks of the Sichuan earthquake and the rate of the Personal Emergency Link calls in Hong Kong.