On skewness and dispersion among convolutions of independent gamma random variables

• Authors: Leila Amiri;Baha-eldin Khaledi;Francisco j. Samaniego
• Venue: Probability in the Engineering and Informational Sciences
• Year: 2011

Let {x(1)≤···≤x(n)} denote the increasing arrangement of the components of a vector x=(x1, …, xn). A vector x∈Rn majorizes another vector y (written $\bf{x} \mathop{\succeq}\limits^{m} \bf{y}$) if $\sum_{i=1}^{j} x_{(i)} \le \sum_{i=1}^{j}y_{(i)}$ for j = 1, …, n−1 and $\sum_{i=1}^{n}x_{(i)} = \sum_{i=1}^{n}y_{(i)}$. A vector x∈R+n majorizes reciprocally another vector y∈R+n (written $\bf{x} \mathop{\succeq}\limits^{rm} \bf{y}$) if $\sum_{i=1}^{j}(1/x_{(i)}) \ge \sum_{i=1}^{j}(1/y_{(i)})$ for j = 1, …, n. Let $X_{\lambda_{i},\alpha},\,i=1,\ldots,n$, be n independent random variables such that $X_{\lambda_{i},\alpha}$ is a gamma random variable with shape parameter α≥1 and scale parameter λi, i = 1, …, n. We show that if $\lambda \mathop{\succeq}\limits^{rm} \lambda^{\ast}$, then $\sum_{i=1}^{n} X_{\lambda_{i},\alpha}$ is greater than $\sum_{i=1}^{n} X_{\lambda^{\ast}_{i},\alpha}$ according to right spread order as well as mean residual life order. We also prove that if $(1/ \lambda_{1}, \ldots,1/ \lambda_{n}) \mathop{\succeq}\limits^{m} \succeq (1/ \lambda_{1}^{\ast}, \ldots, 1/ \lambda_{n}^{\ast})$, then $\sum_{i=1}^{n} X_{\lambda_{i}, \alpha}$ is greater than $\sum_{i=1}^{n} X_{\lambda^{\ast}_{i},\alpha}$ according to new better than used in expectation order as well as Lorenze order. These results mainly generalize the recent results of Kochar and Xu [7] and Zhao and Balakrishnan [14] from convolutions of independent exponential random variables to convolutions of independent gamma random variables with common shape parameters greater than or equal to 1.