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)
Famous quotes containing the word definition:
“Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
—Nadine Gordimer (b. 1923)
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)