Abstract
The geometric process (GP) is a simple and direct approach to modeling of the successive inter-arrival time data set with a monotonic trend. In addition, it is a quite important alternative to the non-homogeneous Poisson process. In the present paper, the parameter estimation problem for GP is considered, when the distribution of the first occurrence time is Power Lindley with parameters and . To overcome the parameter estimation problem for GP, the maximum likelihood, modified moments, modified L-moments and modified least-squares estimators are obtained for parameters a, and . The mean, bias and mean squared error (MSE) values associated with these estimators are evaluated for small, moderate and large sample sizes by using Monte Carlo simulations. Furthermore, two illustrative examples using real data sets are presented in the paper.
Highlights
The Renewal process (RP) is a commonly used method for the statistical analysis of the successive inter-arrival times data set observed from a counting process
The main objective of this study extensively investigates the solution of parameter estimation problem for geometric process (GP) when the distribution of the first arrival time is Power Lindley
The parameter estimation problem has been solved from two points of view as the parametric (ML) and nonparametric (MLS, MM and modified L-moment (MLM))
Summary
The Renewal process (RP) is a commonly used method for the statistical analysis of the successive inter-arrival times data set observed from a counting process. GP is first introduced as a direct approach to modeling of the inter-arrival times data with the monotone trend by Lam [1]. Chan et al [5] investigated the parameter estimation problem of GP by assuming that the distribution of random variable X1 was Gamma. The main objective of this study extensively investigates the solution of parameter estimation problem for GP when the distribution of the first arrival time is Power Lindley. The novelty of this paper is that the distribution of first inter-arrival time is assumed to be Power Lindley for GP and the ML and MLM estimators under this assumption are obtained.
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