The Mean of Marshall–Olkin-Dependent Exponential Random Variables

Abstract The probability distribution of Sd := X1 + · · · + Xd , where the vector (X1, . . . , Xd ) is distributed according to the Marshall–Olkin law, is investigated. Closed-form solutions are derived in the general bivariate case and for d ∈ {2, 3, 4}in the exchangeable subfamily. Our computations can, in principle, be extended to higher dimensions, which, however, becomes cumbersome due to the large number of involved parameters. For the Marshall–Olkin distributions with conditionally independent and identically distributed components, however, the limiting distribution of Sd/d is identified as d tends to infinity. This resultmight serve as a convenient approximation in high-dimensional situations. Possible fields of application for the presented results are reliability theory, insurance, and credit-risk modeling.     «

Abstract The probability distribution of Sd := X1 + · · · + Xd , where the vector (X1, . . . , Xd ) is distributed according to the Marshall–Olkin law, is investigated. Closed-form solutions are derived in the general bivariate case and for d ∈ {2, 3, 4}in the exchangeable subfamily. Our computations can, in principle, be extended to higher dimensions, which, however, becomes cumbersome due to the large number of involved parameters. For the Marshall–Olkin distributions with conditionally indep...     »