The -series process (ASP) is widely used as a monotonic stochastic model in the reliability context. So the parameter estimation problem in an ASP is of importance. In this study parameter estimation problem for the ASP is considered when the distribution of the first occurrence time of an event is assumed to be lognormal. The parameters and of the ASP are estimated via maximum likelihood (ML) method. Asymptotic distributions and consistency properties of these estimators are derived. A test statistic is conducted to distinguish the ASP from renewal process (RP). Further, modified moment (MM) estimators are proposed for the parameters and and their consistency is proved. A nonparametric (NP) novel method is presented to test whether the ASP is a suitable model for data sets. Monte Carlo simulations are performed to compare the efficiencies of the ML and MM estimators. A real life data example is also studied to illustrate the usefulness of the ASP.