Equivariance

Posted on Mar 22, 2022

This post is for Chapter 3 of Lehmann, E. L., & Casella, G. (1998). Theory of point estimation (2nd ed). Springer.

A family of densities $f(x\mid \xi)$, with parameter $\xi$, and a loss function $L(\xi, d)$ are location invariant if, respectively, $f(x’\mid \xi’) = f(x\mid \xi)$ and $L(\xi, d) = L(\xi’, d’)$ whenever $\xi’ = \xi + a$ and $d’ = d+a$. If both the densities and the loss function are location invariant, the problem of estimating $\xi$ is said to be location invariant under the transformations.

An estimator satisfying \(\delta(X_1+a,\ldots,X_n+a) = \delta(X_1,\ldots,X_n) + a\quad \text{for all $a$}\) will be called location equivalent.

Let $X$ be distributed with density \(f(x-\xi) = f(x_1-\xi, \ldots, x_n-\xi)\) and let $\delta$ be equivariant for estimating $\xi$ with loss function $L(\xi, d) = \rho(d-\xi)$. Then the bias, risk, and variance of $\delta$ are all constant (i.e., do not depend on $\xi$.)

In a location invariant estimation problem, if a location equivariant estimator exists which minimizes the constant risk, it is called the minimum risk equivariant (MRE) estimator.

UMVU vs MRE

• when a UMVU estimator exists, it typically minimizes the risk for all convex loss functions, but that for bounded loss functions not even a locally minimum risk unbiased estimator can be expected to exist

in contrast

• unlike UMVU estimator which are frequently inadmissible, the Pitman estimator is admissible under mild assumptions
• the principal area of application of UMVU estimation is that of exponential families, and these have little overlap with location families.
• for location families, UMVU estimators typically do not exist.

scale invariant

An estimator $\delta$ of $\tau^r$ is scale equivariant if

All the usual estimators of $\tau$ are scale equivariant, such as the standard deviation, the mean deviation, the range, and the maximum likelihood estimator.