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

Ma, Chunsheng

J. Stat. Plann. Inference 84, No. 1-2, 11-25 (2000).

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:

60E15

Inequalities; stochastic orderings

Keywords:

Arrangement increasing; Convex order; Decreasing convex order; Laplace order; Likelihood ratio order; Majorization; Reversed hazard rate order; Stochastic dominance; Stochastic order

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