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Invariant Estimator is an intuitively appealing non Bayesian estimator. It is also sometimes called an "equivariant estimator". In the estimation problem we have random vector from space with density function when is from the space . We want to estimate given set of measurements from the distribution . The estimation is denoted by , is a function of the measurements and is in the space . The quality of the result is defined by a loss function which determine a risk function .

Generally speaking invariant estimator is an estimator that obey the 2 following rules:

1. Principle of Rational Invariance: The action taken in a decision problem should not depend on transformation on the measurement used

2. Invariance Principle: If two decision problems have the same formal structure (in terms of , , and ) then the same decision rule should be used in each problem

To define invariant estimator formally we will first set some definitions about groups of transformations:

Invariant Estimation Problem and Invariant Estimator

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A group of transformation of , to be denoted by is a set of (measurable) and onto transformation of into itself, which satisfies the following conditions:

1. If and then

2. If then (

3. ()

and in are equivalent if for some . All the equivalent points form an equivalence class. Such equivalence class is called orbit (in ). The orbit, , is the set . If consist of a single orbit than is said to be transitive.

A family of densities is said to be invariant under the group if, for every and there exists a unique such that has density . will be denoted .

If is invariant under the group than the loss function is said to be invariant under if for every and there exists an such that for all . will be denoted .

is a group of transformations from to itself and is a group of transformations from to itself.

An estimation problem is invariant under if there exists such three groups .

For an estimation problem that is invariant under , estimator is invariant estimator under if for all and .

Properties of Invariant Estimators

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1. The risk function of an invariant estimator is constant on orbits of . Equivalently for all and .

2. The risk function of an invariant estimator with transitive is constant.

For a given problem the invariant estimator with the lowest risk is termed the "best invariant estimator". Best invariant estimator cannot be achieved always. A special case for which it can be achieved is the case when is transitive.

Location Parameter Problem Example

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is a location parameter if the density of is . For and the problem is invariant under . The invariant estimator in this case must satisfy thus it is of the form (). is transitive on so we have here constant risk: . The best invariant estimator is the one that bring the risk to minimum.

In the case that L is squared error

Pitman Estimator

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Given the estimation problem: that has density and loss . This problem is invariant under , and (additive groups).

The best invariant estimator is the one that minimize (Pitman's estimator, 1939).

For the square error loss case we get that

If than

If than and when

References

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  • James O. Berger Statistical Decision Theory and Bayesian Analysis. 1980. Springer Series in Statistics. ISBN 0-387-90471-9.
  • The Pitman estimator of the Cauchy location parameter, Gabriela V. Cohen Freue, Journal of Statistical Planning and Inference 137 (2007) 1900 – 1913