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Moments of some J-shaped distributions

Author

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  • Saralees Nadarajah
  • Samuel Kotz

Abstract

This paper concerns a family of univariate distributions suggested by Topp & Leone in 1955. Topp & Leone provided no motivation for this new family and by way of properties they derived only the first four integer-order moments, i.e. E(Xn) for n=1, r 2, r 3, r 4 . In this paper we provide a motivation for the family of distributions and derive explicit algebraic expressions for: (1) hazard rate function; (2) E(Xn) when n ± 1 is any integer; (3) E(Xn) for n=1, r 2, r … r , r 10 , and (4) E[{X-E(X)} n] , n=2, r 3, r 4 . We also give an expression for the characteristic function and discuss issues on estimation and simulation. The main calculations of this paper use properties of the Gauss hypergeometric function.

Suggested Citation

  • Saralees Nadarajah & Samuel Kotz, 2003. "Moments of some J-shaped distributions," Journal of Applied Statistics, Taylor & Francis Journals, vol. 30(3), pages 311-317.
    • Handle: RePEc:taf:japsta:v:30:y:2003:i:3:p:311-317
    DOI: 10.1080/0266476022000030084
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    Citations

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    Cited by:

    1. MirMostafaee, S.M.T.K., 2014. "On the moments of order statistics coming from the Topp–Leone distribution," Statistics & Probability Letters, Elsevier, vol. 95(C), pages 85-91.
    2. Mustapha Muhammad & Lixia Liu & Badamasi Abba & Isyaku Muhammad & Mouna Bouchane & Hexin Zhang & Sani Musa, 2023. "A New Extension of the Topp–Leone-Family of Models with Applications to Real Data," Annals of Data Science, Springer, vol. 10(1), pages 225-250, February.
    3. Komal Shekhawat & Vikas Kumar Sharma, 2021. "An Extension of J-Shaped Distribution with Application to Tissue Damage Proportions in Blood," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 548-574, November.
    4. Ali Genç, 2012. "Moments of order statistics of Topp–Leone distribution," Statistical Papers, Springer, vol. 53(1), pages 117-131, February.
    5. David E. Giles, 2012. "A Note on Improved Estimation for the Topp-Leone Distribution," Econometrics Working Papers 1203, Department of Economics, University of Victoria.
    6. Rashad Bantan & Mahmoud Elsehetry & Amal S. Hassan & Mohammed Elgarhy & Dreamlee Sharma & Christophe Chesneau & Farrukh Jamal, 2021. "A Two-Parameter Model: Properties and Estimation under Ranked Sampling," Mathematics, MDPI, vol. 9(11), pages 1-16, May.
    7. Ahmad A. Zghoul, 2010. "Order statistics from a family of J-shaped distributions," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 127-136.
    8. Emrah Altun & Gauss M. Cordeiro, 2020. "The unit-improved second-degree Lindley distribution: inference and regression modeling," Computational Statistics, Springer, vol. 35(1), pages 259-279, March.
    9. 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.
    10. Francesca Condino & Filippo Domma & Giovanni Latorre, 2018. "Likelihood and Bayesian estimation of $$P(Y{," Statistical Papers, Springer, vol. 59(2), pages 467-485, June.
    11. 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.
    12. M. E. Ghitany & S. Kotz & M. Xie, 2005. "On some reliability measures and their stochastic orderings for the Topp-Leone distribution," Journal of Applied Statistics, Taylor & Francis Journals, vol. 32(7), pages 715-722.
    13. Saralees Nadarajah, 2009. "Bathtub-shaped failure rate functions," Quality & Quantity: International Journal of Methodology, Springer, vol. 43(5), pages 855-863, September.
    14. Ramadan A. ZeinEldin & Christophe Chesneau & Farrukh Jamal & Mohammed Elgarhy, 2019. "Different Estimation Methods for Type I Half-Logistic Topp–Leone Distribution," Mathematics, MDPI, vol. 7(10), pages 1-23, October.

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