Complete Class Theorems Derived from Conditional Complete Class Theorems

Abstract

Let $(\mathscr{X}, \mathscr{B}_1, \mu)$ and $(\mathscr{Y}, \mathscr{B}_2, \nu)$ be $\sigma$-finite measure spaces and suppose $\Theta$ is a separable metric space. Let $f(x \mid y, \theta)$ be a family of conditional densities on $(\mathscr{X}, \mathscr{B}, \mu).$ Consider an action space $A$ which is a compact metric space with $\mathscr{B}_A$ the Borel $\sigma$-algebra and a loss function $W(\theta, a)$ such that $W(\theta, \bullet)$ is continuous. For any decision rule $\delta: \mathscr{B}_A \times \mathscr{X} \rightarrow \lbrack 0, 1\rbrack,$ assume the risk function $R(\delta, \bullet)$ is continuous on $\Theta.$ Suppose that a set of decision rules $\mathscr{M}_0$ is an essentially complete class for each $y \in \mathscr{Y}$ for the conditional decision problem. Let $\mathscr{M}^\ast$ be the set of decision rules $\eta: \mathscr{B}_A \times (\mathscr{X} \times \mathscr{Y}) \rightarrow \lbrack 0, 1\rbrack$ such that $\eta(\bullet \mid \bullet, y) \in \mathscr{M}_0 \mathrm{a.e.} \lbrack \nu\rbrack.$ Then $\mathscr{M}^\ast$ is an essentially complete class no matter what the family of marginal densities on the space $(\mathscr{Y}, \mathscr{B}_2, \nu).$