Distributions, complete family of

A family of probability measures $\{ \mathbf{P}_\theta : \theta \in \Theta \subset \mathbf{R}^k \}$, defined on a measure space $(\mathfrak{X}, \mathfrak{B})$, for which the unique unbiased estimator of zero in the class of $\mathfrak{B}$-measurable functions on $\mathfrak{X}$ is the function identically equal to zero, that is, if $f({\cdot})$ is any $\mathfrak{B}$-measurable function defined on $\mathfrak{X}$ satisfying the relation \begin{equation}\label{eq:a1} \int_{\mathfrak{X}} f(x) \,\mathrm{d}\mathbf{P}_\theta = 0 \ \ \text{for all}\ \theta\in\Theta\,, \end{equation} then $f(x)=0$ $\mathbf{P}_\theta$-almost-everywhere, for all $\theta\in\Theta$. For example, a family of exponential distributions is complete. If the relation \eqref{eq:a1} is satisfied under the further assumption that $f$ is bounded, then the family $\{ \mathbf{P}_\theta : \theta \in \Theta \}$ is said to be boundedly complete. Boundedly-complete families of distributions of sufficient statistics play a major role in mathematical statistics, in particular in the problem of constructing similar tests with a Neyman structure.

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