A new class of distributions on the whole real line based on the continuous iteration approach

References

• Arslan, O., and A. İ. Genç. 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):481–98. doi:10.1080/02331880802401241.
   Web of Science ®Google Scholar
• Ayebo, A., and T. Kozubowski. 2004. An asymmetric generalization of Gaussian and Laplace laws. Journal of Probability and Statistical Science 1:187–210.
   Google Scholar
• Azzalini, A. 2005. The skew-normal distribution and related multivariate families. Scandinavian Journal of Statistics 32 (2):159–88. doi:10.1111/j.1467-9469.2005.00426.x.
   Web of Science ®Google Scholar
• Balakrishnan, N., and S. Kocherlakota. 1985. On the double Weibull distribution: Order statistics and estimation. Sankhyā, the Indian Journal of Statistics, Series B (1960-2002) 47 (2):161–78.
   Google Scholar
• BenSaïda, A., and S. Slim. 2016. Highly flexible distributions to fit multiple frequency financial returns. Physica A: Statistical Mechanics and Its Applications 442:203–13. doi:10.1016/j.physa.2015.09.021.
   Web of Science ®Google Scholar
• Bindu, P., and K. Sangita. 2015. Double Lomax distribution and its applications. Statistica 75 (3):331–42.
   Web of Science ®Google Scholar
• Cabral, I., F. Caeiro, and M. I. Gomes. 2020. On the comparison of several classical estimators of the extreme value index. Communications in Statistics - Theory and Methods. Advance online publication. doi:10.1080/03610926.2020.1746970.
   Web of Science ®Google Scholar
• Çankaya, M. N., and O. Arslan. 2020. On the robustness properties for maximum likelihood estimators of parameters in exponential power and generalized t distributions. Communications in Statistics - Theory and Methods 49 (3):607–30. doi:10.1080/03610926.2018.1549243.
   Web of Science ®Google Scholar
• Çankaya, M. N., A. Yalçınkaya, Ö. Altındaǧ, and O. Arslan. 2019. On the robustness of an epsilon skew extension for Burr III distribution on the real line. Computational Statistics 34 (3):1247–73. doi:10.1007/s00180-018-0859-y.
   Web of Science ®Google Scholar
• Contreras-Reyes, J. E., M. Maleki, and D. D. Cortés. 2019. Skew-reflected-Gompertz information quantifiers with application to sea surface temperature records. Mathematics 7 (5):403. doi:10.3390/math7050403.
   Web of Science ®Google Scholar
• Dijoux, Y. 2020. Construction of the tetration distribution based on the continuous iteration of the exponential-minus-one function. Applied Stochastic Models in Business and Industry 36 (5):891–916. doi:10.1002/asmb.2538.
   Web of Science ®Google Scholar
• Fernández, C., and M. F. J. Steel. 1998. On Bayesian modeling of fat tails and skewness. Journal of the American Statistical Association 93 (441):359–71.
   Web of Science ®Google Scholar
• Ferreira, J. T. A. S., and M. F. J. Steel. 2006. A constructive representation of univariate skewed distributions. Journal of the American Statistical Association 101 (474):823–9. doi:10.1198/016214505000001212.
   Web of Science ®Google Scholar
• Girard, S. 2004. A Hill type estimator of the Weibull tail-coefficient. Communications in Statistics - Theory and Methods 33 (2):205–34. doi:10.1081/STA-120028371.
   Web of Science ®Google Scholar
• Hanke, M., R. Poulsen, and A. Weissensteiner. 2015. Where would the EUR/CHF exchange rate be without the SNB’s minimum exchange rate policy? Journal of Futures Markets 35 (12):1103–16. doi:10.1002/fut.21704.
   Web of Science ®Google Scholar
• Hoseinzadeh, A., M. Maleki, Z. Khodadadi, and J. E. Contreras-Reyes. 2019. The skew-reflected-Gompertz distribution for analyzing symmetric and asymmetric data. Journal of Computational and Applied Mathematics 349:132– 41. doi:10.1016/j.cam.2018.09.011.
   Web of Science ®Google Scholar
• Hui, C.-H., C.-F. Lo, and T.-W. Fong. 2016. Swiss franc’s one-sided target zone during 2011-2015. International Review of Economics & Finance 44:54–67. doi:10.1016/j.iref.2016.03.004.
   Web of Science ®Google Scholar
• Jones, M. C. 2015. On families of distributions with shape parameters. International Statistical Review 83 (2):175–92. doi:10.1111/insr.12055.
   Web of Science ®Google Scholar
• Jurić, V., T. J. Kozubowski, and M. Perman. 2020. An asymmetric multivariate Weibull distribution. Communications in Statistics - Theory and Methods 49 (18):4394–412. doi:10.1080/03610926.2019.1599949.
   Web of Science ®Google Scholar
• Kotz, S., T. Kozubowski, and K. Podgorski. 2001. The Laplace distribution and generalizations. Boston: Birkhauser Publication.
   Google Scholar
• Lera, S., and D. Sornette. 2016. Quantitative modelling of the EUR/CHF exchange rate during the target zone regime of September 2011 to January 2015. Journal of International Money and Finance 63:28–47. doi:10.1016/j.jimonfin.2016.01.002.
   Web of Science ®Google Scholar
• Ley, C. 2015. Flexible modelling in statistics: Past, present and future. Journal de la Société Française de Statistique 156:176–96.
   Google Scholar
• Linden, M. 2001. A model for stock return distribution. International Journal of Finance & Economics 6 (2):159–69. doi:10.1002/ijfe.149.
   Web of Science ®Google Scholar
• Mac’Oduol, B. V., P. J. van Staden, and R. A. R. King. 2020. Asymmetric generalizations of symmetric univariate probability distributions obtained through quantile splicing. Communications in Statistics - Theory and Methods 49 (18):4413–29. doi:10.1080/03610926.2019.1601219.
   Web of Science ®Google Scholar
• Mcdonald, J., and W. Newey. 1988. Partially adaptive estimation of regression models via the generalized t distribution. Econometric Theory 4 (3):428–57. doi:10.1017/S0266466600013384.
   Web of Science ®Google Scholar
• Mudholkar, G. S., and A. D. Hutson. 2000. The epsilon-skew-normal distribution for analyzing near-normal data. Journal of Statistical Planning and Inference 83 (2):291–309. doi:10.1016/S0378-3758(99)00096-8.
   Web of Science ®Google Scholar
• Rao, A. V. D., and V. L. Narasimham. 1989. Linear estimation in double Weibull distribution. Sankhyā, the Indian Journal of Statistics, Series B (1960-2002) 51 (1):24–64.
   Google Scholar
• Szekeres, G. 1958. Regular iteration of real and complex functions. Acta Mathematica 100 (3–4):203–58. doi:10.1007/BF02559539.
   Web of Science ®Google Scholar
• Szekeres, G. 1962. Fractional iteration of exponentially growing functions. Journal of the Australian Mathematical Society 2 (3):301–20. doi:10.1017/S1446788700026902.
   Google Scholar
• Theodossiou, P. 1998. Financial data and the skewed generalized t distribution. Management Science 44 (12-part-1):1650–61. doi:10.1287/mnsc.44.12.1650.
   Web of Science ®Google Scholar
• Toth, D., and B. Jones. 2019. Against the Norm: Modeling daily stock returns with the Laplace distribution. https://arxiv.org/abs/1906.10325
   Google Scholar
• Walker, P. 1991. Infinitely differentiable generalized logarithmic and exponential functions. Mathematics of Computation 57 (196):723–33. doi:10.1090/S0025-5718-1991-1094963-4.
   Web of Science ®Google Scholar
• Walker, P. L. 1990. The exponential of iteration of exp(x)-1. Proceedings of the American Mathematical Society 110 (3):611–20.
   Web of Science ®Google Scholar
• Wallis, K. F. 2014. The two-piece normal, binormal, or double Gaussian distribution: Its origin and rediscoveries. Statistical Science 29 (1):106–12. doi:10.1214/13-STS417.
   Web of Science ®Google Scholar