Sankhya: The Indian Journal of Statistics

2001, Volume 63, Series A, Pt. 1, pp.128--132

STOCHASTIC INEQUALITIES FOR THE WEIGHTED SUMS OF PAIRWISE I.I.D. GENERALIZED GAMMA RANDOM VARIABLES

By

CHIN-YUAN HU, National Changhua University of Education, Taiwan, Republic of China

and

GWO DONG LIN, Academia Sinica, Taiwan, Republic of China

SUMMARY. Let X1 and X2 be two independent random variables having a common distribution gp,q with density function gp,q(x)= (p/t(q/p))x(q-1), x>0, where p, q > 0.Then we prove that for t>0, the function ft(a1,a2)=P((a1)1/2X1+(a2)1/2X1>=t) is Schur-concave on the set D+= {(a1,a2):a1>= a2=0}. This extends the previous result about the generalized Rayleigh distribution G(2,q) and somewhat answers a question posed by Hitczenko (1998). An inequality for the double generalized gamma distribution is also given.

AMS (1991) subject classification. Primary 60E05, 60E15.

Key words and phrases. Generalized gamma distribution, generalized Rayleigh distribution, double generalized gamma distribution, majorization, Schur-concave function, stochastic order.

Full paper (PDF)