INFERENCE AND APPLICATION IN FINANCE OF $\Gamma$-DISTRIBUTION WITH AT MOST ONE CHANGE-POINT
至多一个变点的$\Gamma$分布的统计推断及在金融中的应用

In this paper the change point of parameter in ${\it \Gamma}$- distribution is considered. Suppose that $X_{1}, X_{2}, \cdots ,X_{n\tau_{0}]},X_{n\tau_{0}]+1},\cdots,X_{n}$ are independent random variables where $X_{1},X_{2}, \cdots ,X_{n\tau_{0}]}$ i.i.d $\sim {\it \Gamma}(x;\nu_{1}, \lambda_{1}),$and $X_{n\tau_{0}]+1}, X_{n\tau_{0}]+2},\cdots, X_{n}\ {\rm i.i.d} \sim {\it \Gamma}(x;\nu_{2}, \lambda_{2}),\ \tau_{0}$ is unknown and called change point. The distribution of the statistic proposed in the paper can be approximated by the first type of extremal distribution . Under mild conditions, the strong consistency and rate of convergence of the estimator for the change point are presented. At the same time, its application are also presented.