# On the linear combination of independent Gumbel random variables

Research output: Other contribution

## Abstract

The exact distribution of the linear combination of p independent Gumbel random variables isobtained as the sum of two independent random variables, the first one corresponding to the sum of a given number of independent log Gamma random variables multiplied by a parameter and the second one corresponding to a shifted Generalized Integer Gamma distribution [1]. Given the complexity of this exact distribution, a near-exact distribution, with a flexible parameter, $\gamma$ , is developed, based on the exact c.f. (characteristic function). This near-exact distribution corresponds to a shifted Generalized Near-Integer Gamma distribution [2]. The expression of its density and cumulative distribution functions does not involve any unsolved integrals, thus allowing for an easy computation of values for the density and cumulative distribution functions with high precision. Numerical studies conducted to assess the quality of this approximation show its very good performance. Furthermore the expressions obtained greatly surpass in manageability the ones in [4].
Original language Unknown 1 Published - 1 Jan 2011