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 :, 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
- The double ratio geometric process.[7] Given a sequence of non-negative random variables , if they are independent and the cdf of is given by for , where and are positive parameters (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
- ^ Lam, Y. (2007). Geometric process and its applications. World Scientific, Singapore MATH. 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. (2018). Doubly geometric processes and applications. Journal of the Operational Research Society, 69(1) 66-77. doi:10.1057/s41274-017-0217-4.
- ^ 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.
- v
- t
- e
Stochastic processes
- Bernoulli process
- Branching process
- Chinese restaurant process
- Galton–Watson process
- Independent and identically distributed random variables
- Markov chain
- Moran process
- Random walk
- Additive process
- Bessel process
- Birth–death process
- Brownian motion
- Cauchy process
- Contact process
- Continuous-time random walk
- Cox process
- Diffusion process
- Dyson Brownian motion
- Empirical process
- Feller process
- Fleming–Viot process
- Gamma process
- Geometric process
- Hawkes process
- Hunt process
- Interacting particle systems
- Itô diffusion
- Itô process
- Jump diffusion
- Jump process
- Lévy process
- Local time
- Markov additive process
- McKean–Vlasov process
- Ornstein–Uhlenbeck process
- Poisson process
- Schramm–Loewner evolution
- Semimartingale
- Sigma-martingale
- Stable process
- Superprocess
- Telegraph process
- Variance gamma process
- Wiener process
- Wiener sausage
- Binomial options pricing model
- Black–Derman–Toy
- Black–Karasinski
- Black–Scholes
- Chan–Karolyi–Longstaff–Sanders (CKLS)
- Chen
- Constant elasticity of variance (CEV)
- Cox–Ingersoll–Ross (CIR)
- Garman–Kohlhagen
- Heath–Jarrow–Morton (HJM)
- Heston
- Ho–Lee
- Hull–White
- Korn-Kreer-Lenssen
- LIBOR market
- Rendleman–Bartter
- SABR volatility
- Vašíček
- Wilkie
- Central limit theorem
- Donsker's theorem
- Doob's martingale convergence theorems
- Ergodic theorem
- Fisher–Tippett–Gnedenko theorem
- Large deviation principle
- Law of large numbers (weak/strong)
- Law of the iterated logarithm
- Maximal ergodic theorem
- Sanov's theorem
- Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
- Cameron–Martin formula
- Convergence of random variables
- Doléans-Dade exponential
- Doob decomposition theorem
- Doob–Meyer decomposition theorem
- Doob's optional stopping theorem
- Dynkin's formula
- Feynman–Kac formula
- Filtration
- Girsanov theorem
- Infinitesimal generator
- Itô integral
- Itô's lemma
- Karhunen–Loève theorem
- Kolmogorov continuity theorem
- Kolmogorov extension theorem
- Lévy–Prokhorov metric
- Malliavin calculus
- Martingale representation theorem
- Optional stopping theorem
- Prokhorov's theorem
- Quadratic variation
- Reflection principle
- Skorokhod integral
- Skorokhod's representation theorem
- Skorokhod space
- Snell envelope
- Stochastic differential equation
- Stopping time
- Stratonovich integral
- Uniform integrability
- Usual hypotheses
- Wiener space
- Actuarial mathematics
- Control theory
- Econometrics
- Ergodic theory
- Extreme value theory (EVT)
- Large deviations theory
- Mathematical finance
- Mathematical statistics
- Probability theory
- Queueing theory
- Renewal theory
- Ruin theory
- Signal processing
- Statistics
- Stochastic analysis
- Time series analysis
- Machine learning
- List of topics
- Category