Theory of Statistics - George Mason University

278 3 Basic Statistical Theory
that is, the random variable is transformed as g(X) = 1 −X. We see that the
transformation ˜g(π) = 1 − π preserves the parameter space, in the sense of
equation (3.109).
Under this new setup, following the same approach that led to the estimator
T(X), we see that T(g(X)) = T(1 − X) is an optimal estimator of
˜g(π) = 1 −π under squared-error loss among the class of unbiased estimators.
Hence, in this case, the squared-error loss function allowed us to develop an
equivariant procedure.
We note that the estimator T(g(X)) = g ∗ (T(X)) = 1 − T(X), and we
have, as in equation (3.111),
L(π, T(X)) = L(˜g(π), g ∗ (T(g(X)))).
In the Bernoulli example above, loss functions of various forms would
have allowed us to develop an equivariant procedure for estimation of the
transformed π. This is not always the case. For some types of transformations
g and ˜g on the sample and parameter spaces, we can develop equivariant
procedures only if the loss function is of some particular form. For example,
in a location family, with transformations of the form g(X) = X + c and
˜g(µ) = µ + c, in order to develop an equivariant procedure that satisfies
equation (3.111) we need a loss function that is a function only of a − g(θ).
************
Following the same approach as above, we see that in a univariate scale
family, with transformations of the form g(X) = cX, in order to develop
an equivariant procedure, we need a loss function that is a function only of
a/g(θ). In order to develop equivariant procedures for a general location-scale
family P(µ,Σ) we need a loss function of the form
L((µ, Σ), a) = Lls(Σ 1/2 (a − µ)). (3.113)
In order to achieve invariance of the loss function for a given group of
transformations G, for each g ∈ G, we need a 1:1 function g ∗ that maps the
decision space onto itself, g ∗ : A ↦→ A. The set of all such g ∗ together with
the induced structure is a group, G ∗ with elements
g ∗ : A ↦→ A, 1 : 1 and onto.
The relationship between G and G ∗ is an isomorphism; that is, for g ∈ G
and g ∗ ∈ G ∗ , there is a function h such that if g ∗ = h(g), then h(g1 ◦ g2) =
h(g1) ◦ h(g2).
Invariance of Statistical Procedures
To study invariance of statistical procedures we will now identify three groups
of transformations G, G, and G ∗ , and the relationships among the groups. This
Theory of Statistics c○2000–2013 James E. Gentle