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

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*Pengarang untuk surat-menyurat; email: eyigit@gazi.edu.tr

 

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