A new estimator for stationary distribution of the inventory model of type (s, S)

Yıl 2015, Cilt: 28 Sayı: 1, 87 - 101, 23.02.2015

Esra Gokpinar Tahir Khaniyev Hamza Gamgam Fikri Gokpinar

Öz

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.

Anahtar Kelimeler

Inventory model of type (s, S), Ergodic distribution, Estimation, Consistency, Asymptotic unbiasedness, Asymptotic normality.

Inventory Model of Type (s, S)

Öz

Kaynakça

• 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