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Two-sided generalized Topp and Leone (TS-GTL) distributions

Author

Listed:
  • Donatella Vicari
  • Johan Rene Van Dorp
  • Samuel Kotz

Abstract

Over 50 years ago, in a 1955 issue of JASA, a paper on a bounded continuous distribution by Topp and Leone [C.W. Topp and F.C. Leone, A family of J-shaped frequency functions, J. Am. Stat. Assoc. 50(269) (1955), pp. 209-219] appeared (the subject was dormant for over 40 years but recently the family was resurrected). Here, we shall investigate the so-called Two-Sided Generalized Topp and Leone (TS-GTL) distributions. This family of distributions is constructed by extending the Generalized Two-Sided Power (GTSP) family to a new two-sided framework of distributions, where the first (second) branch arises from the distribution of the largest (smallest) order statistic. The TS-GTL distribution is generated from this framework by sampling from a slope (reflected slope) distribution for the first (second) branch. The resulting five-parameter TS-GTL family of distributions turns out to be flexible, encompassing the uniform, triangular, GTSP and two-sided slope distributions into a single family. In addition, the probability density functions may have bimodal shapes or admitting shapes with a jump discontinuity at the 'threshold' parameter. We will discuss some properties of the TS-GTL family and describe a maximum likelihood estimation (MLE) procedure. A numerical example of the MLE procedure is provided by means of a bimodal Galaxy M87 data set concerning V-I color indices of 80 globular clusters. A comparison with a Gaussian mixture fit is presented.

Suggested Citation

  • Donatella Vicari & Johan Rene Van Dorp & Samuel Kotz, 2008. "Two-sided generalized Topp and Leone (TS-GTL) distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 35(10), pages 1115-1129.
    • Handle: RePEc:taf:japsta:v:35:y:2008:i:10:p:1115-1129
    DOI: 10.1080/02664760802230583
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    References listed on IDEAS

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    1. Samuel Kotz & J. René Dorp, 2004. "Uneven two-sided power distributions with applications in econometric models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 13(3), pages 285-313, December.
    2. M. Jones, 2004. "Families of distributions arising from distributions of order statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(1), pages 1-43, June.
    3. Saralees Nadarajah & Samuel Kotz, 2003. "Moments of some J-shaped distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 30(3), pages 311-317.
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    Cited by:

    1. Ali Genç, 2012. "Moments of order statistics of Topp–Leone distribution," Statistical Papers, Springer, vol. 53(1), pages 117-131, February.
    2. Francesca Condino & Filippo Domma, 2017. "A new distribution function with bounded support: the reflected generalized Topp-Leone power series distribution," METRON, Springer;Sapienza Università di Roma, vol. 75(1), pages 51-68, April.
    3. Francesca Condino & Filippo Domma & Giovanni Latorre, 2018. "Likelihood and Bayesian estimation of $$P(Y{," Statistical Papers, Springer, vol. 59(2), pages 467-485, June.
    4. Donatella Vicari & Johan Ren� van Dorp, 2013. "On a bounded bimodal two-sided distribution fitted to the Old-Faithful geyser data," Journal of Applied Statistics, Taylor & Francis Journals, vol. 40(9), pages 1965-1978, September.

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