Research Article
BibTex RIS Cite

Parametric Estimation of the Mean Value and Variance Functions in Type II Counter Process with Constant Locking Time

Year 2019, , 28 - 38, 31.05.2019
https://doi.org/10.29233/sdufeffd.437282

Abstract

In
this study, the mean value and variance functions of type II counter process constituted
as a delayed renewal process are obtained analytically. In the applications of
the delayed renewal process, generally, the functional forms of the
distributions are known but some parameters of the distributions are unknown.
In those cases, the mean value and the variance functions must be estimated
from the data. In this study, after obtaining the mean value and variance
functions of type II counter process analytically, depending on the analytical
expressions, we deal with the problem of estimating these functions and some estimators
are proposed. Performance of the estimators is evaluated for small sample sizes
by a simulation study according to bias and mean square error criteria.

References

  • E. W. Frees, “Warranty analysis and renewal function estimation,” Nav. Res. Logist. Q., vol. 33, pp. 361-372, 1986a.
  • E. W. Frees, “Nonparametric Renewal Function Estimation,” Ann. Statist., vol. 14, no. 4, pp. 1366-1378, 1986b.
  • H. Aydoğdu and F. Öztürk, “Estimation of the variance function in renewal processes,” Journal of Applied Statistical Science, vol.7, no. 4, pp. 191-202, 1999.
  • H. Bohman, “A method to calculate the distribution function when the characteristic function is known,” BIT, vol.10, no. 3, pp. 237-242, 1970.
  • S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, 2nd Edition. New York: Academic Press, 1975.
  • W. L. Smith, “Renewal theory and its ramifications,” J. Roy. Stat. Soc. B., vol. 20, pp. 243-302, 1958.
  • G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes. New York:Oxford University Press Inc., 1992.
  • S. Biswas and H. D. Arora, “On an application of Geiger–Muller counter model (Type-II) for optimization relating to hospital administration,” Acta Med. Int., vol. 4, pp. 16-19, 2017.
  • L. Pal and I. Pazsit, “On some problems in the counting statistics of nuclear particles investigation of the dead time problems,” Nucl. Instrum. Methods Phys. Res. A, vol. 693, pp. 26–50, 2012.
  • R. P. Gardner and L. Lianyan, “On extending the accurate and useful counting rate range of GM counter detector systems,” Appl. Radiat. Isot. Vol. 48, no. 10--12, pp. 1605-1615, 1997.

Sabit Bekleme Zamanlı Tip II Sayaç Sürecinde Ortalama Değer ve Varyans Fonksiyonlarının Parametrik Tahmini

Year 2019, , 28 - 38, 31.05.2019
https://doi.org/10.29233/sdufeffd.437282

Abstract

Bu çalışmada gecikmeli bir
yenileme süreci olarak oluşturulan sabit bekleme zamanlı tip II sayaç sürecinin
ortalama değer ve varyans fonksiyonları analitik olarak elde edilmektedir. Gecikmeli
yenileme süreçleri ile ilgili uygulamalarda genellikle dağılım fonksiyonları
şekilsel olarak bilinirken bazı parametreleri bilinmez. Bu durumda ortalama
değer ve varyans fonksiyonları veriden tahmin edilmek zorundadır. Bu çalışmada
tip II sayaç sürecinin ortalama değer ve varyans fonksiyonları analitik olarak
elde edildikten sonra, analitik ifadelerine bağlı biçimde, parametrik olarak bu
fonksiyonların değerlerinin tahmin edilmesi üzerinde durulmakta ve bazı tahmin
ediciler önerilmektedir. Bu tahmin edicilerin performansları küçük örneklem
hacimleri için yan, hata kareler ortalaması kriterlerine göre bir simülasyon
çalışması ile değerlendirilmektedir.

References

  • E. W. Frees, “Warranty analysis and renewal function estimation,” Nav. Res. Logist. Q., vol. 33, pp. 361-372, 1986a.
  • E. W. Frees, “Nonparametric Renewal Function Estimation,” Ann. Statist., vol. 14, no. 4, pp. 1366-1378, 1986b.
  • H. Aydoğdu and F. Öztürk, “Estimation of the variance function in renewal processes,” Journal of Applied Statistical Science, vol.7, no. 4, pp. 191-202, 1999.
  • H. Bohman, “A method to calculate the distribution function when the characteristic function is known,” BIT, vol.10, no. 3, pp. 237-242, 1970.
  • S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, 2nd Edition. New York: Academic Press, 1975.
  • W. L. Smith, “Renewal theory and its ramifications,” J. Roy. Stat. Soc. B., vol. 20, pp. 243-302, 1958.
  • G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes. New York:Oxford University Press Inc., 1992.
  • S. Biswas and H. D. Arora, “On an application of Geiger–Muller counter model (Type-II) for optimization relating to hospital administration,” Acta Med. Int., vol. 4, pp. 16-19, 2017.
  • L. Pal and I. Pazsit, “On some problems in the counting statistics of nuclear particles investigation of the dead time problems,” Nucl. Instrum. Methods Phys. Res. A, vol. 693, pp. 26–50, 2012.
  • R. P. Gardner and L. Lianyan, “On extending the accurate and useful counting rate range of GM counter detector systems,” Appl. Radiat. Isot. Vol. 48, no. 10--12, pp. 1605-1615, 1997.
There are 10 citations in total.

Details

Primary Language Turkish
Subjects Mathematical Sciences
Journal Section Makaleler
Authors

Mustafa Hilmi Pekalp 0000-0002-5183-8394

Halil Aydoğdu 0000-0001-5337-5277

Publication Date May 31, 2019
Published in Issue Year 2019

Cite

IEEE M. H. Pekalp and H. Aydoğdu, “Sabit Bekleme Zamanlı Tip II Sayaç Sürecinde Ortalama Değer ve Varyans Fonksiyonlarının Parametrik Tahmini”, Süleyman Demirel University Faculty of Arts and Science Journal of Science, vol. 14, no. 1, pp. 28–38, 2019, doi: 10.29233/sdufeffd.437282.