Estimation Based on Sequential Order Statistics Coming from the Weibull Distribution
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Mahdy Esmailian, Department of Statistics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran.
In engineering systems, it is usually assumed that the lifetime of the componentsare independent and identically distributed. But with the failure of any component, the pressure comes more to the remaining components. Thus, the distribution of the survivingcomponents would be changed. For modeling this kind of systems, the theoryof sequential order statistics (SOS) may be used. In this paper, SOS with the conditional proportional hazard rates (CPHR) model, coming from the two-parameter Weibull distribution, are used for modeling lifetimes of sequential r-out-of-n: F systems. The Newton-Raphson method to determine the maximum likelihood estimates of parameters is implemented. For illustration purposes, an example is analyzed.
Sequential Order Statistics, Weibull Distribution, Point Estimation, Newton-Raphson Method
[1]
N. Balakrishnan, N. Kannan, C. T. Lin, and H. K. T. Ng. Point and interval estimation for gaussian distribution based on progressively type-ii censored samples. IEEE Transactions on Reliability, 52:90–95, 2003.
[2]
P. Basak, I. Basak, and N. Balakrishnan. Estimation for the three-parameter lognormal distribution based on progressively censored data. Computational Statistics and Data Analysis, 53:3580–3592, 2009.
[3]
I. V. Basawa and B. L. S. Prakasa Rao. Statistical Inference in Stochastic Processes. Academic Press, 1980.
[4]
S. Bedbur. Umpu tests based on sequential order statistics. Journal of Statistical Planning and Inference, 140:2520–2530, 2010.
[5]
Z. Chen. Statistical inference about the shape parameter of the weibull distribution. Statistics and Probability Letters, 36(1):85–90, 1997.
[6]
A. C. Cohen. Maximum likelihood estimation in the weibull distribution based on complete and on censored samples. Technometrics, 7:579–588, 1965.
[7]
E. Cramer and U. Kamps. Sequential order statistics and k-out-of-n systems with sequentially adjusted failure rates. Annals of the Institute of Statistical Mathematics, 48(3):535–549, 1996.
[8]
M. Esmailian and M. Doostparast. Estimation based on sequential order statistics with random removals. Probability And Mathematical Statistics, 34:81–95, 2014.
[9]
B. R. Kale. On the solution of likelihood equations by iteration processes. the multi-parameter case. Biometrika, 49:479–486, 1962.
[10]
U. Kamps. A concept of generalized order statistics. Journal of Statistical Planning and Inference, 48:1–23, 1995.
[11]
U. Kamps. A Concept of Generalized Order Statistics. Teubner, 1995.
[12]
A. Khuri. Advanced calculus with applications in statistics. John Wiley, 2003.
[13]
N. Schenk, M. Burkschat, E. Cramer, and U. Kamps. Bayesian estimation and prediction with multiply type-ii censored samples of sequential order statistics from one-and two-parameter exponential distributions. Journal of Statistical Planning and Inference, 141:1575–1587, 2011.
[14]
R. L. Smith. A comparison of maximum likelihood and bayesian estimators for the three-parameter weibull distribution. Journal of the Royal Statistical Society, Series C, 36(3):358–369, 1987.
[15]
D. R. Thoman, L. J. Bain, and C. E. Antle. Inferences on the parameters of the weibull distribution. Technometrics, 11:445–460, 1969.
[16]
L. A. Weissfeld and H. Schneider. Influence diagnostics for the weibull model fit to censored data. Statistics and Probability Letters, 9(1):67–73, 1990.