Mathematical Statistics I · Sufficiency, completeness, and exponential families
For a family of densities or mass functions with respect to one fixed dominating…
Problem
For a family of densities or mass functions with respect to one fixed dominating measure, the factorization theorem says that \(T\) is sufficient if the joint density has the form \(g_\theta(T(x))h(x)\), with \(h\) independent of \(\theta\). In the usual dominated models here the converse also holds. All support restrictions must be included in that density. Independent counts \(X_i\) have Poisson means \(\lambda t_i\), where the exposures \(t_i>0\) are known and \(\lambda >0\) is unknown. Use factorization to establish sufficiency of \(K=\sum_iX_i\), and verify the conclusion by finding the conditional distribution of the count vector given \(K\). Now suppose the exposures are also unknown and are unrestricted positive parameters. For two counts, show that their total is no longer sufficient for the full parameter vector by examining the conditional probability of \((1,0)\) given total 1.
Hint
Separate factors involving \(\lambda\) from factors involving only data and known exposures.
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