On the robustness of an epsilon skew extension for Burr III distribution on the real line

  • Mehmet Niyazi Çankaya
  • Abdullah YalçınkayaEmail author
  • Ömer Altındaǧ
  • Olcay Arslan
Original Paper


Burr III (BIII) distribution is used in a wide variety of fields, such as lifetime data analysis, reliability theory, and financial literature, and suchlike. It is defined on the positive axis and has two shape parameters, say c and k. These shape parameters make the distribution quite flexible. They also control the tail behaviour of the distribution. In this study, we extend BIII distribution to the real line and also add a skewness parameter, say \(\varepsilon \), with an epsilon skew extension approach. When the parameters c and k have a relationship such that \(ck \le 1 \), it is skew unimodal. Otherwise, it is skew bimodal with the same level of peaks on the negative and positive sides of the real line. Thus, the epsilon skew extension of Burr III (ESBIII) distribution with only three parameters can provide adequate fits for data sets that may have heavy-tailedness, skewness, unimodality or bimodality. A location-scale form of this distribution is also given. Distributional properties are investigated. The maximum likelihood (ML) estimation method for the parameters of ESBIII is considered. The robustness properties of the ML estimators are studied in terms of the boundedness of the influence function. Further, tail behaviour of ESBIII distribution is also examined to explore the robustness of ESBIII distribution against the outliers. The modelling capacity of this distribution is illustrated using two real data examples.


Burr III distribution Epsilon skew extension Asymmetry Robustness Bimodality Unimodality 



We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions. We would like to say our special thanks to the Editorial Board Members. We thank the native speaker who supported us for representation in the formal organization of this paper.


  1. Abd-Elfattah AM, Alharbey AH (2012) Bayesian estimation for Burr distribution type III based on trimmed samples. ISRN Appl Math 250393:1–18MathSciNetzbMATHGoogle Scholar
  2. Abdulah E, Elsalloukh H (2013) Analyzing skewed data with the epsilon skew gamma distribution. J Stat Appl Prob 2(3):195–202Google Scholar
  3. Acıtaş S, Kasap P, Şenoǧlu B, Arslan O (2013) One-step M-estimators: Jones and Faddy’s skewed t-distribution. J Appl Stat 40(7):1–15MathSciNetGoogle Scholar
  4. Ali A, Hasnain SA, Ahmad M (2015) Modified Burr III distribution: properties and applications. Pak J Stat 31(6):697–708MathSciNetGoogle Scholar
  5. Andrade BB, Rathie PN (2016) Fitting asymmetric bimodal data with selected distributions. J Stat Comput Simul 86(16):3205–3224MathSciNetGoogle Scholar
  6. Arellano-Valle RB, Gomez HW, Quintana FA (2005) Statistical inference for a general class of asymmetric distributions. J Stat Plan Inference 128(2):427–443MathSciNetzbMATHGoogle Scholar
  7. Arellano-Valle RB, Cortés MA, Gómez HW (2010) An extension of the epsilon-skew-normal distribution. Commun Stat Theory Methods 39(3):912–922MathSciNetzbMATHGoogle Scholar
  8. Arslan O (2009a) Maximum likelihood parameter estimation for the multivariate skew slash distribution. Stat Probab Lett 79(20):2158–2165MathSciNetzbMATHGoogle Scholar
  9. Arslan O (2009b) An alternative multivariate skew Laplace distribution: properties and estimation. Stat Pap 51(4):865–887MathSciNetzbMATHGoogle Scholar
  10. Arslan O, Genç AI (2009) The skew generalized t distribution as the scale mixture of a skew exponential power distribution and its applications in robust estimation. Statistics 43(5):481–498MathSciNetzbMATHGoogle Scholar
  11. Azimi R, Yaghmaei F (2013) Bayesian estimation for the burr type III distribution under type II Doubly censored data. Int J Adv Stat Probab 1(1):1–3Google Scholar
  12. Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12(2):171–178MathSciNetzbMATHGoogle Scholar
  13. Azzalini A, Capitanio A (2003) Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t distribution. J R Stat Soc Ser B 65(2):367–389MathSciNetzbMATHGoogle Scholar
  14. Bolfarine H, Martínez-Flórez G, Salinas HS (2013) Bimodal symmetric-asymmetric power-normal families. Commun Stat Theory Methods 47(2):259–276MathSciNetzbMATHGoogle Scholar
  15. Box GEP, Tiao GC (1973) Bayesian inference in statistical analysis. Addison-Wesley, ReadingzbMATHGoogle Scholar
  16. Burnham KP, Anderson DR (2002) Model selection and multimodel inference: a practical information-theoretic approach, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  17. Burr IW (1942) Cumulative frequency functions. Ann Math Stat 13(2):215–232MathSciNetzbMATHGoogle Scholar
  18. Burr IW (1973) Parameters for a general system of distributions to match a grid of \(\alpha _3\) and \(\alpha _4\). Commun Stat Theory Methods 2(1):1–21MathSciNetGoogle Scholar
  19. Burr IW, Cislak PJ (1968) On a general system of distributions I. Its curve-shape characteristics II. The sample median. J Am Stat Assoc 63(322):627–635MathSciNetGoogle Scholar
  20. Çankaya MN (2018) Asymmetric bimodal exponential power distribution on the real line. Entropy 20(23):1–19Google Scholar
  21. Çankaya MN, Korbel J (2017) On statistical properties of Jizba–Arimitsu hybrid entropy. Physica A 475:1–10MathSciNetzbMATHGoogle Scholar
  22. Çankaya MN, Bulut YM, Doğru FZ, Arslan O (2015) A bimodal extension of the generalized gamma distribution. Rev Colomb Estad 38(2):353–370MathSciNetGoogle Scholar
  23. Cooray K (2013) Exponentiated sinh Cauchy distribution with applications. Commun Stat Theory Methods 42(21):3838–3852MathSciNetzbMATHGoogle Scholar
  24. Dexter OC (2015) Some skew-symmetric distributions which include the bimodal ones. Commun Stat Theory Methods 44(3):554–563MathSciNetzbMATHGoogle Scholar
  25. Donatella V, Van Dorp JR (2013) On a bounded bimodal two-sided distribution fitted to the old-Faithful geyser data. J Appl Stat 40(9):1965–1978MathSciNetGoogle Scholar
  26. Elsalloukh H (2008) The epsilon-skew Laplace distribution. In: The proceedings of the American statistical association, Biometrics Section, Denever ColoradoGoogle Scholar
  27. Elsalloukh H, Guardiola JH, Young M (2005) The epsilon-skew exponential power distribution family. Far East J Theor Stat 17(1):97–107MathSciNetzbMATHGoogle Scholar
  28. Embrechts P, Kluppelberg C, Mikosch T (1997) Modelling extremal events for insurance and finance. Springer, HeidelbergzbMATHGoogle Scholar
  29. Genç AI (2013) A skew extension of the slash distribution via beta-normal distribution. Stat Pap 54(2):427–442MathSciNetzbMATHGoogle Scholar
  30. Genton MG (2004) Skew-elliptical distributions and their applications: a journey beyond normality. Chapman & Hall/CRC, LondonzbMATHGoogle Scholar
  31. Gómez HW, Torres FJ, Bolfarine H (2007) Large-sample inference for the epsilon-skew-t distribution. Commun Stat Theory Methods 36(1):73–81MathSciNetzbMATHGoogle Scholar
  32. Gove JH, Ducey MJ, Leak WB, Zhang L (2008) Rotated sigmoid structures in managed uneven-aged northern hardwork stands: a look at the Burr Type III distribution. Foresty 81(2):161–176Google Scholar
  33. Gui W (2014) A generalization of the slashed distribution via alpha skew normal distribution. Stat Methods Appl 23(1):1–17MathSciNetGoogle Scholar
  34. Hampel FR, Ronchetti EM, Rousseeuw PJ, Stahel WA (1986) Robust statistics: the approach based on influence functions. Wiley, New YorkzbMATHGoogle Scholar
  35. Hao Z, Singh VP (2009) Entropy-based parameter estimation for extended three-parameter Burr III distribution for low-flow frequency analysis. Trans ASABE 52(4):1193–1202Google Scholar
  36. Hassan MY, El-Bassiouni MY (2016) Bimodal skew-symmetric normal distribution. Commun Stat Theory Methods 45(5):1527–1541MathSciNetzbMATHGoogle Scholar
  37. Huber PJ (1984) Finite sample breakdown of M- and P-estimators. Ann Stat 12(1):119–126MathSciNetzbMATHGoogle Scholar
  38. Jamalizadeh A, Arabpour AR, Balakrishnan N (2011) A generalized skew two-piece skew-normal distribution. Stat Pap 52(2):431–446MathSciNetzbMATHGoogle Scholar
  39. Jizba P, Korbel J (2016) On q-non-extensive statistics with non-Tsallisian entropy. Physica A 444:808–827MathSciNetzbMATHGoogle Scholar
  40. Jones MC (2009) Kumaraswamy’s distribution: a beta-type distribution with some tractability advantages. Stat Methodol 6:70–81MathSciNetzbMATHGoogle Scholar
  41. Jones MC, Faddy MJ (2003) A skew extension of the t-distribution, with applications. J R Stat Soc Ser B 65:159–175MathSciNetzbMATHGoogle Scholar
  42. Lindsay SR, Wood GR, Woollons RC (1996) Modelling the diameter distribution of forest stands using the Burr distribution. J Appl Stat 23(6):609–620Google Scholar
  43. Lucas A (1997) Robustness of the student t based M-estimator. Commun Stat Theory Methods 26(5):1165–1182MathSciNetzbMATHGoogle Scholar
  44. Markovich N (2007) Nonparametric analysis of univariate heavy-tailed data: research and practice, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  45. Mineo AM, Ruggieri M (2005) A software tool for the exponential power distribution: the normalp package. J Stat Softw 12(4):1–24Google Scholar
  46. Mudholkar GS, Hutson AD (2000) The epsilon-skew-normal distribution for analyzing near-normal data. J Stat Plan Inference 83(2):291–309MathSciNetzbMATHGoogle Scholar
  47. Prommier Y, Reinhold W, Sunshine M, Varma S (2018) Genomics and bioinformatics group. Accessed 10 May 2018
  48. Purdom E, Holmes SP (2005) Error distribution for gene expression data. Stat Appl Genet Mol Biol 4(1):2194–6302MathSciNetzbMATHGoogle Scholar
  49. Rathie PN, Silva P, Olinto G (2016) Applications of skew models using generalized logistic distribution. Axioms 5(10):1–26zbMATHGoogle Scholar
  50. Rêgo LC, Cintra RJ, Cordeiro GM (2012) On some properties of the beta normal distribution. Commun Stat Theory Methods 41(20):3722–3738MathSciNetzbMATHGoogle Scholar
  51. Rényi A (1961) On measures of entropy and information. ISRN Applied Mathematics. Hungarian Academy of Sciences, Budapest Hungary, BudapestzbMATHGoogle Scholar
  52. Rodriguez RN (1977) A guide to the Burr type XII distributions. Biometrika 64(1):129–134MathSciNetzbMATHGoogle Scholar
  53. Shams HS, Alamatsaz MH (2013) Alpha–Skew–Laplace distribution. Stat Probab Lett 83(3):774–782MathSciNetzbMATHGoogle Scholar
  54. Shannon CE (1961) Two-way communication channels. In: Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, volume 1: contributions to the theory of statistics, The Regents of the University of CaliforniaGoogle Scholar
  55. Shao Q, Chen YD, Zhang L (2008) An extension of three-parameter Burr III distribution for low-flow frequency analysis. Comput Stat Data Anal 52(3):1304–1314zbMATHGoogle Scholar
  56. Subbotin MT (1923) On the law of frequency of errors. Mat Sb 31:296–301zbMATHGoogle Scholar
  57. Swamee PK, Rathie PN (2007) Invertible alternatives to normal and lognormal distributions. J Hydrol Eng 12(2):218–221Google Scholar
  58. Tadikamalla PR (1980) A look at the Burr and related distributions. Int Stat Rev 48(3):337–344MathSciNetzbMATHGoogle Scholar
  59. Venegas O, Hugo SS, Gallardo DI, Bolfarine H, Gómez HW (2017) Bimodality based on the generalized skew-normal distribution. J Stat Comput Simul 88(1):156–181MathSciNetGoogle Scholar
  60. Wang FK, Keats JB, Zimmer WJ (1996) Maximum likelihood estimation of the Burr XII parameters with censored and uncensored data. Microelectron Reliab 36(3):359–362Google Scholar
  61. Wingo DR (1983a) Maximum Likelihood Methods for fitting the Burr type XII distribution of life test data. Bio Met J 25(1):77–84MathSciNetzbMATHGoogle Scholar
  62. Wingo DR (1983b) Estimating the location of the Cauchy distribution by numerical global optimization. Commun Stat-Simul C 12(2):201–212Google Scholar
  63. Zimmer WJ, Keats JB, Wang FK (1998) The Burr XII distribution in reliability analysis. J Qual Technol 30(4):386–394Google Scholar
  64. Zoraghi N, Abbasi B, Niaki STA, Abdi M (2012) Estimating the four parameters of the Burr III distribution using a hybrid method of variable neighborhood search and iterated local search algorithms. Appl Math Comput 218(19):9664–9675MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Mehmet Niyazi Çankaya
    • 1
    • 2
  • Abdullah Yalçınkaya
    • 3
    Email author
  • Ömer Altındaǧ
    • 4
  • Olcay Arslan
    • 3
  1. 1.Applied Sciences School, Department of International TradingUşakTurkey
  2. 2.Department of Statistics, Faculty of Arts and SciencesUşak UniversityUşakTurkey
  3. 3.Department of Statistics, Faculty of ScienceAnkara UniversityAnkaraTurkey
  4. 4.Department of Statistics, Faculty of Arts and SciencesBilecik Şeyh Edebali UniversityBilecikTurkey

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