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Sains Malaysiana 44(7)(2015): 1041–1051 Asymptotic Properties of the Straight Line Estimator for a Renewal Function (Sifat
Asimptot bagi
Penganggar Garis Lurus untuk Fungsi
Pembaharuan) ESRA GÖKPINAR1*,
TAHIR
KHANIYEV2,3 & HAMZA GAMGAM1 1Gazi University,
Department of Statistics, 06500 Teknikokullar,
Ankara, Turkey 2Department of Industrial
Engineering, TOBB University of Economics
and Technology 06500 Sogutozu, Ankara, Turkey 3Institute of Cybernetics
of Azerbaijan, National Academy of Sciences, Az 1141, Baku Azerbaijan Received: 15 July
2013/Accepted: 5 February 2015 ABSTRACT
In estimation problems in renewal
function, when the distribution is not known, nonparametric estimators
of renewal function are used. Frees (1986a, Warranty
analysis and renewal function estimation, Naval Res. Logist.
Quart, 33, 361-372) proposed the nonparametric estimator of renewal
function for large values of t. Frees’s
estimator is easy to apply in practice. It is a preferred estimator
for large values of t. However, its statistical properties still
have not been investigated in detailed. For this reason, in this
study, we investigate asymptotic properties of this estimator such
as consistency, asymptotic unbiasedness and asymptotic normality.
Also Monte Carlo simulation study is given to assess the performance
of this estimator according to value of renewal function. Simulation
results indicate that in the large values of t, Frees estimator
is sufficiently close to the renewal function for the Gamma distribution
with various parameters. Keywords: Asymptotic normality;
asymptotic unbiasedness; consistency; nonparametric estimator; renewal
function ABSTRAK
Masalah anggaran dalam
fungsi pembaharuan,
apabila pengagihan tidak diketahui, penganggar tidak parametrik fungsi pembaharuan digunakan. Frees (1986a, analisis waranti dan anggaran
fungsi pembaharuan,
Naval Res. Logist. Quart,
33, 361-372) mencadangkan penganggar
tidak parametrik
fungsi pembaharuan bagi nilai besar
t. Penganggar Frees adalah
mudah untuk digunakan
dalam amalan.
Ia penganggar
yang diutamakan bagi
nilai besar t. Walau bagaimanapun, sifat statistiknya masih tidak dikaji
dengan lebih
mendalam. Untuk alasan ini, dalam kajian ini,
kami mengkaji sifat
asimptot penganggar ini seperti konsistensi,
kesaksamaan asimptot
dan kenormalan asimptot. Juga kajian simulasi
Monte Carlo digunakan untuk
menilai prestasi
penganggar ini mengikut nilai fungsi pembaharuan. Hasil simulasi menunjukkan dalam nilai besar
t, penganggar Frees hampir
dengan fungsi pembaharuan
agihan Gama dengan
pelbagai parameter. Kata kunci: Fungsi
pembaharuan; kenormalan
asimptot; ketaksamaan asimptot; konsisten; penganggar tak parametric REFERENCES
Aydoğdu, H. 1997. Estimation in Renewal Process. Ph.D. Thesis, Ankara University,
Ankara, Turkey (unpublished). Bebbington,
M., Davydov, Y. & Zitikis,
R. 2007.
Estimating the renewal function when the second
moment is infinite. Stochastic Models 23: 27-48. Casella,
G. & Berger, R.L. 2002. Statistical Inference. 2nd ed.
Pacific Grove, CA, Duxbury/Thomson Learning. Feller,
W. 1971.
An Introduction to Probability Theory and Its
Applications. Vol. 2. New Jersey:
Wiley. Frees,
E.W. 1986a.
Warranty analysis and renewal function
estimation. Naval Res. Logist. Quart. 33: 361-372.
Frees,
E.W. 1986b.
Nonparametric renewal function estimation.
Ann. Statist. 14(4): 1366-1378. Grübel, R. & Pitts,
S. 1993. Nonparametric estimation in renewal theory 1: The empirical
renewal function. Ann. Statist. 21(3): 1431-1451. Guedon,
Y. & Cocozza-Thivent, C. 2003. Nonparametric
estimation of renewal processes from count data. The Canadian
Journal of Statistics 31(2): 191-223. Hoog,
R.V., Craig, A. & Mckean, J.W. 2005. Introduction to Mathematical Statistics. 6th ed. New Jersey: Pearson Prentice Hall. Karlin,
S. & Taylor, H. 1975. A First Course in
Stochastic Process. New York: Academic Press.
Khaniyev,
T., Kokangül, A. & Aliyev,
R. 2013.
An asymptotic approach for a semi-markovian
inventory model of type (s, S). Applied
Stochastic Models in Business and Industry 29: 439-453.
Khaniyev,
T. & Atalay, K. 2010. On the weak convergence of the ergodic distribution for an inventory
model of type (s, S). Hacettepe
Journal of Mathematical and Statistic 39(4): 599-611. Khaniev,
T. & Mammadova, Z. 2006. On the stationary characteristics of the extended model of type (s,
S) with Gaussian distribution of summands. Journal of
Statistical Computation and Simulation 76(10): 861-874. Lehmann, E.L. 1998.
Elements of Large Samples Theory.
New York: Springer. Markovich,
M.N. & Krieger, R.U. 2006. Nonparametric estimation of the renewal
function by empirical data. Stochastic
Models 22: 175-199. Ross, M.R. 1996.
Stochastic Processes. New
Jersey: John Wiley. Roussas, G.G. 1997. A Course in Mathematical Statistics. 2nd ed. New York: Academic Press. Schneider,
H., Lin, B.S. & O’cinneide, C. 1990. Comparison of nonparametric estimators for the renewal function, J.
Roy. Statist. Soc. Ser. C
39: 55-61. Tijms, H.C. 2003. A First Course in Stochastic Models. New Jersey:
John Wiley. Vardi, Y. 1982. Nonparametric
estimation in renewal processes. Ann. Statist. 10: 772-785.
Zhao,
Q. & Rao, S.S. 1997. Nonparametric renewal function estimation based on
estimated densities. Asia-Pacific J. Oper.
Res. 14: 115-120. *Corresponding author; email: eyigit@gazi.edu.tr |
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