Some Properties and Interval Estimation Based on Maximum Likelihood Estimators of the Crack Lifetime Distribution


  • Monthira Duangsaphon Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, Pathum Thani, Thailand


Asymptotic distribution, bootstrap confidence interval, coverage probability, average width, maximum likelihood estimation


The crack lifetime distribution is used for modeling lifetime data. This paper proposes some properties of the crack lifetime distribution. The confidence intervals of parameters based on the maximum likelihood estimators (MLEs) for crack lifetime distribution are considered. The Fisher information matrix is provided for constructing the asymptotic confidence interval. Moreover, the confidence intervals are computed using the two parametric bootstrap methods: the bootstrap-p and the bootstrap-t methods. Monte Carlo simulations are performed to investigate the performance of the three different interval estimation methods in terms of coverage probabilities and average width. Finally, a real data set is analyzed for illustrative purposes. Results indicate that the asymptotic confidence intervals behave very well for moderate and large sample sizes, while the bootstrap-p intervals work generally well. Moreover, the bootstrap-t intervals also perform quite satisfactorily for small sample sizes.


Download data is not yet available.


Birnbaum ZW, Saunders SC. Estimation for a family of life distributions with applications to fatigue. J Appl Prob. 1969; 6: 328-347.

Bowonrattanaset P, Budsaba K. Some properties of the three-parameter crack distribution. Thail Stat. 2011; 9: 195-203.

Duangsaphon M, Budsaba K, Volodin A. Improve statistical inference for three-parameter crack lifetime distribution. J Prob Stat Sci. 2016; 14: 239-251.

Efron B. The Jackknife, the bootstrap and other resampling plans. CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia. SIAM; 1982.

Gamsmi S, Berzig M. Parameters estimation of the modified Weibull ditribution based on type I censored sample. Appl Math Sci. 2011; 5: 2899-2917.

Hall P. Theoretical comparison of bootstrap confidence intervals. Ann Stat. 1988; 16: 927-953.

Kohansal A, Rezakhah S. Parameter estimation of type-II hybrid censored weighted exponential distribution. Cornel University. 2012 [cited 2012 Mar 1]. Available from: 1203.0094.

Miller RGJ. Survival analysis. New York: Wiley; 1981.

Ng HKT, Kundu D, Balakrishnan N. Point and interval estimations for the two-parameter Birnbaum-Saunders distribution based on type-II censored samples. Comput Stat Data Anal. 2006; 50: 3222-3242.

Panahi H, Asadi S. Estimation of the Weibull distribtion based on type-II censored samples. Appl Math Sci. 2011; 5: 2549-2558.

Pandey BD, Bandyopadhyay P. Bayesian estimation of inverse Gaussian distribution. Cornel University. 2012 [cited 2012 Oct 16]. Available from: https://arxiv. org/abs/1210.4524.