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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.

##### MSC:
 6e+16 Inequalities; stochastic orderings
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##### References:
 [1] Alzaid, A.; Kim, J.S.; Proschan, F., Laplace ordering and its applications, J. appl. probab., 28, 116-130, (1991) · Zbl 0721.60097 [2] Arnold, B.C., 1987. Majorization and the Lorenz Order: A Brief Introduction. Springer, New York. · Zbl 0649.62041 [3] Arnold, B.C., 1991. Preservation and attenuation of inequality as measured by the Lorenz order. In: Mosler, K., Scarsini, M. (Eds.), Stochastic Orders and Decision under Risk, IMS Lecture Notes, Vol. 19, Institute of Mathematical Statistics, Hayward, CA, pp. 25-37. · Zbl 0756.62040 [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 [9] Feller, W., 1971. An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York. · Zbl 0219.60003 [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 [17] Lindsay, B.G., 1995. Mixture Models: Theory, Geometry and Applications. Institute of Mathematical Statistics, Hayward, CA. · Zbl 1163.62326 [18] Ma, C., On peakedness of distributions of convex combinations, J. statist. plan. infer., 70, 51-56, (1998) · Zbl 1067.60500 [19] Marshall, A.W., Olkin, I., 1979. Inequality: Theory of Majorization and Its Applications. Academic Press, New York. · Zbl 0437.26007 [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 [22] Pečarić, J.E., Proschan, F., Tong, Y.L., 1992. Convex Functions, Partial Orderings, and Statistical Applications. Academic Press, San Diego, CA. [23] Pledger, G., Proschan, F., 1971. Comparisons of order statistics and of spacings from heterogeneous distributions. In: Rustagi, J.S. (Ed.), Optimizing Methods in Statistics. Academic Press, New York, pp. 89-113. · Zbl 0263.62062 [24] Rolski, T., 1976. Order relations in the set of probability distribution functions and their applications to queueing theory. Dissertations Math., Vol. 132. Warsaw. · Zbl 0357.60025 [25] Ross, S.M., 1996. Stochastic Processes, 2nd Edition, Wiley, New York. · Zbl 0888.60002 [26] Rychlik, T., 1996. Order statistics of variables with given marginal distributions. In: Rüschendorf, R., Schweizer B., Taylor, M.D. (Eds.), Distributions with Fixed Marginals and Related Topics, IMS Lecture Notes, Vol. 28, Institute of Mathematical Statistics, Hayward, CA, pp. 297-306. [27] Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, San Diego, CA. · Zbl 0806.62009 [28] Shaked, M.; Wong, T., Stochastic orders based on ratios of Laplace transforms, J. appl. probab., 34, 404-419, (1997) · Zbl 0893.60054 [29] Stoyan, D., 1983. Comparison Methods for Queues and Other Stochastic Models. Wiley, New York. · Zbl 0536.60085 [30] Titterington, D.M., Smith, A.F.M., Markov, U.E., 1985. Statistical Analysis of Finite Mixture Distributions. Wiley, New York. · Zbl 0646.62013 [31] Williamson, R.E., Multiply monotone functions and their Laplace transforms, Duke math. J., 23, 189-207, (1956) · Zbl 0070.28501
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