Using the straight line approach of Frees (1986a), we establish estimator for ergodic distribution of inventory model of type ( s , S ) and investigate asymptotic properties of this estimator such as consistency, asymptotic unbiasedness and asymptotic normality.
📌 3 Key Takeaways
🎯
Objective & Hypothesis:We consider inventory model of type ( s , S ) which is used mostly in stock control policy.
📊
Key Findings & Data:It is very important to know characteristics of an inventory model of type ( s, S ), such as stationary distribution.
💡
Conclusion & Impact:Using the straight line approach of Frees (1986a), we establish estimator for ergodic distribution of inventory model of type ( s , S ) and investigate asymptotic properties of this estimator such as consistency, asymptotic unbiasedness and asymptotic normality.
🔬Design:Modeling & Simulation
📍Scope:Comprehensive Sample
⚙️Variables:Inventory model of type (s · Ergodic distribution · Estimation
Abstract
We consider inventory model of type ( s , S ) which is used mostly in stock control policy. It is very important to know characteristics of an inventory model of type ( s, S ), such as stationary distribution. Using the straight line approach of Frees (1986a), we establish estimator for ergodic distribution of inventory model of type ( s , S ) and investigate asymptotic properties of this estimator such as consistency, asymptotic unbiasedness and asymptotic normality.
Gokpinar, E., Khaniyev, T., Gamgam, H., Gokpinar, F. (2015). A new estimator for stationary distribution of the inventory model of type (s, S). Gazi University Journal of Science, Volume: 28 Issue: 1. https://ajindex.com/articles/1269625
Frees, E.W., "Warranty analysis and renewal function estimation", Nav. Res. Log., 33:361-372, (1986a).
Brown, M. and Solomon, H.A., "Second-order approximation for the variance of a renewal-reward process", Stoch. Proc. Appl., 3:301–314, (1975).
Gihman, I.I. and Skorohod, A.V., Theory of Stochastic Processes, II. Springer: Berlin, (1975).
Borovkov, A.A., Stochastic Processes in Queuing Theory, Spinger-Verlag, New York, (1976).
Alsmeyer, G., "Some relations between harmonic renewal measure and certain Şrst passage times", Stat. Probabil. Lett., 12(1):19–27, (1991).
Aras, G. and Woodroofe, M., "Asymptotic expansions for the moments of a randomly stopped average", Ann. Stat., 21:503–519, (1993).
Khaniyev, T., Kesemen, T., Aliyev, R. and Kokangül, A., "Asymptotic expansions for the moments of a semi-Markovian random walk with exponentional distributed interference of chance", Stat. Probabil. Lett., 78(6):785–793, (2008).
Prabhu,N.U., Stochastic Storage Processes, Springer-Verlag, New York, (1981).
Sahin, I., "On the continuous-review (s, S) inventory model under compound renewal demand and random lead times", J. Appl. Probab., 20:213–219, (1983).
Zheng, Y.S. and Federgruen, A., "Computing an optimal (s,S) policy is as easy as a single evaluation of the cost function", Oper. Res., 39:654–665, (1991).
Chen, F. and Zheng, Y., "Waiting time distribution in (T,S) inventory systems", Oper. Res. Lett., 12:145– 151, (1992).
Sethi, S.P. and Cheng, F., "Optimality of (s, S) policies in inventory models with Markovian demand", Oper. Res., 45(6):931–939, (1997).
Janssen, F., Heuts, R. and Kok, T., "On the (R, s, Q) inventory model when demand is modeled as a compound Bernoulli process", Eur. J. Oper. Res., 104, 423–436, (1998).
Heisig, G., "Comparison of (s,S) and (s,nQ) inventory control rules with respect to planning stability", Int. J. Prod., 73:59–82, (2001).
Gavirneni, S., "An efficient heuristic for inventory control when the customer is using a (s; S) policy", Oper. Res. Lett., 28:187–192, (2001).
Khaniev, T. and Mammadova, Z., "On the stationary characteristics of the extended model of type (s,S) with Gaussian distribution of summands", J. Stat. Comput. Sim., 76(10):861–874, (2006).
Khaniyev, T. and Atalay, K. "On the weak convergence of the ergodic distribution for an inventory model of type (s, S)", Hacet. J. Math. Stat., 39(4):599- 611, (2010).
Khaniyev, T., Kokangül, A. and Aliyev, R., "An asymptotic approach for a semi-markovian inventory model of type (s, S)", Appl. Stoch. Model. in Bus., 29:439-453, (2013).
Vardi, Y., "Nonparametric estimation in renewal processes", Ann. Stat., 10: 772-785, (1982).
Frees, E.W., "Nonparametric renewal function estimation", Ann. Stat., 14(4), 1366-1378, (1986b).
Grübel, R. and Pitts, S., "Nonparametric estimation in renewal theory 1: the empirical renewal function", Ann. Stat., 21(3):1431-145, (1993).
Zhao, Q. and Rao, S.S., "Nonparametric renewal function estimation based on estimated densities", Asia-Pac. J. Oper. Res., 14:115-12, (1997). Guedon,
"Nonparametric estimation of renewal processes from count data", Can. J. Stat., 3(12), (2003). C., Markovich,
"Nonparametric estimation of the renewal function by N. empirical data", Stoch. Models, 22:175-199, (2006).
Bebbington, M., Davydov, Y. and Zitikis, R., "Estimating the renewal function when the second moment is infinite", Stoch. Models, 23:27-48, (2007).
Necir, A., Rassoul A. and Meraghni D., “POT- Based Estimation of the Renewal Function of Interoccurrence Times of Heavy Tailed Risks”, J. Probab. Stat., (2010).
Abdelaziz, R., “Estimating of the Renewal Function with heavy-tailed claims”, World Acad. Sci. Eng. Tech., 6:2-21, (2012).
Nasirova, T.I.,Yapar, D. and Khaniev, T., "Probability Characteristics of the Inventory Level in (s,S) Model", Cybemetics and Systems Analysis, 34(5):689-695, (1998).
Feller, W. An introduction to probability theory and its applications, Vol. 2. Second edition, New York: Wiley, (1971).
Lehmann, E.L., Elements of large samples theory, New York, Springer, (1998).
Casella, G. and Berger, R.L., Statistical Inference. 2nd edn. Pacific Grove, CA, Duxbury/Thomson Learning, (2002). Appendix A
Proof. The covariance between X k and X m is obtained as follows: ( ) ( E X Cov X,Xm= k, ) ( ) k k m. Xm− µm =E X Xm− µ µ. Here E X X m k term is obtained as
Prabhu,N.U., Stochastic Storage Processes, Springer-Verlag, New York, (1981).