Nonparametric estimation in αα-series processes

A counting process {N(t),t≥0} with the interoccurrence times X1,X2,…X1,X2,… is an αα-series process if there exists a real number αα such that (kαXk)k=1,2,…(kαXk)k=1,2,… forms a renewal process. The nonparametric inference problem in an αα-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 αα-series process. Some nonparametric estimators for three important parameters of the αα-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.

► We propose some nonparametric estimators for the parameters of an αα-series process.
► The consistency and asymptotic normality properties are investigated.
► Some suggestions on the choice of the estimators are made.
► The coal-mining disasters data set is used for application purposes.