Nonparametric estimation in [formula omitted]-series processes

Abstract

A counting process { N ( t ) , t ≥ 0 } with the interoccurrence times X 1 , X 2 , … is an α -series process if there exists a real number α such that ( k α X k ) k = 1 , 2 , … forms a renewal process. The nonparametric inference problem in an α -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 α -series process. Some nonparametric estimators for three important parameters of the α -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.