Jointly sufficient statistics of a multi-parameter exponential family

by Xiaomi   Last Updated September 20, 2018 02:19 AM

Let $f_X$ be a joint density function that comes from an $s$-parameter exponential family with sufficient statistics $(T_1, T_2, \dots, T_s)$ so that the density $f_X$ can be expressed as

$$f_{X|\theta}(x) = h(x) \exp \left(\sum_{i=1}^s T_i(x)\eta_i(\theta) - A(\theta) \right)$$

I have two questions:

  1. Expressed in this form, is it correct to say the statistics $T_1,\dots,T_s$ are jointly sufficient, as opposed to independently sufficient?

  2. Based on them being jointly sufficient, is it correct to say any unbiased estimator $\tau (X)$ such that $E[\tau(X)|T_i,T_j] = \theta_k$, for some $i,j,k$, must be UMVUE of $\theta_k \in \theta$?

I'm trying to understand the difference between being sufficient, and jointly sufficient

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