Convex orders for linear combinations of random variables. (English) Zbl 1131.60303

Summary: Linear combinations \(\sum_{i=1}^kb_iX_i\) and \(\sum_{i=1}^ka_iX_i\) of random variables \(X_1,\dots,X_k\) are ordered in the sense of the decreasing convex order and the Laplace order, where \((b_1,\dots,b_k)\) is majorized by \((a_1,\dots,a_k)\), when the underlying random variables are independent but possibly nonidentically distributed, and the joint density is arrangement increasing, respectively. Finite mixture distributions \(\sum_{i=1}^ka_iF_iX_i(x)\) and \(\sum_{i=1}^kb_iF_iX_i(x)\) are compared in the sense of the usual stochastic order, the convex order and higher-order stochastic dominance. The comparison between \(\sum_{i=1}^k b_iX_i\) and \(\sum_{i=1}^ka_iX_i\) is also studied for binary random variables \(I_{a_i}\), \(I_{b_i}\) \((i=1,\dots,k)\). Some applications in economics and reliability are described.


60E15 Inequalities; stochastic orderings
Full Text: DOI


[1] Alzaid, A.; Kim, J. S.; Proschan, F., Laplace ordering and its applications, J. Appl. Probab., 28, 116-130 (1991) · Zbl 0721.60097
[4] Arnold, B. C.; Villaseñor, J. A., Lorenz ordering of mean and medians, Statist. Probab. Lett., 4, 47-49 (1986) · Zbl 0587.60017
[5] Bock, M. E.; Diaconis, P.; Huffer, H. W.; Perlman, M. D., Inequalities for linear combinations of gamma random variables, Canad. J. Statist., 15, 387-395 (1987) · Zbl 0653.60017
[6] Caballé, J.; Pomansky, A., Mixed risk aversion, J. Econom. Theory, 71, 485-513 (1996) · Zbl 0877.90009
[7] Capéraà, P., Tail ordering and asymptotic efficiency of rank tests, Ann. Statist., 16, 470-478 (1988) · Zbl 0638.62043
[8] Eaton, M. L.; Olshen, R. A., Random quotients and the Behrens-Fisher problem, Ann. Math. Statist., 43, 1852-1860 (1972) · Zbl 0255.62024
[10] Fishburn, P. C., Continua of stochastic dominance relations for bounded probability distributions, J. Math. Econom., 3, 295-311 (1976) · Zbl 0352.60015
[11] Fishburn, P. C., Continua of stochastic dominance relations for unbounded probability distributions, J. Math. Econom., 7, 271-285 (1980) · Zbl 0449.90006
[12] Kanter, M., Probabilities for convex sets and multidimensional concentration functions, J. Multivariate Anal., 6, 222-236 (1976) · Zbl 0347.60043
[13] Karlin, S. J.; Novikoff, A., Generalized convex inequalities, Pacific J. Math., 13, 1251-1279 (1963) · Zbl 0126.28102
[14] Lam, D., The dynamics of population growth, differential fertility, and inequality, Amer. Econom. Rev., 76, 1103-1116 (1986)
[15] Landsberger, M.; Meilijson, I., Demand for risky financial assets: a portfolio analysis, J. Econom. Theory, 50, 204-213 (1990) · Zbl 0723.90006
[16] Levy, H., Stochastic dominance and expected utility: survey and analysis, Manage. Sci., 38, 555-593 (1992) · Zbl 0764.90004
[18] Ma, C., On peakedness of distributions of convex combinations, J. Statist. Plan. Infer., 70, 51-56 (1998) · Zbl 1067.60500
[20] Marshall, A. W.; Proschan, F., An inequality for convex functions involving majorization, J. Math. Anal. Appl., 12, 87-90 (1965) · Zbl 0145.28601
[21] O’Cinneide, C. A., Phase-type distributions and majorization, Ann. Appl. Prob., 1, 219-227 (1991) · Zbl 0729.60069
[28] Shaked, M.; Wong, T., Stochastic orders based on ratios of Laplace transforms, J. Appl. Probab., 34, 404-419 (1997) · Zbl 0893.60054
[31] Williamson, R. E., Multiply monotone functions and their Laplace transforms, Duke Math. J., 23, 189-207 (1956) · Zbl 0070.28501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.