Geometric process

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  :${\displaystyle \{X_{k},k=1,2,\dots \}}$, if they are independent and the cdf of ${\displaystyle X_{k}}$ is given by ${\displaystyle F(a^{k-1}x)}$ for ${\displaystyle k=1,2,\dots }$, where ${\displaystyle a}$ is a positive constant, then ${\displaystyle \{X_{k},k=1,2,\ldots \}}$ 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:${\displaystyle \{X_{k},k=1,2,\dots \}}$, if they are independent and the cdf of ${\displaystyle {\frac {X_{k}}{k^{a}}}}$ is given by ${\displaystyle F(x)}$ for ${\displaystyle k=1,2,\dots }$, where ${\displaystyle a}$ is a positive constant, then ${\displaystyle \{X_{k},k=1,2,\ldots \}}$ is called an α- series process.
• The threshold geometric process. ^[4] A stochastic process ${\displaystyle \{Z_{n},n=1,2,\ldots \}}$ is said to be a threshold geometric process (threshold GP), if there exists real numbers ${\displaystyle a_{i}>0,i=1,2,\ldots ,k}$ and integers ${\displaystyle \{1=M_{1}<M_{2}<\cdots <M_{k}<M_{k+1}=\infty \}}$ such that for each ${\displaystyle i=1,\ldots ,k}$, ${\displaystyle \{a_{i}^{n-M_{i}}Z_{n},M_{i}\leq n<M_{i+1}\}}$ forms a renewal process.
• The doubly geometric process. ^[5] Given a sequence of non-negative random variables :${\displaystyle \{X_{k},k=1,2,\dots \}}$, if they are independent and the cdf of ${\displaystyle X_{k}}$ is given by ${\displaystyle F(a^{k-1}x^{h(k)})}$ for ${\displaystyle k=1,2,\dots }$, where ${\displaystyle a}$ is a positive constant and ${\displaystyle h(k)}$ is a function of ${\displaystyle k}$ and the parameters in ${\displaystyle h(k)}$ are estimable, and ${\displaystyle h(k)>0}$ for natural number ${\displaystyle k}$, then ${\displaystyle \{X_{k},k=1,2,\ldots \}}$ is called a doubly geometric process (DGP).
• The semi-geometric process. ^[6] Given a sequence of non-negative random variables ${\displaystyle \{X_{k},k=1,2,\dots \}}$, if ${\displaystyle P\{X_{k}<x|X_{k-1}=x_{k-1},\dots ,X_{1}=x_{1}\}=P\{X_{k}<x|X_{k-1}=x_{k-1}\}}$ and the marginal distribution of ${\displaystyle X_{k}}$ is given by ${\displaystyle P\{X_{k}<x\}=F_{k}(x)(\equiv F(a^{k-1}x))}$, where ${\displaystyle a}$ is a positive constant, then ${\displaystyle \{X_{k},k=1,2,\dots \}}$ is called a semi-geometric process
• The double ratio geometric process. ^[7] Given a sequence of non-negative random variables ${\displaystyle \{Z_{k}^{D},k=1,2,\dots \}}$, if they are independent and the cdf of ${\displaystyle Z_{k}^{D}}$ is given by ${\displaystyle F_{k}^{D}(t)=1-\exp\{-\int _{0}^{t}b_{k}h(a_{k}u)du\}}$ for ${\displaystyle k=1,2,\dots }$, where ${\displaystyle a_{k}}$ and ${\displaystyle b_{k}}$ are positive parameters (or ratios) and ${\displaystyle a_{1}=b_{1}=1}$. We call the stochastic process the double-ratio geometric process (DRGP).

References

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. (2018). Doubly geometric processes and applications. Journal of the Operational Research Society, 69(1) 66-77. 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.
7. Wu, S. (2022) The double ratio geometric process for the analysis of recurrent events. Naval Research Logistics, 69(3) 484-495.