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:
“... if, as women, we accept a philosophy of history that asserts that women are by definition assimilated into the male universal, that we can understand our past through a male lens—if we are unaware that women even have a history—we live our lives similarly unanchored, drifting in response to a veering wind of myth and bias.”
—Adrienne Rich (b. 1929)
“Beauty, like all other qualities presented to human experience, is relative; and the definition of it becomes unmeaning and useless in proportion to its abstractness. To define beauty not in the most abstract, but in the most concrete terms possible, not to find a universal formula for it, but the formula which expresses most adequately this or that special manifestation of it, is the aim of the true student of aesthetics.”
—Walter Pater (1839–1894)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)