Nonparametric estimation in [alpha]-series processes

A counting process with the interoccurrence times X1,X2,... is an [alpha]-series process if there exists a real number [alpha] such that (k[alpha]Xk)k=1,2,... forms a renewal process. The nonparametric inference problem in an [alpha]-series process is taken into consideration. The Mann test is applied for trend analysis and a graphical technique is presented in order to test whether the data come from an [alpha]-series process. Some nonparametric estimators for three important parameters of the [alpha]-series process are obtained by using a linear regression method. The consistency and asymptotic normality properties are investigated. The performances of the estimators are evaluated by a simulation study. Some suggestions on the choice of the estimators are made based on the theoretical and simulation results. Further, the method is illustrated through a real-life example.