Equivariant estimator
A statistical point estimator that preserves the structure of the problem of statistical estimation relative to a given group of one-to-one transformations of a sampling space.
Suppose that in the realization of a random vector , the components
of which are independent, identically distributed random variables taking values in a sampling space
,
, it is necessary to estimate the unknown true value of the parameter
. Next, suppose that on
acts a group of one-to-one transformations
such that
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In turn, the group generates on the parameter space
a so-called induced group of transformations
, the elements of which are defined by the formula
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Let be a group of one-to-one transformations on
such that
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Under these conditions it is said that a point estimator of
is an equivariant estimator, or that it preserves the structure of the problem of statistical estimation of the parameter
with respect to the group
, if
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The most interesting results in the theory of equivariant estimators have been obtained under the assumption that the loss function is invariant with respect to .
References
[1] | S. Zachs, "The theory of statistical inference" , Wiley (1971) |
[2] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) |
Equivariant estimator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Equivariant_estimator&oldid=15622