Completeness (statistics)

In statistics, completeness is a property of a statistic in relation to a parameterised model for a set of observed data.

A complete statistic T is one for which any proposed distribution on the domain of T is predicted by one or more prior distributions on the model parameter space. In other words, the model space is 'rich enough' that every possible distribution of T can be explained by some prior distribution on the model parameter space. In contrast, a sufficient statistic T is one for which any two prior distributions will yield different distributions on T. (This last statement assumes that the model space is identifiable, i.e. that there are no 'duplicate' parameter values. This is a minor point.)

Put another way: assume that we have an identifiable model space parameterised by ${\displaystyle \theta }$, and a statistic ${\displaystyle T}$ (which is effectively just a function of one or more i.i.d. random variables drawn from the model). Then consider the map ${\displaystyle f:p_{\theta }\mapsto p_{T|\theta }}$ which takes each distribution on model parameter ${\displaystyle \theta }$ to its induced distribution on statistic ${\displaystyle T}$. The statistic ${\displaystyle T}$ is said to be complete when ${\displaystyle f}$ is surjective, and sufficient when ${\displaystyle f}$ is injective.