Application of discrete Burr XII distribution in the analysis of animal production data

Josmar Mazucheli, Ricardo Puziol Oliveira, Danielle Peralta, Isabele P. Emanuelli


In animal production, the models that mimicry the biological reality are of great importance for optimization and sustainability of the productive system. The continuous Burr XII distribution is widely used in survival data analysis, however, the same does not occur with its discrete version, recently proposed in the literature. The purpose of this work is to use the discrete Burr XII distribution, obtained by the discretization method proposed by Nakagawa and Osaki (1975), in the analysis of data related to animal production. The data analyzed describe the time, in days, from birth to first laying of yellow quail (Coturnix coturnix japonica) submitted to two diets. For this purpose the discretized versions of five distributions were used: the discrete Burr XII, the discrete Weibull, the discrete gamma, the discrete inverse-Gaussian and the discrete log-normal. For all distributions, the parameter estimates were obtained by the maximum likelihood method. Despite the similarity between the estimates it is natural to choose the discrete given the nature of the data and assuming the discrete distribution, it could be calculated exactly, for example, the probability of the time to the first posture, which is not possible if a continuous distribution is assumed. Thus, among the discrete distributions, the chi-square goodness-of-fit test showed that the Burr XII distribution was the only one indicated to describe the behavior of the data considered.


Animal production; Discretization; Likelihood function; Model selection; Probability function; Survival analysis


Akdoǧan, Y., Kuş, Klnacl, I., 2014. Point estimation of parameters in discrete Burr distribution based on type I censored sample. Journal of the Turkish Statistical Association

(69), 80–86.

Bakouch, H. S., Jazi, M. A., Nadarajah, S., 2014. A new discrete distribution. Statistics

(1), 200–240.

Bracquemond, C., Gaudoin, O., 2003. A survey on discrete lifetime distributions. International Journal of Reliability, Quality and Safety Engineering 10 (01), 69–98.

Burr, I. W., 1942. Cumulative frequency functions. Annals of Mathematical Statistics 13,


Chakraborty, S., 2015a. Generating discrete analogues of continuous probability distributions - A survey of methods and constructions. Journal of Statistical Distributions and

Applications 2 (1), 1–30.

Chakraborty, S., 2015b. A new discrete distribution related to generalized Gamma distribution and its properties. Communications in Statistics - Theory and Methods 44 (8),


Chakraborty, S., Chakravarty, D., 2012. Discrete Gamma distributions: properties and parameter estimations. Communications in Statistics - Theory and Methods 41 (18),


Collett, D., 2003. Modelling Survival Data in Medical Research, 2nd Edition. Chapaman and Hall, New York.

Davison, A. C., 2003. Statistical Models. Cambridge: Cambridge University Press, New 292 York.

Davison, A. C., Hinkley, D. V., 1997. Bootstrap Methods and Their Applications. Cambridge: Cambridge University Press, New York.

Delignette-Muller, M. L., Dutang, C., 2015. tdistrplus: An R package for tting distributions. Journal of Statistical Software 64 (4).

Efron, B., 1979. Bootstrap methods: Another look at the Jackknife. Annals of Statistics (1).

Efron, B., Tibshirani, R. J., 1993. An introduction to the Bootstrap. Vol. 57 of Monographs on Statistics and Applied Probability. Chapman and Hall, New York.

Ghosh, T., Roy, D., Chandra, N. K., 2013. Reliability approximation through the discretization of random variables using reversed hazard rate function. International Journal of Mathematical, Computational, Statistical, Natural and Physical Engineering 304 7 (4), 96-100.

Good, I. J., 1953. The population frequencies of species and the estimation of population parameters. Biometrika 40 (3-4), 237-264.

Hamada, M. S., Wilson, A. G., Reese, C. S., Martz, H. F., 2008. Bayesian Reliability. Springer Series in Statistics. Springer, New York.

Held, L., Sabanes Bove, D., 2014. Applied Statistical Inference -

Likelihood and Bayes. Springer, Heidelberg.

Kalbeisch, J. D., Prentice, R. L., 2002. The Statistical Analysis of Failure Time Data, 2nd Edition. Wiley, New York, NY.

Kamari, H., Bevrani, H., Kamary, K., 2015. Bayesian estimate 313 of discrete Burr distribution with two parameters. Research & Reviews: Journal of Statistics and Mathematical Sciences 1 (2), 62{68.

Kemp, A. W., 1997. Characterizations of a discrete Normal distribution. Journal of Statistical Planning and Inference 63 (2), 223 { 229.

Khorashadizadeh, M., Roknabadi, R. A. H., G. R. Borzadaran, M. G. R., 2013. Characterization of life distributions using Log-odds rate in discrete aging. Communications in Statistics - Theory and Methods 42 (1), 76{87.

Klein, J. P., Moeschberger, M. L., 1997. Survival Analysis: Techniques for Censored and Truncated Data. Springer-Verlag, New York.

Kotz, S., Y., L., Pensky, M., 2003. The Stress-Strength Model and its Generalizations: Theory and Applications. Academic Press.

Krishna, H., Pundir, P. S., 2009. Discrete Burr and discrete Pareto distributions. Statistical Methodology 6 (2), 177 { 188.

Kulasekera, K. B., Tonkyn, D. W., 1992. A new discrete distribution, with applications to survival, dispersal and dispersion. Communications in Statistics - Simulation and Computation 21 (2), 499{518.

Lai, C., 2013. Issues concerning constructions of discrete lifetime models. Quality Technology & Quantitative Management 10 (2), 251{262.

Lai, C., 2014. Discrete Weibull Distributions and Their Generalizations. Springer Berlin Heidelberg, Berlin, Heidelberg, pp. 97{113.

Lawless, J. F., 2003. Statistical Models and Methods for Lifetime Data, 2nd Edition. Wiley Series in Probability and Statistics. John Wiley & Sons, Hoboken, NJ.

Lee, E. T.,Wang, J.W., 2003. Statistical Methods for Survival Data Analysis, 3rd Edition. Wiley Series in Probability and Statistics. Wiley-Interscience [John Wiley & Sons],

Hoboken, NJ.

Meeker, W. Q., Escobar, L. A., 1998. Statistical Methods for 339 Reliability Data. John Wiley & Sons, New York.

Mrode, R. A. a., 2014. Linear Models for the Prediction of Animal Breeding Values, 3rd Edition. CABI, Nosworthy Way, UK.

Nakagawa, T., Osaki, S., 1975. The discrete Weibull distribution. IEEE Transactions on Reliability 5, 300{301.

Para, B. A., Jan, T. R., 2016. On discrete three parameter Burr type XII and discrete Lomax distributions and their applications to model count data from medical science. Biometrics & Biostatistics International Journal 4 (2), 1{15.

Pearson, K., 1895. Contributions to the mathematical theory of evolution. Philosophical Transactions of the Royal Society of London. A, 343{414.

Peralta, D., Mazucheli, J., Emanuelli, I. P., Rossi, R. M., 2017. Modelagem do tempo até a primeira postura de codornas. Revista Brasileira de Biometria 1 (35), 1{10.

R Core Team, 2016. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. URL

Rodriguez, R. N., 1977. A guide to the Burr type XII distributions. Biometrika 64 (1).

Roy, D., 2004. Discrete Rayleigh distribution. IEEE Transactions on Reliability 53 (2), 255{260.

Roy, D., Dasgupta, T., 2001. A discretizing approach for evaluating reliability of complex systems under stress-strength model. IEEE Transactions on Reliability 50 (2), 145{150.

Rupert, M., 2011. Survival Analysis. Vol. 66. John Wiley & Sons.

Sato, H., Ikota, M., Sugimoto, A., Masuda, H., 1999. A new defect distribution metrology with a consistent discrete exponential formula and its applications. Semiconductor Manufacturing, IEEE Transactions on 12 (4), 409{418.

Stacy, E. W., 1962. A generalization of the Gamma distribution. 364 The Annals of Mathematical Statistics 33 (3), 1187{1192.

Stein, W. E., Dattero, R., 1984. A new discrete Weibull distribution. IEEE Transactions on Reliability 33 (2), 196{197.

Tadikamalla, P. R., 1980. A look at the Burr and related distributions. International Statistical Review. Revue International de Statistique 48 (3), 337{344.

Voinov, V., Nikulin, M., Balakrishnan, N., 2013. Chi-Squared Goodness of Fit Tests with Applications. Academic Press.

Zimmer, W. J., Keats, J. B., Wang, F. K., 1998. The Burr XII distribution in reliability. Journal of Quality Technology 30 (4), 389{394.