Choosing joint distributions so that the variance of the sum is small
Journal of Multivariate Analysis
An Introduction to Copulas
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Following the results of Ruschendorf and Uckelmann (2002) [20], we introduce the completely mixable distributions on R and prove that the distributions with monotone density and moderate mean are completely mixable. Using this method, we solve the minimization problem min"X"""i"~"PEf(X"1+...+X"n) for convex functions f and marginal distributions P with monotone density. Our results also provide valuable implications in variance minimization, bounds for the sum of random variables and risk theory.