### Estimation in Inverse Weibull Distribution Based on Randomly Censored Data

#### Abstract

This article deals with the estimation of the parameters and reliability characteristics in inverse Weibull (IW) distribution based on the random censoring model. The censoring distribution is also taken as an IW distribution. Maximum likelihood estimators of the parameters, survival and failure rate functions are derived. Asymptotic confidence intervals of the parameters based on the Fisher information matrix are constructed. Bayes estimators of the parameters, survival and failure rate functions under squared error loss function using non-informative and gamma informative priors are developed. Furthermore, Bayes estimates are obtained using Tierney-Kadane's approximation method and Markov chain Monte Carlo (MCMC) techniques. Also, highest posterior density (HPD) credible intervals of the parameters based on MCMC techniques are constructed. A simulation study is conducted to compare the performance of various estimates. Finally, a randomly censored real data set supports the estimation procedures developed in this article.

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H. AKAIKE (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19, no. 6, pp. 716–723.

F. G. AKGUL, B. SENOGLU (2018). Comparison of estimation methods for inverse Weibull distribution. In M. TEZ, D. VON ROSEN (eds.), Trends and Perspectives in Linear Statistical Inference. Springer, Cham, Switzerland, pp. 1–22.

F. G. AKGUL, B. SENOGLU, T. ARSLAN (2016). An alternative distribution to Weibull for modeling the wind speed data: Inverse Weibull distribution. Energy Conversion and Management, 114, pp. 234–240.

N. BALAKRISHNAN, R. AGGARWALA (2000). Progressive Censoring: Theory, Methods and Applications. Springer Science & Business Media, New York.

N. BRESLOW, J. CROWLEY (1974). A large sample study of the life table and product limit estimates under random censorship. The Annals of Statistics, pp. 437–453.

A. CHATURVEDI, N. KUMAR, K. KUMAR (2018). Statistical inference for the reliability functions of a family of lifetime distributions based on progressive Type II right censoring. Statistica, 78, no. 1, pp. 81–101.

M.-H. CHEN, Q.-M. SHAO (1999). Monte Carlo estimation of Bayesian credible and HPD intervals. Journal of Computational and Graphical Statistics, 8, no. 1, pp. 69–92.

H. A. DAVID, M. L. MOESCHBERGER (1978). The Theory of Competing Risks. C. Griffin, London.

R. GARG, M. DUBE, K. KUMAR, H. KRISHNA (2016). On randomly censored generalized inverted exponential distribution. American Journal of Mathematical and Management Sciences, 35, no. 4, pp. 361–379.

A. GELMAN, J. B. CARLIN, H. S. STERN, D. B. DUNSON, A. VEHTARI, D. B. RUBIN (2013). Bayesian Data Analysis. Chapman and Hall/CRC, Boca Raton, FL.

M. GHITANY, S. AL-AWADHI (2002). Maximum likelihood estimation of Burr XII distribution parameters under random censoring. Journal of Applied Statistics, 29, no. 7, pp. 955–965.

J. P. GILBERT (1962). Random Censorship. Ph.D. thesis, University of Chicago, Department of Statistics, Chicago.

E. L. KAPLAN, P. MEIER (1958). Nonparametric estimation from incomplete observations. Journal of the American statistical association, 53, no. 282, pp. 457–481.

J. A. KOZIOL, S. B. GREEN (1976). A Cramér-Von Mises statistic for randomly censored data. Biometrika, 63, no. 3, pp. 465–474.

H. KRISHNA, M. DUBE, R. GARG (2019). Estimation of stress strength reliability of inverse Weibull distribution under progressive first failure censoring. Austrian Journal of Statistics, 48, no. 1, pp. 14–37.

H. KRISHNA, N. GOEL (2018). Classical and Bayesian inference in two parameter exponential distribution with randomly censored data. Computational Statistics, 33, no. 1, pp. 249–275.

H. KRISHNA, VIVEKANAND, K. KUMAR (2015). Estimation in Maxwell distribution with randomly censored data. Journal of Statistical Computation and Simulation, 85, no. 17, pp. 3560–3578.

K. KUMAR (2018). Classical and Bayesian estimation in log-logistic distribution under random censoring. International Journal of System Assurance Engineering and Management, 9, no. 2, pp. 440–451.

K. KUMAR, R. GARG (2014). Estimation of the parameters of randomly censored generalized inverted Rayleigh distribution. International Journal of Agricultural and Statistical Sciences, 10, no. 1, pp. 147–155.

K. KUMAR, R. GARG, H. KRISHNA (2017). Nakagami distribution as a reliability model under progressive censoring. International Journal of System Assurance Engineering and Management, 8, no. 1, pp. 109–122.

D. KUNDU, H. HOWLADER (2010). Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data. Computational Statistics & Data Analysis, 54, no. 6, pp. 1547–1558.

J. F. LAWLESS (2003). Statistical Models and Methods for Lifetime Data, vol. 362. John Wiley & Sons, New Jersey.

R CORE TEAM (2018). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.

C. ROBERT, G. CASELLA (2004). Monte Carlo Statistical Methods. Springer Science & Business Media, New York.

G. SCHWARZ (1978). Estimating the dimension of a model. The Annals of Statistics, 6, no. 2, pp. 461–464.

S. K. SINGH, U. SINGH, D. KUMAR (2013). Bayesian estimation of parameters of inverse Weibull distribution. Journal of Applied statistics, 40, no. 7, pp. 1597–1607.

K. SULTAN, N. ALSADAT, D. KUNDU (2014). Bayesian and maximum likelihood estimations of the inverse Weibull parameters under progressive Type-II censoring. Journal of Statistical Computation and Simulation, 84, no. 10, pp. 2248–2265.

L. TIERNEY, J. B. KADANE (1986). Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81, no. 393, pp. 82–86.

P. W. ZEHNA (1966). Invariance of maximum likelihood estimators. Annals of Mathematical Statistics, 37, no. 3, p. 744.

G. ZHENG, J. L. GASTWIRTH (2001). On the Fisher information in randomly censored data. Statistics & probability letters, 52, no. 4, pp. 421–426.

DOI: 10.6092/issn.1973-2201/8414