Abstract
In this paper, the moments of a class of reward processes defined on a discrete-time semi-Markov process and the asymptotic behaviors of the corresponding empirical estimators have been investigated. Some known results concerning the asymptotic distribution and properties of semi-Markov kernel have been obtained by a different approach. By using the empirical estimator of the semi-Markov kernel and the mentioned approach, the estimators for the moments of the reward process have been introduced and their asymptotic properties have been established. As a consequence of the strong consistency and asymptotic normality, the confidence intervals have also been obtained. A numerical example illustrates the results.
Similar content being viewed by others
References
Anderson TW, Goodman LA (1957) Statistical inference about Markov chain. Ann math Statist 28:89–110
Barbu V, Boussmart M, Limnios N (2004) Discrete time semi-Markov model for reliability and survival analysis. Communications in Statistics – Theory and Methods 33(11):2833–2868
Barbu V, Limnios N (2006) Empirical estimator for discrete-time semi-Markov processes with applications in reliability. Journal of Nonparametric statistics 18:483–498
Cinlar E (1969) Markov renewal theory. Adv Appl Prob 1:123–187
D’Amico G (2009) Nonparametric estimation of the accumulated reward for semi-Markov chains. SORT 33(2):159–170
D’Amico G (2010) Measuring the quality of life through Markov reward process: Analysis and inference. Environmetrics 21:208–220
D’Amico G, Guillen M, Manca R (2013) Semi-Markov Disability Models. Communications in Statistics: Theory and Methods 42(16):2172–2188
D’Amico G, Janssen J, Manca R (a 2011) A Non-Homogeneous Semi-Markov Reward Model for the Credit Spread Computation. Int J Theo Appl Finance 14 (2):221–238
D’Amico G, Manca R, Salvi G (2013) A Semi-Markov Modulated Interest Rate Model. Statistics and Probability Letters 83:2094–2102
Janssen J, Manca R (2006) Applied Semi-Markov processes. Springer, New York
Khorshidian K, Soltani AR (2002) Asymptotic behavior of multivariate reward processes with nonlinear reward functions. Bulletin of the Iranian Mathematical society 28(2):1–17
Khorshidian K (2008) Strong law large numbers of semi-Markov reward processes. Asian Journal of Mathematics and Statistics 1(1):57–62
Khorshidian K (2009) Central Limit Theorem for Nonlinear Semi-Markov Reward processes. Stoch Anal Appl 27(4):656–670
Limnios N (2004) A functional central limit theorem for the empirical estimator of a semi-Markov kernel. J Nonparametric statistics 16(1–2):13–18
Ouhbi B, Limnios N (1999) Nonparametric estimation for Markov processes based on its hazard rate. Stat Infer Stochat Proc 2(2):151–173
Ouhbi B, Limnios N (2003) Nonparametric reliability estimation of semi-Markov processes. J Statist Plann Infer 109(1–2):155–165
Pyke R (1961a) Markov renewal process: definitions and preliminary properties. Ann Math Statist 32:1231–1242
Sadek A, Limnios N (2002) Asymptotic properties for maximum likelihood estimators for reliability and failure rates of Markov chains. Communication in Statistics-Theory and Methods 31(10):1837–1861
Soltani AR (1996) Reward processes with nonlinear reward function. J Appl Prob 33:1101–1017
Soltani AR, Khorshidian K (1998) Reward processes for semi-Markov processes: Asymptotic behavior. J Appl Prob 35:833–842
Soltani AR, Khorshidian K, Ghafaripour A (2010) Prediction for reward processes. Stoch Model 26:242–255
Van Der Vart AW (2000). Asymptotic statistics, Cambridge series in statistical and probability mathematics, 3, Cambridge University Press
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Khorshidian, K., Negahdari, F. & Mardnifard, H.A. Estimation for Discrete-time Semi-Markov Reward Processes: Analysis and Inference. Methodol Comput Appl Probab 18, 885–900 (2016). https://doi.org/10.1007/s11009-015-9468-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-015-9468-1
Keywords
- Discrete-time semi-Markov processes
- Reward process
- Empirical estimator
- Delta method
- Asymptotic properties