一类特殊的指数族分布的参数估计

Estimation of the Parameter on a Class of Special Exponential Distribution Family

设随机变量X有密度函数p(x) =T′(x)β exp(- T(x)β ) , a <x<b ,其中 ,β>0为未知参数 .分别在平方损失和熵损失下 ,研究了参数 β的估计问题 .特别地 ,当T(x)满足 :T(cx) =T(c)T(x) (如T(x) =xm,m >0 )时 ,导出了在变换群G ={gc:gc(x) =T(c)x}下的最优同变估计 ,并说明了它们的可容许性 .

Let random variable X’ s density is p(x)=T′(x)β exp (-T(x)β), a<x<b, where β(β>0) is unknown parameter. In this paper, the estimations of parameter β are given both under square loss and entropy loss function. Especially, when T(x) satisfies: T(cx)=T(c)T(x) (for example, T(x)=x m,m>0), the minimum risk equivariant estimation is studied on the transformation group G{g c:g c(x)=T(c)x}, and their admissibility is illustrated.