Geometric process
This article may be too technical for most readers to understand.(August 2020) |
In probability, statistics and related fields, the geometric process is a counting process, introduced by Lam in 1988.[1] It is defined as
The geometric process. Given a sequence of non-negative random variables :, if they are independent and the cdf of is given by for , where is a positive constant, then is called a geometric process (GP).
The GP has been widely applied in reliability engineering[2]
Below are some of its extensions.
- The α- series process.[3] Given a sequence of non-negative random variables:, if they are independent and the cdf of is given by for , where is a positive constant, then is called an α- series process.
- The threshold geometric process.[4] A stochastic process is said to be a threshold geometric process (threshold GP), if there exists real numbers and integers such that for each , forms a renewal process.
- The doubly geometric process.[5] Given a sequence of non-negative random variables :, if they are independent and the cdf of is given by for , where is a positive constant and is a function of and the parameters in are estimable, and for natural number , then is called a doubly geometric process (DGP).
- The semi-geometric process.random variables, if and the marginal distribution of is given by , where is a positive constant, then is called a semi-geometric process
- The double ratio geometric process.random variables, if they are independent and the cdf of is given by for , where and are positiveparameters(or ratios) and . We call the stochastic process the double-ratio geometric process (DRGP).
References
- ^ Lam, Y. (1988). Geometric processes and replacement problem. Acta Mathematicae Applicatae Sinica. 4, 366–377
- ISBN 978-981-270-003-2.
- ^ Braun, W. J., Li, W., & Zhao, Y. Q. (2005). Properties of the geometric and related processes. Naval Research Logistics (NRL), 52(7), 607–616.
- ^ Chan, J.S., Yu, P.L., Lam, Y. & Ho, A.P. (2006). Modelling SARS data using threshold geometric process. Statistics in Medicine. 25 (11): 1826–1839.
- .
- ^ Wu, S., Wang, G. (2017). The semi-geometric process and some properties. IMA J Management Mathematics, 1–13.
- ^ Wu, S. (2022) The double ratio geometric process for the analysis of recurrent events. Naval Research Logistics, 69(3) 484-495.