Geometric process

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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.[6] Given a sequence of non-negative random variables , if and the marginal distribution of is given by , where is a positive constant, then is called a semi-geometric process

References[edit]

  1. ^ Lam, Y. (1988). Geometric processes and replacement problem. Acta Mathematicae Applicatae Sinica. 4, 366–377
  2. ^ Lam, Y. (2007). Geometric process and its applications. World Scientific, Singapore MATH. ISBN 978-981-270-003-2.
  3. ^ Braun, W. J., Li, W., & Zhao, Y. Q. (2005). Properties of the geometric and related processes. Naval Research Logistics (NRL), 52(7), 607–616.
  4. ^ 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.
  5. ^ Wu, S. (2017). Doubly geometric processes and applications. Journal of the Operational Research Society, 1–13. doi:10.1057/s41274-017-0217-4.
  6. ^ Wu, S., Wang, G. (2017). The semi-geometric process and some properties. IMA J Management Mathematics, 1–13.