Geometric process

{{Technical|date=August 2020}}

In probability, statistics and related fields, the geometric process is a counting process, introduced by Lam in 1988.Lam, Y. (1988). [https://dx.doi.org/10.1007/BF02007241 Geometric processes and replacement problem]. Acta Mathematicae Applicatae Sinica. 4, 366–377 It is defined as

The geometric process. Given a sequence of non-negative random variables :$\backslash \{X\_k,k=1,2,\; \backslash dots\backslash \}$, if they are independent and the cdf of $X\_k$ is given by $F(a^\{k-1\}x)$ for $k=1,2,\; \backslash dots$, where $a$ is a positive constant, then $\backslash \{X\_k,k=1,2,\backslash ldots\backslash \}$ is called a geometric process (GP).

The GP has been widely applied in reliability engineeringLam, Y. (2007). Geometric process and its applications. World Scientific, Singapore MATH. {{ISBN|978-981-270-003-2}}.

Below are some of its extensions.

• The α- series process.Braun, W. J., Li, W., & Zhao, Y. Q. (2005). [https://dx.doi.org/10.1002/nav.20099 Properties of the geometric and related processes]. Naval Research Logistics (NRL), 52(7), 607–616. Given a sequence of non-negative random variables:$\backslash \{X\_k,k=1,2,\; \backslash dots\backslash \}$, if they are independent and the cdf of $\backslash frac\{X\_k\}\{k^a\}$ is given by $F(x)$ for $k=1,2,\; \backslash dots$, where $a$ is a positive constant, then $\backslash \{X\_k,k=1,2,\backslash ldots\backslash \}$ is called an α- series process.
• The threshold geometric process.Chan, J.S., Yu, P.L., Lam, Y. & Ho, A.P. (2006). [https://dx.doi.org/10.1002/sim.2376 Modelling SARS data using threshold geometric process]. Statistics in Medicine. 25 (11): 1826–1839. A stochastic process $\backslash \{Z\_n,\; n\; =\; 1,2,\; \backslash ldots\backslash \}$ is said to be a threshold geometric process (threshold GP), if there exists real numbers $a\_i\; >\; 0,\; i\; =\; 1,2,\; \backslash ldots\; ,\; k$ and integers such that for each $i\; =\; 1,\; \backslash ldots\; ,\; k$, forms a renewal process.
• The doubly geometric process.Wu, S. (2018). [https://dx.doi.org/10.1057/s41274-017-0217-4 Doubly geometric processes and applications]. Journal of the Operational Research Society, 69(1) 66-77. {{doi|10.1057/s41274-017-0217-4}}. Given a sequence of non-negative random variables :$\backslash \{X\_k,k=1,2,\; \backslash dots\backslash \}$, if they are independent and the cdf of $X\_k$ is given by $F(a^\{k-1\}x^\{h(k)\})$ for $k=1,2,\; \backslash dots$, where $a$ is a positive constant and $h(k)$ is a function of $k$ and the parameters in $h(k)$ are estimable, and $h(k)>0$ for natural number $k$, then $\backslash \{X\_k,k=1,2,\backslash ldots\backslash \}$ is called a doubly geometric process (DGP).
• The semi-geometric process.Wu, S., Wang, G. (2017). [https://academic.oup.com/imaman/article/doi/10.1093/imaman/dpx002/3829520/The-semigeometric-process-and-some-properties?guestAccessKey=eaf4f88c-24d4-4791-9e28-607b8a460d12 The semi-geometric process and some properties]. IMA J Management Mathematics, 1–13. Given a sequence of non-negative random variables $\backslash \{X\_k,\; k=1,2,\backslash dots\backslash \}$, if and the marginal distribution of $X\_k$ is given by , where $a$ is a positive constant, then $\backslash \{X\_k,\; k=1,2,\backslash dots\backslash \}$ is called a semi-geometric process
• The double ratio geometric process.Wu, S. (2022) [https://onlinelibrary.wiley.com/doi/full/10.1002/nav.22021 The double ratio geometric process for the analysis of recurrent events]. Naval Research Logistics, 69(3) 484-495. Given a sequence of non-negative random variables $\backslash \{Z\_k^D,k=1,2,\; \backslash dots\backslash \}$, if they are independent and the cdf of $Z\_k^D$ is given by $F\_k^D(t)=1-\backslash exp\backslash \{-\backslash int\_0^\{t\}\; b\_k\; h(a\_k\; u)\; du\backslash \}$ for $k=1,2,\; \backslash dots$, where $a\_k$ and $b\_k$ are positive parameters (or ratios) and $a\_1=b\_1=1$. We call the stochastic process the double-ratio geometric process (DRGP).