Theory of Statistics - George Mason University

256 3 Basic Statistical Theory
define the action space as A = {0, 1}, in which 0 represents not rejecting and
1 represents rejecting.
If we observe X, we take the action T(X) = a ∈ A. An action or a decision
may be the assignment of a specific value to an estimator, that is, an estimate,
or it may be to decide whether or not to reject a statistical hypothesis.
Decision Rules
Given a random variable X with associated measurable space (X, FX) and an
action space A with a σ-field FA, a decision rule is a function, T, from X to
A that is measurable FX/FA.
A decision rule is also often denoted by δ or δ(X).
Randomized Decision Rules
Sometimes the available data, that is, the realization of X, does not provide
sufficient evidence to make a decision. In such cases, of course, it would be
best to obtain more data before making a decision. If a decision must be made,
however, it may be desirable to choose an action randomly, perhaps under a
probability model that reflects the available evidence. A randomized decision
rule is a function δ over X × FA such that for every A ∈ FA, δ(·, A) is a Borel
function, and for every x ∈ X, δ(x, ·) is a probability measure on (A, FA).
To evaluate a randomized decision rule requires the realization of an additional
random variable. As suggested above, this random variable may not
be independent of the data. Randomized decision rules are rarely appropriate
in actual applications, but an important use of randomized decision rules is
to evaluate properties of statistical procedures. In the development of statistical
theory, we often use randomized decision rules to show that certain
deterministic rules do or do not have certain properties.
Loss Function
A loss function, L, is a mapping from P×A to [0, ∞[. The value of the function
at a given distribution P for the action a is L(P, a). More commonly, we refer
to the loss function associated with a given nonrandomized decision rule T(X)
as composition L(P, T(X)). For a given rule T, we may also denote the loss
function as LT(P). If the class of distributions is indexed by a parameter θ,
we may use the equivalent notation L(θ, T) or LT (θ).
Given a loss function L(P, a), the loss function associated with a given
randomized decision rule δ(X, A) is
L(P, δ(X, A)) = Eδ(X,·)(L(P, Y ))
where Y is a random variable corresponding to the probability measure δ(X, ·).
Theory of Statistics c○2000–2013 James E. Gentle