## Brazilian Journal of Probability and Statistics

### The beta Burr III model for lifetime data

#### Abstract

For the first time, the beta Burr III distribution is introduced as an important model for problems in several areas such as actuarial sciences, meteorology, economics, finance, environmental studies, survival analysis and reliability. The new distribution can be expressed as a linear combination of Burr III distributions and then it has tractable properties for the moments, generating and quantile functions, mean deviations, reliability and entropies. The density of its order statistics can be given in terms of an infinite linear combination of Burr III densities. The beta Burr III model is modified for the possibility of long-term survivors. We define a log-beta Burr III regression model to analyze censored data. The estimation of parameters is approached by maximum likelihood and the observed information matrix is derived. The proposed models are applied to three real data sets.

#### Article information

Source
Braz. J. Probab. Stat., Volume 27, Number 4 (2013), 502-543.

Dates
First available in Project Euclid: 9 September 2013

https://projecteuclid.org/euclid.bjps/1378729985

Digital Object Identifier
doi:10.1214/11-BJPS179

Mathematical Reviews number (MathSciNet)
MR3105041

Zentralblatt MATH identifier
1298.62074

#### Citation

Gomes, Antonio E.; da-Silva, Cibele Q.; Cordeiro, Gauss M.; Ortega, Edwin M. M. The beta Burr III model for lifetime data. Braz. J. Probab. Stat. 27 (2013), no. 4, 502--543. doi:10.1214/11-BJPS179. https://projecteuclid.org/euclid.bjps/1378729985

#### References

• Akinsete, A., Famoye, F. and Lee, C. (2008). The beta-Pareto distribution. Statistics 42, 547–563.
• Al-Dayian, G. R. (1999). Burr type III distribution: Properties and estimation. The Egyptian Statistical Journal 43, 102–116.
• Barreto-Souza, W., Santos, A. H. S. and Cordeiro, G. M. (2010). The beta generalized exponential distribution. Journal of Statistical Computation and Simulation 80, 159–172.
• Berkson, J. and Gage, R. P. (1952). Survival curve for cancer patients following treatment. Journal of the American Statistical Association 47, 501–515.
• Burr, I. W. (1942). Cumulative frequency distributions. Annals of Mathematical Statistics 13, 215–232.
• Chen, G. and Balakrishnan, N. (1995). A general purpose approximate goodness-of-fit test. Journal of Quality Technology 27, 154–161.
• Cordeiro, G. M. and Lemonte, A. J. (2011a). The $\beta$-Birnbaum–Saunders distribution: An improved distribution for fatigue life modeling. Computational Statistics and Data Analysis 55, 1445–1461.
• Cordeiro, G. M. and Lemonte, A. J. (2011b). The beta Laplace distribution. Statistics and Probability Letters 81, 973–982.
• Eugene, N., Lee, C. and Famoye, F. (2002). Beta-normal distribution and its applications. Communication in Statistics—Theory and Methods 31, 497–512.
• Farewell, V. T. (1982). The use of mixture models for the analysis of survival data with long-term survivors. Biometrics 38, 1041–1046.
• Feigl, P. and Zelen, M. (1965). Estimation of exponential probabilities with concomitant information. Biometrics 21, 826–838.
• Fischer, M. J. and Vaughan, D. (2010). The beta-hyperbolic secant distribution. Austrian Journal of Statistics 39, 245–258.
• Fleming, T. R., O’Fallon, J. R., O’Brien, P. C. and Harrington, D. P. (1980). Modified Kolgomorov–Smirnov test procedures with application to arbitrarily right censored data. Biometrics 36, 607–626.
• Gove, J. H., Ducey M. J., Leak, W. B. and Zhang, L. (2008). Rotated sigmoid structures in managed uneven-aged northern hardwood stands: A look at the Burr type III distribution. Forestry 81, 161–176.
• Gradshteyn, I. S. and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products. New York: Academic Press.
• Hose, G. C. (2005). Assessing the need for groundwater quality guidlines for pesticides using the species sensitivity distribution approach. Human and Ecological Risk Assessment 11, 951–966.
• Hoskings, J. R. M. (1990). $L$-moments: Analysis and estimation of distributions using linear combinations of order statistics. Journal of the Royal Statistical Society, Series B 52, 105–124.
• Ibrahim, J. G., Chen, M. H. and Sinha, D. (2001). Bayesian Survival Analysis. New York: Springer.
• Jones, M. C. (2004). Family of distributions arising from distribution of order statistics. Test 13, 1–43.
• Kenney, J. F. and Keeping, E. S. (1962). Mathematics of Statistics, Part 1. Princeton, NJ: Van Nostrand, pp. 101–102.
• Khan, M. S. (2010). The beta inverse Weibull distribution. International Transactions in Mathematical Sciences and Computer 3, 113–119.
• Klugman, S. A., Panjer, H. H. and Willmot, G. E. (1998). Loss Models. New York: Wiley.
• Kotz, S., Lumelskii, Y. and Pensky, M. (2003). The Stress-Strength Model and Its Generalizations: Theory and Applications. River Edge, NJ: World Scientific.
• Lee, C., Famoye and F. Olumolade, O. (2007). Beta-Weibull distribution: Some properties and applications to censored data. Journal of Modern Applied Statistical Methods 6, 173–186.
• Lindsay, S. R., Wood, G. R. and Woollons R. C. (1996). Modelling the diameter distribution of forest stands using the Burr distribution. Journal of Applied Statistics 23, 609–620.
• Maller, R. and Zhou, X. (1996). Survival Analysis with Long-term Survivors. New York: Wiley.
• Mielke, P. W. (1973). Another family of distributions for describing and analyzing precipitation data. Journal of Applied Meteorology 12, 275–280.
• Mielke, P. W. and Johnson, E. S. (1973). Three-parameter kappa distribution maximum likelihood estimates and likelihood ratio test. Monthly Weather Review 101, 701–707.
• Mokhlis, N. A. (2005). Reliability of a stress-strength model with Burr type III distributions. Communications in Statistics—Theory and Methods 34, 1643–1657.
• Nadarajah, S. and Gupta, A. K. (2004). The beta Fréchet distribution. Far East Journal of Theoretical Statistics 14, 15–24.
• Nadarajah, S. and Kotz, S. (2004). The beta Gumbel distribution. Mathematical Problems in Engineering 10, 323–332.
• Nadarajah, S. and Kotz, S. (2006). The beta exponential distribution. Reliability Engineering and System Safety 91, 689–697.
• Ortega, E. M. M., Rizato, H. and Demétrio, C. G. B. (2009). The generalized log-gamma mixture model with covariates: Local influence and residual analysis. Statistics Methods and Applications 18, 305–331.
• Paranaíba, P. F., Ortega, E. M. M., Cordeiro, G. M. and Pescim, R. R. (2011). The beta Burr XII distribution with application to lifetime data. Computational Statistics and Data Analysis 55, 1118–1136.
• Pescim, R. R., Demétrio, C. G. B, Cordeiro, G. M., Ortega, E. M. M. and Urbano, M. R. (2010). The beta generalized half-normal distribution. Computational Statistics and Data Analysis 54, 945–957.
• Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I. (1986). Integrals and Series, Vol. 1. Amsterdam: Gordon and Breach Science Publishers.
• Rodriguez, R. N. (1980). Multivariate Burr III distributions, Part I. Theoretical properties. Research Publication GMR-3232, General Motors Research Laboratories, Warren, MI.
• Shao, Q. (2000). Estimation for hazardous concentrations based on NOEC toxicity data: An alternative approach. Environmetrics 11, 583–595.
• Shao, Q., Chen, Y. D. and Zhang, L. (2008). An extension of three-parameter Burr III distribution for low-flow frequency analysis. Computational Statistics and Data Analysis 52, 1304–1314.
• Sherrick, B. J., Garcia, P. and Tirupattur, V. (1996). Recovering probabilistic information from option markets: Test of distributional assumptions. The Journal of Future Markets 16, 545–560.
• Silva, G. O., Ortega, E. M. M. and Cordeiro, G. M. (2010). The beta modified Weibull distribution. Lifetime Data Analysis 16, 409–430.
• Sy, J. P. and Taylor, M. M. G. (2000). Estimation in a proportional hazards cure model. Biometrics 56, 227–336.