Abstract
In the present, we introduce and study the \(\mathcal{G-}\)inhomogeneous Markov system of high order, which is a more general in many respects stochastic process than the known inhomogeneous Markov system. We define the inhomogeneous superficial razor cut mixture transition distribution model extending for the homogeneous case the idea of the mixture transition model. With the introduction of the appropriate vector stochastic process and the establishment of relationships among them, we study the asymptotic behaviour of the \(\mathcal{G-}\)inhomogeneous Markov system of high order. In the form of two theorems, the asymptotic behaviour of the inherent \(\mathcal{G-}\)inhomogeneous Markov chain and the expected and relative expected population structure of the \(\mathcal{G-}\)inhomogeneous Markov system of high order, are provided under assumptions easily met in practice. Finally, we provide an illustration of the present results in a manpower system.
Similar content being viewed by others
References
Adke SR, Deshmukh SR (1988) Limit distribution of high order Markov chain. J R Stat Soc B (1):105–108
Anderson TWW, Goodman LA (1957) Statistical inference about Markov chains. Ann Math Stat 28:89–110
Barbu V, Boussemart M, Limnios N (2004) Discrete-time semi Markov model for reliability and survival analysis. Commun Stat Theory Methods 33:2833–2868
Bartholomew DG (1963) A multistage renewal process. J R Stat Soc B 25:150–168
Bartholomew DG (1982) Stochastic models for social processes, 3rd edn. Wiley, New York
Bertchtold A (1998) Chaines de Markov et modeles de transition: applications aux sciences sociales. HERMES, Paris
Bertchtold A (2001) Estimation in the mixture transition distribution model. J Time Ser Anal 22:379–397
Bertchtold A, Raftery AE (2002) The mixture transition distribution model fo high-order Markov chains and non-Gaussian time series. Stat Sci 17(3):328–356
Bielecki TR, Rutkowski M (2004) Credit risk: modeling, valuation and hedging. Springer, New York
Bollerslev T, Chou RY, Kroner KF (1992) ARCH modeling in finance: a review of the theory and empirical evidence. J Econom 52:5–59
Craig P (1989) Time series analysis of directional data. Ph.D. thesis, Dept. Statistics, Trinity College, Dublin
Crooks GE (2000) Path ensemble averages in system driven far from equilibrium. Phys Rev E 61(3):2361–2366
De Freyter T (2006) Modelling heterogeneity in manpower planning: dividing the personel system into more homogeneous subgroups. Appl Stoch Model Bus 22(4):321–334
Foucher Y, Mathew E, Saint Pierre P, Durand J-F, Daures JP (2005) A semi-Markov model based on generalized Weibull distribution with an illustration for HIV disease. Biom J 47(6):1–9
Gantmacher FR (1959) Applications of the theory of matrices. Interscience, New York
Hamilton JD (1994) Time series analysis. Princeton University Press, Princeton
Horn RA, Johnson CR (1990) Matrix analysis. Cambridge University Press, Cambridge
Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, Cambridge
Huang CC, Isaacson D, Vinograde B (1976) The rate of convergence of certain nonhomogeneous Markov chains. Z Wahrscheinlichkeitstheor Verw Geb 35(2):141–146
Iosifescu M (1980) Finite Markov processes and their applications. Wiley, New York
Isaakson DL, Madsen RW (1985) Markov chains. Theory and applications. Krieger, Malabar
Limnios N, Oprisan G (2003) An introduction to semi-Markov processes with application to reliability. In: Shanbhag DN, Rao CR (eds) Handbook of statistics, vol 21. Elsevier, Amsterdam, pp 515–556
Mathew E, Foucher Y, Dellamonica P, Daures, J-P (2006) Parametric and non-homogeneous semi-Markov process for HIV control. Working paper. Archet Hospital, Nice
McClean SI, Gribbin JO (1987) Estimation for incomplete manpower data. Appl Stoch Models Data Anal 3:13–25
McClean SI, Gribbin JO (1991) A non parametric competing risks model for manpower planning. Appl Stoch Models Data Anal 7:327–341
McClean SI, Scotney B, Robinson S (2003) Conceptual clustering of heterogeneous gene expression sequences. Artif Intell Rev 20:53–73
MacDonald IL, Zucchini W (1997) Hidden Markov and other models for discrete-valued time series. Chapman and Hall, London
Nilakantan K, Raghavendra BG (2005) Control aspects in proportionality Markov manpower systems. Appl Math Model 29(1):85–116
Ocana-Riola R (2005) Non-homogeneous Markov processes for biomedical data analysis. Biom J 47(3):369–376
Pegram GGS (1980) An autoregressive model for multilag Mrarkov chains. J Appl Probab 17:350–362
Patoucheas PD, Stamou G (1993) Non homogeneous Markovian models in ecological modelling: a study of the zoobenthos dynamics in Thermaikos Gulf, Greece. Ecol Model 66:197–215
Perez-Ocon R, Castro JER (2004) A semi-Markov model in biomedical studies. Commun Stat Theory Methods 33(3):437–455
Raftery AE (1985) A model for high-order Markov chains. J R Stat Soc B 47:528–539
Raftery AE, Tavare S (1994) Estimation and modelling repeated patterns in high order Markov chains with the mixture transition distribution model. Appl Stat Sci 1:403–423
Seneta E (1981) Non negative matrices and Markov chains. Springer, New York
Sen Gupta A, Ugwuowo FI (2006) Modelling multi-stage processes through multivariate distributions. J Appl Stat 33(2):175–187
Saint Pierre P (2005) Modelles multi-etats de type Markovien et application a l asthme. Ph.D. thesis, Universite Montpellier I
Tweedie RL (1981) Operator geometric stationary distributions for Markov chains with applications to queuing models. Adv Appl Probab 14:368–391
Vasileiou A, Vassiliou P-CG (2006) An inhomogeneous Semi-Markov model for the term structure of credit risk spreads. Adv Appl Probab 38:171–198
Vassiliou P-CG (1976) A Markov chain model for watage in manpower systems. Oper Res Q 27:57–70
Vassiliou P-CG (1978) A high order non-linear Markovian model for promotion in manpower systems. J R Stat Soc A 137:584–595
Vassiliou P-CG (1981) On the limiting behaviour of a nonhomogeneous Markov system. Biometrika 68(2):557–564
Vassiliou P-CG (1982) Asymptotic behavior of Markov systems. J Appl Probab 19:815–857
Vassiliou P-CG (1997) The evolution of the theory of non-homogeneous Markov systems. Appl Stoch Models Data Anal 13(3–4):159–176
Vassiliou P-CG, Papadopoulou AA (1992) Nonhomogeneous semi-Markov systems and maintainability of the state sizes. J Appl Probab 29(3):519–534
Vassiliou P-CG, Tsaklidis G (1989) The rate of convergence of the vector of variances and covariances in non-homogeneous Markov systems. J Appl Probab 27:776–783
Ugwuowo FI, McClean SI (2000) Modelling heterogeneity in a manpower system: a review. Appl Stoch Models Bus Ind 2:99–110
Yadavalli VSS, Natarajan R, Udayabhaskaran S (2002) Optimal training policy for promotion - stochastic models of manpower systems. Electron Publ 13(1):13–23
Young A, Almond G (1961) Predicting distributions of staff. Comput J 3:246–250
Young A, Vassiliou P-CG (1974) A non-linear model on the promotion of staff. J R Stat Soc A 137:584–595
Author information
Authors and Affiliations
Corresponding author
Additional information
T. P. Moysiadis was supported by a research assistanship I.K.Y.
Rights and permissions
About this article
Cite this article
Vassiliou, PC.G., Moysiadis, T.P. \(\boldsymbol{\mathcal{G}-}\)Inhomogeneous Markov Systems of High Order. Methodol Comput Appl Probab 12, 271–292 (2010). https://doi.org/10.1007/s11009-009-9143-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-009-9143-5
Keywords
- Non homogeneous Markov chains
- Markov population systems
- Asymptotic behaviour
- Mixture transition distribution models