Nonparametric Renewal Function Estimation and Smoothing by Empirical Data

We consider an estimate of the renewal function (rf) using a limited number of independent observations of the interarrival times for an unknown interarrival-time distribution (itd). The nonparametric estimate is derived from the rf-representation as series of distribution functions (dfs) of consecutive arrival times using a finite summation and approximations of the latter by empirical dfs. Due to the limited number of observed interarrival times the estimate is accurate just for closed time intervals [0, t]. An important aspect is the selection of an optimal number of terms k of the finite sum. Here two methods are proposed: (1) an a priori choice of k as function of the sample size l which provides almost surely (a.s.) the uniform convergence of the estimate to the rf for lightand heavy-tailed itds if the time interval is not too large, and (2) a data-dependent selection of k by the plot of the proposed estimate against k for a fixed time t. To evaluate both the efficiency of the estimate and the selection method of k, a Monte Carlo study is performed.