Exponentiated Weibull distribution, introduced as an extension of the Weibull distribution, is characterized by bathtub shaped, unimodal failure rates besides a broader class of monotone failure rates. In this paper, we derive maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators and three other estimators of the probability density function and the cumulative distribution function of the exponentiated Weibull distribution and compare their performances through numerical simulations. Simulation studies show that the MLE is more efficient than the others. Analysis of a real data set is presented for illustrative purposes.
References
Achcar, J. A., Ortiz-Rodríguez, G. and Rodrigues, E. R. (2009). Estimating the number of ozone peaks in Mexico City using a non-homogeneous Poisson model and a Metropolis–Hastings algorithm. International Journal of Pure and Applied Mathematics 53, 1–20.Achcar, J. A., Ortiz-Rodríguez, G. and Rodrigues, E. R. (2009). Estimating the number of ozone peaks in Mexico City using a non-homogeneous Poisson model and a Metropolis–Hastings algorithm. International Journal of Pure and Applied Mathematics 53, 1–20.
Ashour, S. K. and Afify, W. M. (2007). Statistical analysis of exponentiated Weibull family under type I progressive interval with random removals. Applied Sciences Research 3, 1851–1863.Ashour, S. K. and Afify, W. M. (2007). Statistical analysis of exponentiated Weibull family under type I progressive interval with random removals. Applied Sciences Research 3, 1851–1863.
Ashour, S. K. and Afify, W. M. (2008). Estimation of the parameters of exponentiated Weibull family with type II progressive interval censoring with random removals. Applied Sciences Research 4, 1428–1442.Ashour, S. K. and Afify, W. M. (2008). Estimation of the parameters of exponentiated Weibull family with type II progressive interval censoring with random removals. Applied Sciences Research 4, 1428–1442.
Bagheri, S. F., Alizadeh, M., Baloui Jamkhaneh, E. and Nadarajah, S. (2014). Evaluation and comparison of estimations in the generalized exponential-Poisson distribution. Journal of Statistical Computation and Simulation. 84, 2345–2360.Bagheri, S. F., Alizadeh, M., Baloui Jamkhaneh, E. and Nadarajah, S. (2014). Evaluation and comparison of estimations in the generalized exponential-Poisson distribution. Journal of Statistical Computation and Simulation. 84, 2345–2360.
Burnham, K. P. and Anderson, D. R. (2004). Multimodel inference: Understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.Burnham, K. P. and Anderson, D. R. (2004). Multimodel inference: Understanding AIC and BIC in model selection. Sociological Methods and Research 33, 261–304.
Dixit, U. J. and Jabbari Nooghabi, M. (2011). Efficient estimation in the Pareto distribution with the presence of outliers. Statistical Methodology 8, 340–355.Dixit, U. J. and Jabbari Nooghabi, M. (2011). Efficient estimation in the Pareto distribution with the presence of outliers. Statistical Methodology 8, 340–355.
Fang, Y. (2011). Asymptotic equivalence between cross-validations and Akaike information criteria in mixed-effects models. Journal of Data Science 9, 15–21.Fang, Y. (2011). Asymptotic equivalence between cross-validations and Akaike information criteria in mixed-effects models. Journal of Data Science 9, 15–21.
Gurler, S. and Capar, S. (2011). An algorithm for mean residual life computation of $(n-k+1)$-out-of-$n$ systems: An application of exponentiated Weibull distribution. Applied Mathematics and Computation 217, 7806–7811.Gurler, S. and Capar, S. (2011). An algorithm for mean residual life computation of $(n-k+1)$-out-of-$n$ systems: An application of exponentiated Weibull distribution. Applied Mathematics and Computation 217, 7806–7811.
Jaheen, Z. F. and Al Harbi, M. M. (2011). Bayesian estimation based on dual generalized order statistics from the exponentiated Weibull model. Journal of Statistical Theory and Applications 10, 591–602.Jaheen, Z. F. and Al Harbi, M. M. (2011). Bayesian estimation based on dual generalized order statistics from the exponentiated Weibull model. Journal of Statistical Theory and Applications 10, 591–602.
Kao, J. H. K. (1958). Computer methods for estimating Weibull parameters in reliability studies. Transaction of IRE-Reliability and Quality Control 13, 15–22.Kao, J. H. K. (1958). Computer methods for estimating Weibull parameters in reliability studies. Transaction of IRE-Reliability and Quality Control 13, 15–22.
Kim, C., Jung, J. and Chung, Y. (2011). Bayesian estimation for the exponentiated Weibull model type II progressive censoring. Statistical Papers 52, 53–70.Kim, C., Jung, J. and Chung, Y. (2011). Bayesian estimation for the exponentiated Weibull model type II progressive censoring. Statistical Papers 52, 53–70.
Mudholkar, G. S. and Hutson, A. D. (1996). The exponentiated Weibull family: Some properties and a flood data application. Communications Statistics—Theory and Methods 25, 3059–3083.Mudholkar, G. S. and Hutson, A. D. (1996). The exponentiated Weibull family: Some properties and a flood data application. Communications Statistics—Theory and Methods 25, 3059–3083.
Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-real data. IEEE Transaction Reliability 42, 299–302.Mudholkar, G. S. and Srivastava, D. K. (1993). Exponentiated Weibull family for analyzing bathtub failure-real data. IEEE Transaction Reliability 42, 299–302.
Mudholkar, G. S., Srivastava, D. K. and Freimes, M. (1995). The exponentiated Weibull family: A reanalysis of the bus-motor-failure data. Technometrics 37, 436–445.Mudholkar, G. S., Srivastava, D. K. and Freimes, M. (1995). The exponentiated Weibull family: A reanalysis of the bus-motor-failure data. Technometrics 37, 436–445.
Nassar, M. M. and Eissa, F. H. (2004). Bayesian estimation for the exponentiated Weibull model. Communications in Statistics—Theory and Methods 33, 2343–2362. MR2104117 10.1081/STA-200031447Nassar, M. M. and Eissa, F. H. (2004). Bayesian estimation for the exponentiated Weibull model. Communications in Statistics—Theory and Methods 33, 2343–2362. MR2104117 10.1081/STA-200031447
Nichols, M. D. and Padgett, W. J. (2006). A bootstrap control chart for Weibull percentiles. Quality and Reliability Engineering International 22, 141–151.Nichols, M. D. and Padgett, W. J. (2006). A bootstrap control chart for Weibull percentiles. Quality and Reliability Engineering International 22, 141–151.
Schwarz, G. E. (1978). Estimating the dimension of a model. Annals of Statistics 6, 461–464. MR468014 10.1214/aos/1176344136 euclid.aos/1176344136
Schwarz, G. E. (1978). Estimating the dimension of a model. Annals of Statistics 6, 461–464. MR468014 10.1214/aos/1176344136 euclid.aos/1176344136
Singh, U., Gupta, P. K. and Upadhaya, S. K. (2002). Estimation of exponentiated Weibull shape parameters under LINEX loss function. Communications in Statistics—Simulation and Computation 31, 523–537.Singh, U., Gupta, P. K. and Upadhaya, S. K. (2002). Estimation of exponentiated Weibull shape parameters under LINEX loss function. Communications in Statistics—Simulation and Computation 31, 523–537.
Singh, U., Gupta, P. K. and Upadhaya, S. K. (2005a). Estimation of parameters for exponentiated Weibull family under type II censoring scheme. Computational Statistics and Data Analysis 48, 509–523.Singh, U., Gupta, P. K. and Upadhaya, S. K. (2005a). Estimation of parameters for exponentiated Weibull family under type II censoring scheme. Computational Statistics and Data Analysis 48, 509–523.
Singh, U., Gupta, P. K. and Upadhaya, S. K. (2005b). Estimation of three parameters exponentiated Weibull distribution under type II censoring. Journal of Statistical Planning and Inference 134, 350–372.Singh, U., Gupta, P. K. and Upadhaya, S. K. (2005b). Estimation of three parameters exponentiated Weibull distribution under type II censoring. Journal of Statistical Planning and Inference 134, 350–372.
Singh, U., Gupta, P. K. and Upadhaya, S. K. (2006). Some point estimates for the shape parameters of exponentiated Weibull family. Journal of the Korean Statistical Society 35, 63–77.Singh, U., Gupta, P. K. and Upadhaya, S. K. (2006). Some point estimates for the shape parameters of exponentiated Weibull family. Journal of the Korean Statistical Society 35, 63–77.
Swain, J., Venkatraman, S. and Wilson, J. (1988). Least squares estimation of distribution function in Johnson’s translation system. Journal of Statistical Computation and Simulation 29, 271–297.Swain, J., Venkatraman, S. and Wilson, J. (1988). Least squares estimation of distribution function in Johnson’s translation system. Journal of Statistical Computation and Simulation 29, 271–297.
Wang, M. and Rennolls, K. (2005). Tree diameter distribution modelling: Introducing the logit-logistic distribution. Canadian Journal of Forest Research 35, 1305–1313.Wang, M. and Rennolls, K. (2005). Tree diameter distribution modelling: Introducing the logit-logistic distribution. Canadian Journal of Forest Research 35, 1305–1313.
Zhang, T., Xie, M., Tang, L. C. and Ng, S. H. (2005). Reliability and modeling of systems integrated with firmware and hardware. International Journal of Reliability, Quality and Safety Engineering 12, 227–239.Zhang, T., Xie, M., Tang, L. C. and Ng, S. H. (2005). Reliability and modeling of systems integrated with firmware and hardware. International Journal of Reliability, Quality and Safety Engineering 12, 227–239.