Abstract
In this
paper, we consider the stress-strength reliability for record data when the distribution of
random stress and strength have the type I extreme-value distribution.
First, classical inference methods, namely uniformly minimum variance unbiased
estimate (UMVUE) and maximum likelihood estimate (MLE), are used for . Second,
Bayesian inference of are considered for gamma priors assumption.
When the common parameter of stress and strength variables is known, the exact
Bayes estimate and Bayesian credible interval of are obtained. Markov Chain Monte Carlo (MCMC)
method are used to derive the Bayes estimate and highest probability density
(HPD) credible interval of when the common parameter is unknown. Finally,
Monte Carlo simulations are performed to compare the performance of the
obtained estimates. A real data set about the weather temperature is analyzed
to illustrate the performances of the derived estimators in the paper.
References
- Referenc1: Murthy, D.N.P, Xie, M, Jiang R. (2003). Weibull Models. Wiley. New York.
- Referenc2: Lai, C.D., Xie, M. (2003). A modified Weibull distribution. IEEE Transactions on Reliability, 52, 33-37.
- Referenc3: Chandler, K.N. (1952). The distribution and frequency of record values. Journal of the Royal Statistical Society, Series B, 14, 220-228.
- Referenc4: Arnold, B.C., Balakrishnan, N., Nagaraja, H.N. (1998). Records. John Wiley & Sons, New York.
- Referenc5: Ahsanullah, M., Nevzorov, V. (2015). Records via probability theory. Atlantis Press.
- Referenc6: Birnbaum, Z.W. (1956). On a use of Mann-Whitney statistics. In Proceedings of 3rd Berkeley Symposium on Mathematical Statistics and Probability, 1, 13-17.
- Referenc7: Kotz, S., Lumelskii, Y., Pensky, M. (2003). The Stress-Strength Model and its Generalizations: Theory and Applications. World Scientific, Singapore.
- Referenc8: Tarvirdizade, B., Gharehchobogh, H.K. (2015). Inference on Pr(X>Y) based on record values from the Burr Type X distribution. Hacettepe Journal of Mathematics and Statistics, 45, 267-278.
- Referenc9: Basirat, M., Baratpour, S., Ahmadi , J. (2016). On estimation of stress–strength parameter using record values from proportional hazard rate models. Communications in Statistics - Theory and Methods, 45, 5787-5801.
- Referenc10: Kızılaslan, F., Nadar, M. (2017). Statistical inference of P(X<Y) for the Burr Type XII distribution based on records. Hacettepe Journal of Mathematics and Statistics, 46, 713-742.
- Referenc11:Rasethuntsa, T. R., Nadar, M. (2018). Stress–strength reliability of a non-identical-component-strengths system based on upper record values from the family of Kumaraswamy generalized distributions. Statistics, 52, 684-716.
- Referenc12: Çetinkaya, Ç., Genç, A.İ. (2019). Stress–strength reliability estimation under the standard two-sided power distribution. Applied Mathematical Modelling, 65, 72-88.
- Referenc13: Basirat, M., Baratpour, S., Ahmadi, J. (2015). Statistical inferences for stress-strength in the proportional hazard models based on progressive type-ii censored samples. Journal of Statistical Computational and Simulation, 85, 431-449.
- Referenc14: Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B. (2003). Bayesian Data Analysis. Chapman & Hall, London.
- Referenc15: Chen, M.H., Shao, Q.M. (1999). Monte carlo estimation of Bayesian credible and hpd intervals. Journal of Computational Graphical and Statistics, 8, 69-92.
I.Tip Uçdeğer Dağılımından Gelen Rekor Değerler İçin Stres Dayanıklılık Modelinin Güvenilirliğinin Tahmini
Abstract
Bu çalışmada, stres Y ve dayanıklılık X rastgele
değişkenleri I. Tip uçdeğer dağılımına sahip olduğunda rekor değerler için
stres dayanıklılık modelinin güvenilirliği ele alınmıştır. İlk olarak için klasik
yaklaşım yani değişmez en küçük varyanslı yansz minimum varyans tahmin edici ve
en çok olabilirlik tahmin edicisi kullanılmıştır. Sonra, önsellerin gamma
dağılımına sahip olması varsayımı altın için Bayes
yaklaşımı ele alınmıştır. Stres ve dayanıklılık değişkenlerinin ortak
parametresi biliniyorken, nin kesin Bayes
tahmin edicisi ve Bayes güven aralığı elde edilmiştir. Stres ve dayanıklılık
değişkenlerinin ortak parametresi bilinmiyorken, ’nin Bayes tahmin edicisi ve en yüksek olasılık
yoğunluklu Bayes güven aralığı Markov Zinciri Monte Carlo (MCMC) metodu ile
elde edilmiştir. Son olarak elde edilen tahmin edicilerin performanslarını
karşılaştırmak için Monte Carlo simülasyonu gerçekleştirildi. Elde edilen
tahmin edicilerin performanslarını göstermek için hava sıcaklıkları ile ilgili
gerçek veri seti analiz edilmiştir.
References
- Referenc1: Murthy, D.N.P, Xie, M, Jiang R. (2003). Weibull Models. Wiley. New York.
- Referenc2: Lai, C.D., Xie, M. (2003). A modified Weibull distribution. IEEE Transactions on Reliability, 52, 33-37.
- Referenc3: Chandler, K.N. (1952). The distribution and frequency of record values. Journal of the Royal Statistical Society, Series B, 14, 220-228.
- Referenc4: Arnold, B.C., Balakrishnan, N., Nagaraja, H.N. (1998). Records. John Wiley & Sons, New York.
- Referenc5: Ahsanullah, M., Nevzorov, V. (2015). Records via probability theory. Atlantis Press.
- Referenc6: Birnbaum, Z.W. (1956). On a use of Mann-Whitney statistics. In Proceedings of 3rd Berkeley Symposium on Mathematical Statistics and Probability, 1, 13-17.
- Referenc7: Kotz, S., Lumelskii, Y., Pensky, M. (2003). The Stress-Strength Model and its Generalizations: Theory and Applications. World Scientific, Singapore.
- Referenc8: Tarvirdizade, B., Gharehchobogh, H.K. (2015). Inference on Pr(X>Y) based on record values from the Burr Type X distribution. Hacettepe Journal of Mathematics and Statistics, 45, 267-278.
- Referenc9: Basirat, M., Baratpour, S., Ahmadi , J. (2016). On estimation of stress–strength parameter using record values from proportional hazard rate models. Communications in Statistics - Theory and Methods, 45, 5787-5801.
- Referenc10: Kızılaslan, F., Nadar, M. (2017). Statistical inference of P(X<Y) for the Burr Type XII distribution based on records. Hacettepe Journal of Mathematics and Statistics, 46, 713-742.
- Referenc11:Rasethuntsa, T. R., Nadar, M. (2018). Stress–strength reliability of a non-identical-component-strengths system based on upper record values from the family of Kumaraswamy generalized distributions. Statistics, 52, 684-716.
- Referenc12: Çetinkaya, Ç., Genç, A.İ. (2019). Stress–strength reliability estimation under the standard two-sided power distribution. Applied Mathematical Modelling, 65, 72-88.
- Referenc13: Basirat, M., Baratpour, S., Ahmadi, J. (2015). Statistical inferences for stress-strength in the proportional hazard models based on progressive type-ii censored samples. Journal of Statistical Computational and Simulation, 85, 431-449.
- Referenc14: Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B. (2003). Bayesian Data Analysis. Chapman & Hall, London.
- Referenc15: Chen, M.H., Shao, Q.M. (1999). Monte carlo estimation of Bayesian credible and hpd intervals. Journal of Computational Graphical and Statistics, 8, 69-92.
Details
| Primary Language | English |
|---|---|
| Journal Section | Research Articles |
| Authors | |
| Publication Date | September 1, 2019 |
| Published in Issue | Year 2019 Volume: 31 Issue: 3 |