Minimum-variance Unbiased Estimator - Definition

Definition

Consider estimation of based on data i.i.d. from some member of a family of densities, where is the parameter space. An unbiased estimator of is UMVU if ,

for any other unbiased estimator

If an unbiased estimator of exists, then one can prove there is an essentially unique MVUE. Using the Rao–Blackwell theorem one can also prove that determining the MVUE is simply a matter of finding a complete sufficient statistic for the family and conditioning any unbiased estimator on it.

Further, by the Lehmann–Scheffé theorem, an unbiased estimator that is a function of a complete, sufficient statistic is the UMVU estimator.

Put formally, suppose is unbiased for, and that is a complete sufficient statistic for the family of densities. Then

is the MVUE for

A Bayesian analog is a Bayes estimator, particularly with minimum mean square error (MMSE).

Read more about this topic:  Minimum-variance Unbiased Estimator

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)

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)

    “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)