至多一个变点的$Gamma$分布的统计推断及在金融中的应用

对至多一个变点的Γ分布,即X1,X2…,Xn为一列相互独立的随机变量序列,且X1,X2,…,X[nΥ0]i．i．d～Γ(x;ν1,λ1),X[nΥ0] 1,X[nΥ0] 2,…,Xn i．i．d～Γ(x;ν2,λ2),其中Υ0未知,称Υ0为该序列的变点．在利用第一型极值分布逼近文中提出统计量的分布的基础上,给出了变点Υ0估计(?)的相合性及强弱收敛速度．最后给出了在金融序列上的应用．

INFERENCE AND APPLICATION IN FINANCE OF $Gamma$-DISTRIBUTION WITH AT MOST ONE CHANGE-POINT

In this paper the change point of parameter in ${it Gamma}$- distribution is considered. Suppose that $X_{1}, X_{2}, cdots ,X_{[ntau_{0}]},X_{[ntau_{0}]+1},cdots,X_{n}$ are independent random variables where $X_{1},X_{2}, cdots ,X_{[ntau_{0}]}$ i.i.d $sim {it Gamma}(x;nu_{1}, lambda_{1}),$and $X_{[ntau_{0}]+1}, X_{[ntau_{0}]+2},cdots, X_{n} {rm i.i.d} sim {it Gamma}(x;nu_{2}, lambda_{2}), tau_{0}$ 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.