Definition
Consider a random variable X whose probability distribution belongs to a parametric family of probability distributions Pθ parametrized by θ.
Formally, a statistic s is a measurable function of X; thus, a statistic s is evaluated on a random variable X, taking the value s(X), which is itself a random variable. A given realization of the random variable X(ω) is a data-point (datum), on which the statistic s takes the value s(X(ω)).
The statistic s is said to be complete for the distribution of X if for every measurable function g (which must be independent of θ) the following implication holds:
- E(g(s(X))) = 0 for all θ implies that Pθ(g(s(X)) = 0) = 1 for all θ.
The statistic s is said to be boundedly complete if the implication holds for all bounded functions g.
Read more about this topic: Completeness (statistics)
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