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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 Diserahkan: 15
Julai 2013/Diterima: 5 Februari 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 RUJUKAN
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*Pengarang untuk surat-menyurat; email: eyigit@gazi.edu.tr
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