Pitman estimator
An equivariant estimator for the shift parameter with respect to a group of real shifts, having minimal risk with respect to a quadratic loss function.
Let the components of a random vector
be independent random variables having the same probability law, with probability density belonging to the family
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and with
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for any . Also, let
be the group of real shifts operating in the realization space
of
:
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In this case, the task of estimating is invariant with respect to the quadratic loss function
if one uses an equivariant estimator
of
, i.e.
for all
. E. Pitman [1] has shown that the equivariant estimator
for the shift parameter
with respect to the group
that has minimal risk with respect to the quadratic loss function takes the form
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where , and
is the
-th order statistic of the observation vector
. The Pitman estimator is unbiased (cf. Unbiased estimator); it is a minimax estimator in the class of all estimators for
with respect to the quadratic loss function if all equivariant estimators for
have finite risk function [2].
Example 1. If
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i.e. ,
, has exponential distribution with unknown shift parameter
, then the Pitman estimator
for
is
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and its variance is .
Example 2. If
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i.e. ,
, has normal distribution
with unknown mathematical expectation
, then the arithmetic mean
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is the Pitman estimator.
References
[1] | E.J. Pitman, "The estimation of the location and scale parameters of a continuous population of any given form" Biometrika , 30 (1939) pp. 391–421 |
[2] | M.A. Girshick, L.J. Savage, "Bayes and minimax estimates for quadratic loss functions" J. Neyman (ed.) , Proc. 2-nd Berkeley Symp. Math. Statist. Prob. , Univ. California Press (1951) pp. 53–73 |
[3] | S. Zachs, "The theory of statistical inference" , Wiley (1971) |
Pitman estimator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pitman_estimator&oldid=48182