Likelihood Functions; Monte Carlo Method; Models, Statistical; Multidisciplinary
Abstract :
[en] In this paper, we develop a generator to propose new continuous lifetime distributions. Thanks to a simple transformation involving one additional parameter, every existing lifetime distribution can be rendered more flexible with our construction. We derive stochastic properties of our models, and explain how to estimate their parameters by means of maximum likelihood for complete and censored data, where we focus, in particular, on Type-II, Type-I and random censoring. A Monte Carlo simulation study reveals that the estimators are consistent. To emphasize the suitability of the proposed generator in practice, the two-parameter Fréchet distribution is taken as baseline distribution. Three real life applications are carried out to check the suitability of our new approach, and it is shown that our extension of the Fréchet distribution outperforms existing extensions available in the literature.
Disciplines :
Mathematics Engineering, computing & technology: Multidisciplinary, general & others
Author, co-author :
Aslam, Muhammad ; Quaid-i-Azam University, Islamabad, Pakistan
LEY, Christophe ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) ; Department of Applied Mathematics, Computer Science and Statistics, Gent University, Ghent, Belgium
Hussain, Zawar; Department of Social & Allied Sciences, Cholistan University of Veterinary & Animal University, Bahwalpur, Pakistan
Shah, Said Farooq ; Department of Statistics, University of Peshawar, Peshawar, Pakistan
Marshall AW, Olkin I. A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families. Biometrika. 1997; 84:641–652.
Gupta RC, Gupta PI, Gupta RD. Modeling failure time data by Lehmann alternatives. Commun. Stat. Theory Methods. 1998; 27:887–904.
Eugene N, Lee C, Famoye F. Beta-normal distribution and its applications. Commun. Stat. Theory Methods. 2002; 31:497–512.
Cordeiro GM, de Castro M. A new family of generalized distributions. J. Stat. Comput. Sim. 2011; 81:883–898.
Zografos K, Balakrishnan N. On families of beta and generalized gamma generated distributions and associated inference. Stat. Methodol. 2009; 6:344–362.
Maurya SK, Kaushik A, Singh RK, Singh SK, Singh U. A new method of proposing distribution and its application to real data. Imp. J. Interdiscip. Res. 2016; 2 (6):1331–1338.
Weibull W. Wide applicability. J. Appl. Mech. 1951; 103(730): 293–7.
Meeker WQ, Escobar LA. Statistical Methods for Reliability Data. John Wiley & Sons Inc., New York; 1998.
Patti S, Biganzoli E, Boracchi P. Review of the Maximum Likelihood Functions for Right Censored Data. A New Elementary Derivation. COBRA Preprint Series, 2007; 21.
Chen F, Chen S. Injury severities of truck drivers in single-and multi-vehicle accidents on rural highways. Accid. Anal. Prev. 2011; 43(5), 1677–1688. https://doi.org/10.1016/j.aap.2011.03.026 PMID: 21658494
Chen F, Ma X, Chen S. Refined-scale panel data crash rate analysis using random-effects tobit model. Accid. Anal. Prev. 2014; 73, 323–332. https://doi.org/10.1016/j.aap.2014.09.025 PMID: 25269099
Ma X, Chen F, Chen S. Modeling crash rates for a mountainous highway by using refined-scale panel data. Transprot. Res. Rec. 2015; 2515(1), 10–16.
Zeng Q, Wen H, Huang H, Pei X, Wong SC. A multivariate random-parameters Tobit model for analyzing highway crash rates by injury severity. Accid. Anal. Prev. 2017; 99, 184–191. https://doi.org/10.1016/j.aap.2016.11.018 PMID: 27914307
Dong B, Ma X, Chen F, Chen S. Investigating the differences of single-vehicle and multivehicle accident probability using mixed logit model. J. Adv. Transprot. 2018.
Zeng Q, Guo Q, Wong SC, Wen H, Huang H, Pei X. Jointly modeling area-level crash rates by severity: a Bayesian multivariate random-parameters spatio-temporal Tobit regression. Transportmetrica. A. 2019; 15(2), 1867–1884.
Panchang VG, Gupta RC. On the Determination of Three–Parameter Weibull MLE’s. Commun. Stat-Simul. C. 1989; 18(3):1037–57.
Pushpa Gupta L, Gupta RC, Lvin SJ. Numerical methods for the maximum likelihood estimation of weibull parameters. J. Stat. Comput. Sim. 1998; 62(1–2):1–7.
Cheng RC, Traylor L. Characterization of material strength properties using probabilistic mixture models. WIT Transactions on Modelling and Simulation. 1970; 12.
Bjerkedal T. Acquisition of Resistance in Guinea Pies infected with Different Doses of Virulent Tubercle Bacilli. Am. J. Hyg. 1960; 72 (1):130–48.
Zeng Q, Wen H, Huang H, Pei X, Wong SC. Incorporating temporal correlation into a multivariate random parameters Tobit model for modeling crash rate by injury severity. Transportmetrica. A. 2018; 14 (3), 177–191.
Khan AA, Aslam M, Hussain Z, Tahir M. Comparison of loss functions for estimating the scale parameter of log-normal distribution using non-informative priors. Hacettepe Journal of Mathematics and Statistics. 2015; 45(6): 1831–1845.
Kersey JH, Weisdorf D, Nesbit ME, Lebien TW, Woods WG, McGlave PB, et al. Comparison of autologous and autologous bone marrow transplantation for treatment of high-risk refractory acutelymphoblastic leukemia. N. Engl. J. Med. 1987; 317:461–467. https://doi.org/10.1056/ NEJM198708203170801 PMID: 3302708
Ramos PL, Nascimento D, Louzada F. The Long Term Frechet distribution: Estimation, Properties and its Application. 2017. arXiv preprint arXiv:1709.07593.
