On skewness and dispersion among convolutions of independent gamma random variables Article

Amiri, L, Khaledi, BE, Samaniego, FJ. (2011). On skewness and dispersion among convolutions of independent gamma random variables . PROBABILITY IN THE ENGINEERING AND INFORMATIONAL SCIENCES, 25(1), 55-69. 10.1017/S0269964810000240

cited authors

  • Amiri, L; Khaledi, BE; Samaniego, FJ

authors

abstract

  • Let {x(1)≤⋯≤x(n)?} denote the increasing arrangement of the components of a vector x=(x⋯n). A vector x ∈ Rn majorizes another vector y (written x m y) if ∑i=1j x(i)} ≤ ∑i=1 j y (i) for j = 1,⋯, n-1 and ∑ i=1nx(i) = ∑i=1n y(i). A vector x isin; R+n majorizes reciprocally another vector y isin; R+n (written x y) if ∑i=1j (1/x(i)) ≥ ∑ i=1j (1/y(i)) for j = 1,⋯, n. Let X λi,α, i=1,⋯,n, be n independent random variables such that Xλi,α is a gamma random variable with shape parameter α ≥ 1 and scale parameter λi, i = 1,⋯, n. We show that if λ rm λ *, then ∑i=1n X λi,α is greater than ∑i=1n Xλ*α according to right spread order as well as mean residual life order. We also prove that if (1/ λ1 z.ast;,⋯,1λ1n*) m(1/ λ1z.ast;,⋯, 1λ1n*)$, then ∑i=1n Xλα, is greater than ∑i=1n Xλi*α according to new better than used in expectation order as well as Lorenze order. These results mainly generalize the recent results of Kochar and Xu [7] and Zhao and Balakrishnan [14] from convolutions of independent exponential random variables to convolutions of independent gamma random variables with common shape parameters greater than or equal to 1. © 2011 Cambridge University Press.

publication date

  • January 1, 2011

published in

Digital Object Identifier (DOI)

start page

  • 55

end page

  • 69

volume

  • 25

issue

  • 1