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A Study Based on Using Pearson Distribution Family on Reliability Analysis

Yıl 2013, Cilt: 15 Sayı: 4, 693 - 703, 20.03.2014

Öz

Abstract

All products or systems that we use in daily life, degrade in time so, they ultimately fail. It is very crucial for manufacturers to forecast the reasons of the failures before. With that perspective reliability analysis is carried out to determine the potential lifetime of products. Failure time’s distribution is the basis of the reliability analysis. While determining the proper distribution, some statistical methods can be used. Cumulative distribution, reliability function, hazard function, mean residual life, variance residual life are most common tools to determine proper distribution in reliability analysis. At the same time failure distributions can be characterized by using relations between these functions.

Pearson Differantial Equation System includes many distributions which are also used in reliability analysis commonly. Because of this it plays a very important role in reliability analysis. In this study, Pearson Differantial Equation System's cubic denominator structure which derives asymmetric distribution will be handled. Then conditional moments and asymmetry measures will be analysed for that structure.

Keywords: Reliability Analysis, Pearson Differantial Equation System, Conditional Moments.

Kaynakça

  • Asadi, M. (1998). Characterization of the pearson system of distributions based on reliability measures. Statistical Papers, 39 (1): 347-360.
  • Elderton, W. P. (1953). Frequency curves and correlation. London: Charles and Edwin Layton.
  • Fisz, M. (1967). Probability theory and mathematical statistics. New York: John Wiley and Sons Inc.
  • Glanzel, W., Telcs, A. ve Schubert, A. (1984). Characterization by truncated moments and ıts application to pearson-type distributions. Zeitschrift für Wahrscheinlichkeitsheorie and Verwandte Gebiete, 66 (2): 173-183.
  • Glanzel, W. (1991). Characterization trough some conditional moments of pearson-type distributions and discrete analogues. The Indian Journal of Statistics, 53 (1): 17-24.
  • Gupta, R. C. ve Bradley, D. M. (2003). Representing the mean residual life in terms of the failure rate. Mathematical and Computer Modelling, 37 (12): 1271-1280.
  • Hogg, R. V. ve Craig, A. T. (1995). Introduction to mathematical statistics. Hong Kong: Higher Education Press.
  • Kotz, S. (1974). Characterizations of statistical distributions: a supplement to recent surveys. International Statistical Review, 42 (1): 39-65.
  • Lawless, J. (2003). Statistical models and methods for lifetime data. New Jersey: John Wiley and Sons Inc.
  • Nair, N. U. ve Sankaran, P. G. (1991). Characterization of the pearson family of distributions. IEE Transactions on Reliability, 40 (1): 75-77.
  • Nair, N. U. ve Sankaran, P. G. (2000). On some reliability aspects of pearson family of distributions. Statistical Papers, 41 (1): 109-117.
  • Navarro, J., Franco, M. ve Ruiz, J. M. (1998). Characterization through moments of the residual life and conditional spacings. The Indian Journal of Statistics, 60 (1): 36-48.
  • Osaki, S. ve Li, X. (1988). Characterizations of gamma and negative binomial distributions. IEE Transactions on Reliability, 37 (4): 379-382.
  • Papathanasiou, V. (1995). A characterization of the pearson system of distributions and the associated orthogonal polynomials. Annals of the Institute of Statistical Mathematics, 47 (1): 171-176.
  • Pearson, K. (1916). Mathematical contributions to the theory of evolution. xıx. second supplement to a memoir on skew variation. Philosophical Transactions of the Royal Society of London, Series A, 216 (1): 429-457.
  • Rausand, M. ve Hoyland, A. (2004). System reliability theory. New Jersey: John Wiley and Sons Inc.
  • Sankaran, P. G. (1992). Characterization of Probability Distributions by Reliability Concepts. Unpublished Doctoral Dissertation. Cochin University of Science and Technology, Department of Statistics, India.
  • Sankaran, P. G., Nair, N. U. ve Sindu, T. K. (2003). A generalized pearson system useful in reliability analysis. Statistical Papers, 44 (1): 125-130.
  • Saraçoğlu, B. ve Çevik, F. (1995). Matematiksel istatistik. Ankara: Gazi Büro Kitabevi.
  • Shakil, M., Kibria, B. M. ve Singh, J. N. (2010). A new family of distributions based on the generalized pearson differantial equation with some applications. Austrian Journal of Statistics, 39 (3): 259-278.
  • Sindu, T. K. (2002). An Extended Pearson System Useful in Reliability Analysis. Unpublished Doctoral Dissertation. Cochin University of Science and Technology, Department of Statistic, India.
  • Stuart, A. ve Ord, J. K., (1987). Kendall’s advanced theory of statistics. New York: Oxford University Press.
  • Şehirlioğlu, A. K. (2011). Pearson dağılış ailesi. Yayınlanmamış Ders Notları. Dokuz Eylül Üniversitesi İktisadi ve İdari Bilimler Fakültesi, İzmir.
  • Ünlü, M. (2013). Pearson dağılış ailesinin güvenilirlik analizinde kullanılması üzerine bir çalışma. Yayınlanmamış Yüksek Lisans Tezi. Dokuz Eylül Üniversitesi Sosyal Bilimler Enstitüsü, İzmir.
  • Wasserman, G. (2002). Reliability verification, testing, and analysis in engineering design. New York: Marcel Dekker Inc.

Pearson Dağılış Ailesinin Güvenilirlik Analizinde Kullanılması Üzerine Bir Çalışma

Yıl 2013, Cilt: 15 Sayı: 4, 693 - 703, 20.03.2014

Öz

Öz

Günlük hayatımızda kullandığımız tüm ürünler veya sistemler zaman içinde yıpranmakta ve bunun sonucunda da bozulmaktadır. Üreticiler açısından bu olası yıpranma ve bozulmaların sebeplerinin önceden bilinmesi hayati önem taşımaktadır. Bu bakış açısıyla ürünlerin potansiyel yaşamlarının belirlenmesi amacına yönelik güvenilirlik analizi çalışmaları yapılmaktadır. Güvenilirlik analizinin temelinde hata sürelerinin dağılımı vardır. Uygun dağılım belirlenirken çeşitli istatistiksel araçlardan yararlanılabilir. Güvenilirlik analizinde genellikle kümülatif dağılım fonksiyonu, güvenilirlik fonksiyonu, hazard fonksiyonu, ortamla artık yaşam fonksiyonu ve artık yaşam varyansı bu dağılımı belirlemede kullanılan en yaygın araçlardır. Aynı zamanda hata dağılışları bu fonksiyonlar arasındaki ilişkilerden yararlanılarak karakterize edilebilmektedir. Pearson diferansiyel denklem sistemi, güvenilirlik analizinde kullanılan birçok dağılışı içerisinde barındırmaktadır. Bu nedenle güvenilirlik analizinde önemli bir yeri vardır. Bu çalışmada Pearson diferansiyel denklem sisteminin, asimetrik dağılım türeten kübik paydalı bir yapısı ele alınacaktır. Daha sonra bu yapı için koşullu momentler ile asimetri ölçüleri incelenecektir.

Anahtar Kelimeler: Güvenilirlik Analizi, Kübik Paydalı Pearson Diferansiyel Denklem Sistemi, Koşullu Momentler.

 

Kaynakça

  • Asadi, M. (1998). Characterization of the pearson system of distributions based on reliability measures. Statistical Papers, 39 (1): 347-360.
  • Elderton, W. P. (1953). Frequency curves and correlation. London: Charles and Edwin Layton.
  • Fisz, M. (1967). Probability theory and mathematical statistics. New York: John Wiley and Sons Inc.
  • Glanzel, W., Telcs, A. ve Schubert, A. (1984). Characterization by truncated moments and ıts application to pearson-type distributions. Zeitschrift für Wahrscheinlichkeitsheorie and Verwandte Gebiete, 66 (2): 173-183.
  • Glanzel, W. (1991). Characterization trough some conditional moments of pearson-type distributions and discrete analogues. The Indian Journal of Statistics, 53 (1): 17-24.
  • Gupta, R. C. ve Bradley, D. M. (2003). Representing the mean residual life in terms of the failure rate. Mathematical and Computer Modelling, 37 (12): 1271-1280.
  • Hogg, R. V. ve Craig, A. T. (1995). Introduction to mathematical statistics. Hong Kong: Higher Education Press.
  • Kotz, S. (1974). Characterizations of statistical distributions: a supplement to recent surveys. International Statistical Review, 42 (1): 39-65.
  • Lawless, J. (2003). Statistical models and methods for lifetime data. New Jersey: John Wiley and Sons Inc.
  • Nair, N. U. ve Sankaran, P. G. (1991). Characterization of the pearson family of distributions. IEE Transactions on Reliability, 40 (1): 75-77.
  • Nair, N. U. ve Sankaran, P. G. (2000). On some reliability aspects of pearson family of distributions. Statistical Papers, 41 (1): 109-117.
  • Navarro, J., Franco, M. ve Ruiz, J. M. (1998). Characterization through moments of the residual life and conditional spacings. The Indian Journal of Statistics, 60 (1): 36-48.
  • Osaki, S. ve Li, X. (1988). Characterizations of gamma and negative binomial distributions. IEE Transactions on Reliability, 37 (4): 379-382.
  • Papathanasiou, V. (1995). A characterization of the pearson system of distributions and the associated orthogonal polynomials. Annals of the Institute of Statistical Mathematics, 47 (1): 171-176.
  • Pearson, K. (1916). Mathematical contributions to the theory of evolution. xıx. second supplement to a memoir on skew variation. Philosophical Transactions of the Royal Society of London, Series A, 216 (1): 429-457.
  • Rausand, M. ve Hoyland, A. (2004). System reliability theory. New Jersey: John Wiley and Sons Inc.
  • Sankaran, P. G. (1992). Characterization of Probability Distributions by Reliability Concepts. Unpublished Doctoral Dissertation. Cochin University of Science and Technology, Department of Statistics, India.
  • Sankaran, P. G., Nair, N. U. ve Sindu, T. K. (2003). A generalized pearson system useful in reliability analysis. Statistical Papers, 44 (1): 125-130.
  • Saraçoğlu, B. ve Çevik, F. (1995). Matematiksel istatistik. Ankara: Gazi Büro Kitabevi.
  • Shakil, M., Kibria, B. M. ve Singh, J. N. (2010). A new family of distributions based on the generalized pearson differantial equation with some applications. Austrian Journal of Statistics, 39 (3): 259-278.
  • Sindu, T. K. (2002). An Extended Pearson System Useful in Reliability Analysis. Unpublished Doctoral Dissertation. Cochin University of Science and Technology, Department of Statistic, India.
  • Stuart, A. ve Ord, J. K., (1987). Kendall’s advanced theory of statistics. New York: Oxford University Press.
  • Şehirlioğlu, A. K. (2011). Pearson dağılış ailesi. Yayınlanmamış Ders Notları. Dokuz Eylül Üniversitesi İktisadi ve İdari Bilimler Fakültesi, İzmir.
  • Ünlü, M. (2013). Pearson dağılış ailesinin güvenilirlik analizinde kullanılması üzerine bir çalışma. Yayınlanmamış Yüksek Lisans Tezi. Dokuz Eylül Üniversitesi Sosyal Bilimler Enstitüsü, İzmir.
  • Wasserman, G. (2002). Reliability verification, testing, and analysis in engineering design. New York: Marcel Dekker Inc.
Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil Türkçe
Bölüm Makaleler
Yazarlar

Ali Şehirlioğlu

Mustafa Ünlü Bu kişi benim

Yayımlanma Tarihi 20 Mart 2014
Gönderilme Tarihi 20 Ocak 2015
Yayımlandığı Sayı Yıl 2013 Cilt: 15 Sayı: 4

Kaynak Göster

APA Şehirlioğlu, A., & Ünlü, M. (2014). Pearson Dağılış Ailesinin Güvenilirlik Analizinde Kullanılması Üzerine Bir Çalışma. Dokuz Eylül Üniversitesi Sosyal Bilimler Enstitüsü Dergisi, 15(4), 693-703.