Distribución de Burr XII con datos truncados y censurados: Estimación de máxima verosimilitud basada en Método de Newton-Raphson
Resumen
Este trabajo muestra una metodología para estimar por máxima verosimilitud (ML) los parámetros de una distribución Burr XII cuando los datos son simultáneamente truncados por la izquierda y censurados por la derecha. Dado que las ecuaciones de ML no tienen solución definitiva en estas condiciones, se considera un procedimiento iterativo basado en el método de Newton-Raphson. La coincidencia de percentiles se utiliza para establecer valores iniciales en el algoritmo. Los resultados basados en simulaciones y análisis de datos reales indican que la alternativa propuesta tiene un buen desempeño.
Palabras clave
Distribución Burr XII, datos censurados, datos truncados, máxima verosimilitud
Citas
- Burr I. Cumulative frequency functions. Annals of Mathematical Statistics. 1942; 2(13): 215-232.
- Hogg R, Klugman S. Loss Distributions. New York: John Wiley & Sons; 2009.
- Klugman S, Panjer H, Willmot G. Loss Models: From Data to Decisions. New York: John Wiley &
- Sons; 2012.
- Embrechts P, Klüppelberg C, Mikosch T. Modelling Extremal Events: for Insurance and Finance. New
- York: Springer Science & Business Media; 2013.
- Albrecher H, Beirlant J, Teugels J. Actuarial and Statistical Aspect. New York: John Wiley & Sons;
- Zimmer W, Keats J, Wang, F. The Burr XII distribution in reliability analysis. Journal of Quality
- Technology. 1998; 30(4):386-394.
- Rao G, Aslam M, Kundu D. Burr-XII distribution parametric estimation and estimation of reliability of
- multicomponent stress-strength. Communications in Statistics-Theory and Methods. 2015; 44(23):4953–
- Wingo D. Maximum likelihood methods for fitting the Burr type XII distribution to life test data.
- Biometrical Journal. 1983; 25(1):77-84.
- Shao Q, Zhou X. A new parametric model for survival data with long-term survivor. Statistics in
- Medicine. 2004; 23(22):3525-3543.
- Kleiber C, Kotz S. Statistical Size Distributions in Economics and Actuarial Sciences. New York: John
- Wiley & Sons; 2003.
- Beirlant J, Goegebeur Y, Verlaak R, Vynckier P. Burr regression and portfolio segmentation. Insurance:
- Mathematics and Economics. 1998; 23(3):231-250.
- Yourstone S, ZimmerW. Non-normality and the design of control charts for averages. Decision Sciences.
- ; 23(5):1099-1113.
- Chou C, Chen C, Liu H. Acceptance control charts for non-normal data. Journal of Applied Statistics.
- ; 32(1):25-36.
- Wingo D. Maximum likelihood methods for fitting the Burr type XII distribution to multiply (progressively)
- censored life test data. Metrika. 1993; 40(1):203-210.
- Wang F, Keats J, Zimmer W. Maximum likelihood estimation of the Burr XII parameters with censored
- and uncensored data. Microelectronics Reliability. 1996; 36(3):359-362.
- Ghitany M, Al-Awadhi S. Maximum likelihood estimation of Burr XII distribution parameters under
- random censoring. Journal of Applied Statistics. 2002; 29(7):955-965.
- A. Abuzaid. The estimation of the Burr-XII parameters with middle-censored data. SpringerPlus vol. 4,
- no. 1, pp 1-10, 2015.
- Aslam M, Mahmood Y, Lio Y, Tsai T, Khan M. Double acceptance sampling plans for Burr type XII
- distribution percentiles under the truncated life test. Journal of the Operational Research Society. 2012;
- (7):1010-1017.
- Tsai T, Lio Y, Jiang N, Lin Y, Fan Y. Economical sampling plans with warranty based on truncated data
- from Burr type XII distribution. Journal of the Operational Research Society. 2015; 66(9):1511-1518.
- Balakrishnan N, Mitra D. Left truncated and right censored Weibull data and likelihood inference with
- an illustration. Computational Statistics & Data Analysis. 2012; 56(12):4011-4025.
- Balakrishnan N, Mitra D. EM-based likelihood inference for some lifetime distributions based on left
- truncated and right censored data and associated model discrimination. South African Statistical Journal.
- ; 48(12):125-171.
- Kundu D, Mitra D. A. Analysis of left truncated and right censored competing risks data. Computational
- Statistics & Data Analysis. 2017; 108:12-26.
- Kundu D, Mitra D. Statistical inference based on left truncated and interval censored data from loglocation-
- scale family of distributions. Communications in Statistics Simulation and Computation. 2021;
- (4):1073-1093.
- Afify A, Cordeiro G, Ortega E, Yousof H, Butt N. The four-parameter Burr XII distribution: Properties,
- regression model, and applications. Communications in Statistics-Theory and Methods. 2018;
- (11):2605-2624.
- Paranaíba P, Ortega E, Cordeiro G, Pascoa M. The Kumaraswamy Burr XII distribution: theory and
- practice. Journal of Statistical Computation and Simulation. 2021; 83(11):2117-2143.
- Olapade A. On a six-parameter generalized Burr XII distribution. arXiv preprint 0806.1579:1–8.
- El-Bassiouny A, Abdo N. Reliability properties of seven parameters Burr XII distribution. Computational
- Methods in Science and Technology. 2010; 16:127-133.
- Adeyemi S, Adebanji A. Moments of order statistics from doubly truncated Burr XII distribution: A
- complementary note with applications. Journal of Statistical Research. 2015; 41(1):37-50.
- Gomes A, da-Silva C, Cordeiro, G. Two extended Burr models: Theory and practice. Journal of Applied
- Statistics. 2015; 44(8):1706-1734.
- Mead M. A new generalization of Burr XII distribution. Journal of Statistics: Advances in Theory and
- Applications. 2014; 12(2):53-73.
- Lanjoni B, Ortega E, Cordeiro G. Extended Burr XII regression models: theory and applications. Journal
- of Agricultural, Biological, and Environmental Statistics. 2016; 21(1):203-224.
- Shakhatreh M, Al-Masri A. On the weighted Burr XII distribution: theory and practice. Electronic
- Journal of Applied Statistical Analysis. 2015; 13(1):229-255.
- Süli E, Mayers, D. An Introduction to Numerical Analysis. 2003, Cambridge University Press.
- Efron B, Hinkley, D. Assessing the accuracy of the maximum likelihood estimator: Observed versus
- expected Fisher information. Biometrika. 1978; 65(3): 457-483
- Watkins, A. A note on expected fisher information for the Burr XII distribution. Microelectronics
- Reliability. 1997; 37(12), 1849-1852.
- Brazauskas, V. Information matrix for Pareto (IV), Burr, and related distributions. Communications in
- Statistics-Theory and Methods. 2003; 32(2), 315-325.
- Dutang C, Goulet V, Pigeon M. actuar: An R package for actuarial science, Journal of Statistical
- Software. 2008; 25(7):1-37.
- R Core Team. R. A Language and Environment for Statistical Computing. R Foundation for Statistical
- Computing. 2018; https://www.R-project.org/.
- Okasha M, Matter M. On the three-parameter Burr type XII distribution and its application to heavy
- tailed lifetime data. Journal of Statistics: Advances in Theory and Applications. 2014; 12(2):53-73.
- Lee E, Wang, J. Statistical Methods for Survival Data Analysis. 2003, ohn Wiley & Sons.
- Al-Shomrani A, Shawky A, Arif O, Aslam, M. Log-logistic Distribution for Survival Data Analysis
- Using MCMC. SpringerPlus. 2016; 5:1-16.