Theory of Statistics - George Mason University

3.1 Inferential Information in Statistics 217
of the random variables. In a parametric setting the group G of transformations
of the random variable can be associated with a group G of transformations
of the parameter. Likewise, we consider a group of transformations on
the estimator, G ∗ . For g ∈ G and g ∗ ∈ G ∗ an estimator T(X) is equivariant if
T(g(X)) = g ∗ (T(X)). (3.22)
Some people use the terms “invariant” and “invariance” for equivariant
and equivariance, but I prefer the latter terms unless, indeed there is no
change in the statistical procedure.
For equivariant or invariant statistical procedures, there are issues that
relate to other properties of the estimator that must be considered (see, for
example, the discussion of L-invariance on page 262). We will discuss the
equivariance property of statistical procedures in more detail in Section 3.4.
Uniform Properties
If the goodness of an estimator does not depend on the parameter, we say
the estimator is uniformly good (and, of course, in this statement we would
be more precise in what we mean by “good”). 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.
Unbiasedness, by definition, is a uniform property. We will see, however,
that many other desirable properties cannot be uniform.
Statements of Probability Associated with Statistics
Although much of the development of inferential methods emphasizes the
expected value of statistics, often it is useful to consider the probabilities
of statistics being in certain regions. Pitman closeness is an example of the
use of probabilities associated with estimators. Two other approaches involve
the probabilities of various sets of values that the statistics may take on.
These approaches lead to statistical tests of hypotheses and determination
of confidence sets. These topics will be discussed in Section 3.5, and more
thoroughly in later chapters.
3.1.2 Sufficiency, Ancillarity, Minimality, and Completeness
There are important properties of statistics, such as sufficiency and complete
sufficiency, that determine the usefulness of those statistics in statistical inference.
These general properties often can be used as guides in seeking optimal
statistical procedures.
Theory of Statistics c○2000–2013 James E. Gentle