Abstract
In the modeling of successive arrival times with a monotone trend, the alpha-series process provides quite successful results. Both selecting the distribution of the first arrival time and making an optimal statistical inference play a crucial role in the modeling performance of the alpha-series process. In this study, when the distribution of the first arrival time is the generalized Rayleigh, the problem of statistical inference for the , , and parameters of the alpha-series process is considered. Further, in order to obtain optimal modeling performance from the mentioned alpha-series process, various estimators for the model parameters are obtained by employing different estimation methodologies such as maximum likelihood, modified maximum spacing, modified least-squares, modified moments, and modified L-moments. By a series of Monte Carlo simulations, the estimation efficiencies of the obtained estimators are evaluated through the different sample sizes. Finally, two real datasets are analyzed to illustrate the importance of modeling with the alpha-series process.
Highlights
Modeling the failure times of an engineering product or a system is quite important in terms of reliability
Let X1, X2, · · ·, Xn be a random sample of size n from an alpha-series process (ASP) with the generalized Rayleigh distribution, and we indicate the estimation of the parameter α as α NL
We have investigated the solution of the statistical inference problem for the ASP with the generalized Rayleigh distribution
Summary
Modeling the failure times of an engineering product or a system is quite important in terms of reliability. It is a general approach to use the renewal process in modeling the non-trending times of successive failures (successive arrival times) of repairable systems. In most cases, successive arrival times for repairable systems may include a trend due to the effects of accumulated wear, aging, or unknown reasons such as changing the maintenance unit and quality of replacement parts. In this case, it would be more appropriate to consider a model with monotonic behavior, which takes into account the trend in the data [1]. Lam [2] introduced the geometric process (GP) for modeling the successive arrival times with a monotone trend. There is a wide range of studies to show the main characteristics of the GP and its performance in the modeling of successive arrival times with a trend; see [1,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]
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