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Concept
Completeness (statistics)
Summary
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 , and a statistic (which is effectively just a function of one or more i.i.d. random variables drawn from the model). Then consider the map which takes each distribution on model parameter to its induced distribution on statistic . The statistic is said to be complete when is surjective, and sufficient when is injective.
Consider a random variable X whose probability distribution belongs to a parametric model Pθ parametrized by θ.
Say T is a statistic; that is, the composition of a measurable function with a random sample X1,...,Xn.
The statistic T is said to be complete for the distribution of X if, for every measurable function g,:
The statistic T is said to be boundedly complete for the distribution of X if this implication holds for every measurable function g that is also bounded.
The Bernoulli model admits a complete statistic. Let X be a random sample of size n such that each Xi has the same Bernoulli distribution with parameter p. Let T be the number of 1s observed in the sample, i.e. . T is a statistic of X which has a binomial distribution with parameters (n,p). If the parameter space for p is (0,1), then T is a complete statistic. To see this, note that
Observe also that neither p nor 1 − p can be 0.
Official source
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