Üçgensel Müdahaleli Yarı- Markov (s,S) Tipli Modellerin Momentleri İçin Tahmin Ediciler

Yıl 2018, , 430 - 441, 31.07.2018

Tülay Kesemen , Melek Eriş Büyükkaya , Aslı Bektaş Kamışlık

https://doi.org/10.17714/gumusfenbil.396957

Öz

Bu çalışmada üçgensel müdahaleli
(s,S) tipli yarı-Markov bir envanter(stok kontrol) model yenileme süreci ele
alınmış, ve bu sürecin ergodik dağılımının momentleri için tahmin edici problemi
araştırılmıştır. Yenileme süreçlerinde, yenilemeler arasında geçen sürenin
dağılımı tam olarak bilinmediğinde yenileme fonksiyonu için tahmin ediciler literatürde
mevcuttur. Bu çalışmanın ana motivasyonu ise, Frees (1986 b) tarafından
yenileme fonksiyonu için önerilmiş tahmin edicidir. Bu amaçla Frees (1986 b)’
in yaklaşımı kullanılarak ele alınan modelin ergodik dağılımının momentleri
için istatistiksel bir tahmin edici bulunmuş ve bu tahmin edici için sırasıyla
tutarlılık, asimptotik yansızlık ve asimptotik normallik gibi istatistiksel
özellikler araştırılmıştır.

Anahtar Kelimeler

Asimptotik Normallik, Asimptotik Yansızlık, İstatistiksel tahmin edici, (sS) tipli yarı-Markow envanter model, Tutarlılık

Estimators of the Moments for a Semi- Markovian Model of Type (s,S) With a Triangular Distributed Interference of Chance

Öz

In this study inventory
model of type (s,S) with triangular interference of chance has been
investigated by using renewal process and the statistical estimator problem for
moments of ergodic distribution of the process which define the model has been
obtained. Different estimators for the renewal function are available in the
literature, when the distribution which defines the time between renewals is
not known precisely. The main motivation of this study is the publication by
Frees (1986 b) in which he proposed a statistical estimator for the renewal
function. By using the estimator by Frees (1986 b) we obtained a new estimator
for the moments of ergodic distribution function of the process describing
inventory model of type (s,S). Moreover statistical properties such as consistency,
asymptotic unbiasedness and asymptotic normality were investigated for the
obtained estimator, respectively.

Anahtar Kelimeler

Inventory model of type (s S), Asymptotic Normality, Asymptotic Unbiasedness, Statistical estimators, Consistency

Kaynakça

• Bebbington, M., Davydov, Y. ve Zitikis, R., 2007. Estimating the Renewal Function When the Second Moment is Infinite, Stochastic Models, 23(1), 27–48.
• Casella, G. ve Berger, R.L. 2002. Statistical Inference (Second edition). Pacific Grove, CA, Duxbury/ Thomson Learning.
• Feller, W., 1971. An Introduction to Probability Theory and Its Application 2, Wiley, New York, 669 p.
• Frees, E.W., 1986 a. Warranty Analysis and Renewal Function Estimation. Nav. Res. Logist. Q., 33, 361-372.
• Frees, E.W., 1986 b. Nonparametric Renewal Function Estimation. Annals of Statistics, 14(4), 1366-1378.
• Gökpınar, E., Khaniyev, T. ve Gamgam, H., 2015a. Asymptotic Properties of the Straight Line Estimator for a Renewal Function. Sains Malaysiana. 44(7), 1041-1051.
• Gökpınar, E., Khaniyev, T., Gamgam, H. ve Gökpınar, F., 2015b. A New Estimator for Stationary Distribution of the Inventory Model of Type (s,S). Gazi University Journal of Science. 28(1), 87-101.
• Gökpınar, E., Khaniyev, T., Gamgam, H. ve Gökpınar, F., 2016a. Estimators of the Moments for the Invertory Models of Type (s,S) with a Discrete Interference of Chance. Journal of the Turkish Statistical Association. Vol. 9, No. 1, pp. 17-28.
• Gökpınar, E., Khaniyev, T., Gamgam, H. ve Gökpınar, F., 2018. Estimators of the Moments for the Invertory Models of Type (s,S). Iranian Journal of Science and Technology, Transactions A: Science. Vol. 42, No. 1, pp. 5-12.
• Gökpınar, E., 2016b. Yenileme Sürecinde Tahmin Ediciler ve Uygulamaları. Doktora Tezi, Gazi Üniversitesi Fen Bilimleri Enstitüsü. Ankara, 111 s.
• Grübel, R. ve Pitts, S., 1993. Nonparametric Estimation in Renewal Theory 1: the Empirical Renewal Function. Annals of Statistics, 21(3), 1431-145.
• Guedon, Y. ve Cocozza-Thivent, C., 2003. Nonparametric Estimation of Renewal Processes from Count Data. The Canadian Journal of Statistics, 12(3), 191-223.
• Kesemen, T., Bektaş Kamışlık, A., Küçük, Z. ve Şenol, E., 2016. Inventory Model of Type (s,S) with Subexponential Weibull Distributed Demand. Journal of the Turkish Statistical Association, 9 (3), 81-92.
• Khaniyev, T.A. ve Atalay, K.D., 2010. On the Weak Convergence of the Ergodic Distribution for an Inventory Model of Type (s,S)”. Hacettepe Journal of Mathematics and Statistics, 9 (4), 599-611.
• Khaniyev, T.A., Kokangul, A. ve Aliyev, R.T., 2013. An Asymptotic Approach for a Semi-Markovian Inventory Model of Type (s,S). Applied Stochastic Models in Business and Industry, 29 (5), 439 – 453.
• Markovich N.M. ve Krieger, U.R., 2006. Nonparametric Estimation of the Renewal Function by Empirical Data. Stochastic Models, 22 (2), 175–199.
• Necir, A., Rassoul, A. ve Meraghni, D., 2010. POT-Based Estimation of the Renewal Function of Interoccurrence Times of Heavy-Tailed Risks. Journal of Probability and Statistics, doi:10.1155/2010/965672.
• Schneider, H., Lin, B.S. ve O’Cinneide, C., 1990. Comparison of Nonparametric Estimators for the Renewal Function. Applied Statistics, 39(1), 55-61.
• Sgibnev, M.S., 1981. On the Renewal Theorem in the Case of Infinite Variance, Sibirskit Mathematicheskit Zhurnal, 22(5), 178-189. Vardi, Y., 1982. Nonparametric Estimation in Renewal Processes. Annals of Statistics, 10(3), 772-785.
• Zhao, Q. ve Rao, S., 1997. Nonparametric renewal function estimation based on estimated densities. Asia-Pacific Journal of Operational Research, 14 (1), 115–126.