Theory of Statistics - George Mason University

262 3 Basic Statistical Theory
the sample median is L-unbiased under a squared-error loss, but it is not
admissible under that loss, while the sample mean is L-unbiased under an
absolute-error loss, but it is not admissible.
This is the basis for defining unbiasedness for statistical tests and confidence
sets.
Unbiasedness for estimators has a simple definition. For squared-error loss
for estimating g(θ), if T is L-unbiased, then, and only then, T is an unbiased
estimator of g(θ). Of course, in this case, the loss function need not be
considered and the requirement is just Eθ(T(X)) = g(θ).
L-Invariance
On page 217 we referred to equivariant estimators in parametric transformation
group families (see Section 2.6). We mentioned that associated with the
group G of transformations of the random variable is a group, G, of transformations
of the parameter and a group of transformations on the estimator,
G ∗ .
In a decision-theoretic approach, the relevance of equivariance depends
not only on the family of distributions, but also on the equivariance of the
loss function. In the loss function L(P, T(X)), the first argument under a
transformation can be thought of as a map PX → Pg(X) or equivalently as a
map Pθ → P¯g(θ). The statistical decision procedure T(X) is L-invariant for
a given loss function L if for each g ∈ G, there exists a unique g ∗ ∈ G ∗ , such
that
L(PX, T(X)) = L(Pg(X), g ∗ (T(X))), (3.99)
or equivalently for each ¯g ∈ G,
L(Pθ, T(X)) = L(P¯g(θ), g ∗ (T(X))). (3.100)
The g ∗ in these expressions is the same as in equation (3.22). We will often
require that statistical procedures be equivariant, in the sense that the
quantities involved (the estimators, the confidence sets, and so on) change in
a manner that is consistent with changes in the parametrization. The main
point of this requirement, however, is to ensure L-invariance, that is, invariance
of the loss. We will discuss equivariance of statistical procedures in more
detail in Section 3.4.
Uniformly Minimizing the Risk
All discussions of statistical inference are in the context of some family of
distributions, and when we speak of a “uniform” property, we mean a property
that holds for all members of the family.
If we have the problem of estimating g(θ) under some given loss function
L, it is often the case that for some specific value of θ, say θ1, one particular
Theory of Statistics c○2000–2013 James E. Gentle