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2 results

  • The r-monotonicity of generalized Bernstein polynomials

  • Part of
  • Laiyi Zhu, Zhiyong Huang
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 55 / Issue 3 / October 2012
    Published online by Cambridge University Press:
    26 July 2012, pp. 797-807
    • Article
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  • Let f ∊ C[0, 1] and let the Bn(f, q; x) be generalized Bernstein polynomials based on the q-integers that were introduced by Phillips. We prove that if f is r-monotone, then Bn(f, q; x) is r-monotone, generalizing well-known results when q = 1 and the results when r = 1 and r = 2 by Goodman et al. We also prove a sufficient condition for a continuous function to be r-monotone.

  • Dispersive ordering of distributions

  • Moshe Shaked
  • Journal:
    Journal of Applied Probability / Volume 19 / Issue 2 / June 1982
    Published online by Cambridge University Press:
    14 July 2016, pp. 310-320
    Print publication:
    June 1982

  • Two distributions, F and G, are said be ordered in dispersion if F-1(β)-F-1(α)≦G-1(β)-G-1(α) whenever 0<α <β <1. This relation has been studied by Saunders and Moran (1978). The purpose of this paper is to study this partial ordering in detail. Few characterizations of this concept are given. These characterizations are then used to show that some particular pairs of distributions are ordered by dispersion. In addition to it some proofs of results of Saunders and Moran (1978) are simplified. Furthermore, the characterizations of this paper can be used to throw a new light on the meaning of the underlying partial ordering and also to derive various inequalities. Several examples illustrate the methods of this paper.