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:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“It’s a rare parent who can see his or her child clearly and objectively. At a school board meeting I attended . . . the only definition of a gifted child on which everyone in the audience could agree was “mine.””
—Jane Adams (20th century)