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Polya-Type, Schur-Concave and Related Probability Distributions

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Transactions of the Eighth Prague Conference

Part of the book series: Czechoslovak Academy of Sciences ((TPCI,volume 8A))

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Abstract

The objective of this paper is to give a unified and an extended structure of the above classes of distributions as well as their many applications. The behaviour of these families is investigated under most commonly occurring functional operations such as closure under convolution, a passage to a limit in the weak sense, reversal and mixing properties. The useful and smooth properties of unimodality, strong unimodality and their variants are found to hold for some subclasses of these applicable probability distributions.

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References

  1. A-Hameed, M.S. and Proschan, F. (1973): Nonstationary shock models. Stoch. Proc. Appl., 1, 383–404.

    Article  MathSciNet  Google Scholar 

  2. Ahmad, R. (1974): On the structure of symmetric sample testing: a distribution-free approach. Ann. Inst. Statist. Math., 26, 233–245.

    Article  MATH  MathSciNet  Google Scholar 

  3. Ahmad, R. (1975): Some characterizations of the exchangeable processes and distribution-free tests in, Statistical Distributions in Scientific Work (eds. G.P. Patil et al.), Vol. 3, D. Reidel Pub. Co., Dordrecht, 237–248

    Google Scholar 

  4. Ahmad, R. and Abouammoh, A.M. (1977a): On the structure and applications of infinite divisibility, stability and symmetry in stochastic inference, in Recent Developments in Statistics (eds. J.R. Barra et al.), North-Holland, Amsterdam 1977, 303–317.

    Google Scholar 

  5. Ahmad, R. and Abouammoh, A.M. (1977b): On infinite A-divisibility. To appear.

    Google Scholar 

  6. Barndorff-Nielsen, O. (1976): Plausibility inference. J. Roy. Statist. Soc. Ser. B, 38, 103–131.

    MATH  MathSciNet  Google Scholar 

  7. Barlow, R.E. and Proschan, F. (1975); Mathematical Theory of Reliability and Life Testing. Holt, Rinehart and Winston.

    Google Scholar 

  8. Birnbaum, Z.W., Esary, J.D. and Marshall, A.W. (1966): Stochastic characterization for components and systems. Ann. Math. Statist., 37, 316–325.

    Article  MathSciNet  Google Scholar 

  9. Block, H.W. and Savits, T.H. (1976): The IFRA closure problem. Ann. prob., 4, 1030–1032.

    Article  MATH  MathSciNet  Google Scholar 

  10. Borell, C. (1975): Convex set in d-space.Period. Math. Hungar., 6, 111–136.

    Article  MathSciNet  Google Scholar 

  11. Ghurye, S.G. and Wallace, D.L. (1959): A convolution class of Monotone Likelihood ratio families. Ann. Math. Statist., 30, 1158–1164.

    Article  MATH  MathSciNet  Google Scholar 

  12. Hardy, G.H., Littlewood, C-.E. and Polya, A. (1952): Inequalities. 2nd ed., Cambridge Univ. press. Cambridge.

    MATH  Google Scholar 

  13. Hewitt, E. and Savage, L.J. (1955): Symmetric measures on Cartesian products. Trans. Amer. Math. Soc., 80, 470–501.

    Article  MATH  MathSciNet  Google Scholar 

  14. Hollander, M., Proschan, F. and Sethuraman, J. (1977): Functional decreasing in transposition and their applications in ranking problems. Ann. Statist., 5, 722–734.

    Article  MATH  MathSciNet  Google Scholar 

  15. Johnson, N.L. and Kotz, S. (1972): Continuous Multivariate Distributions. -2. Houghton Mifflin Co.

    MATH  Google Scholar 

  16. Kanter, M. (1977): Unimodality and dominance for symmetric random vectors. Trans. Amer. Math. Soc., 227, 65–85.

    Article  MathSciNet  Google Scholar 

  17. Karlin, S. (1956): Decision theory for Polya type distributions, case of two actions, I. Proceeding of the third Berkeley symposium on Prob, and Statist., Vol. 1, Univ. of California press, 115–129. (1957): Polya type distributions, Ann. Math. Statist., 23, 231–308.

    Google Scholar 

  18. Lehmann, E.L. (1959): Testing Statistical Hypotheses. Wiley, New York.

    MATH  Google Scholar 

  19. Lukacs, E. (1970). Characteristic Functions. 2nd ed. Griffin, London.

    MATH  Google Scholar 

  20. Mudholkar, G.S. (1966): The integral of an invariant unimodal function over an invariant convex set - an inequality and applications. Proc. Amer. Math. Soc., 17, 1327–1333.

    MATH  MathSciNet  Google Scholar 

  21. Marshall, A.W. and Olkin, I. (1974): Majorization in Multivariate distributions. Ann. Statist., 2, 1189–1200.

    Article  MATH  MathSciNet  Google Scholar 

  22. Parthasarathy, K.R. (1967): Probability Measures on Metric Spaces. Academic Press. New York.

    MATH  Google Scholar 

  23. Prekopa, A. (1973): On logarithmic concave measures and functions. Acta Sci. Math. (Szeged), 34, 335–343.

    MATH  MathSciNet  Google Scholar 

  24. Proschan, F. and Sethuraman, J. (1977); Schur functions in statistics I. The preservation theorem. Ann. Statist., 5, 256–262.

    MATH  MathSciNet  Google Scholar 

  25. Rinott, Y. (1976): On convexity of measures. Ann. Prob., 4, 1020–1026.

    Article  MATH  MathSciNet  Google Scholar 

  26. Schur, I. (1923): Ubereine Klasse notl Mittelbildungenmit Anmendungen auf die Determinantentheorie. Sitzber. Berl. Math. Ges, 22, 9–20.

    Google Scholar 

  27. Shaked, M. (1977): A concept of positive dependence for exchangeable random variables. Ann. Statist., 5, 505–515.

    Article  MATH  MathSciNet  Google Scholar 

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© 1978 ACADEMIA, Publishing House of the Czechoslovak Academy of Sciences, Prague

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Ahmad, R., Abouammoh, A.M. (1978). Polya-Type, Schur-Concave and Related Probability Distributions. In: Transactions of the Eighth Prague Conference. Czechoslovak Academy of Sciences, vol 8A. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-9857-5_5

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  • DOI: https://doi.org/10.1007/978-94-009-9857-5_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-9859-9

  • Online ISBN: 978-94-009-9857-5

  • eBook Packages: Springer Book Archive

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