Theory of Statistics - George Mason University

7
Statistical Hypotheses and Confidence Sets
In a frequentist approach to statistical hypothesis testing, the basic problem
is to decide whether or not to reject a statement about the distribution of
a random variable. The statement must be expressible in terms of membership
in a well-defined class. The hypothesis can therefore be expressed by
the statement that the distribution of the random variable X is in the class
PH = {Pθ : θ ∈ ΘH}. An hypothesis of this form is called a statistical hypothesis.
The basic paradigm of statistical hypothesis testing was described in Section
3.5.1, beginning on page 286. We first review some of those ideas in Section
7.1, and then in Section 7.2 we consider the issue of optimality of tests.
We first consider the Neyman-Pearson Fundamental Lemma, which identifies
the optimal procedure for testing one simple hypothesis versus another simple
hypothesis. Then we discuss tests that are uniformly optimal in Section 7.2.2.
As we saw in the point estimation problem, it is often not possible to develop
a procedure that is uniformly optimal, so just as with the estimation
problem, we can impose restrictions, such as unbiasedness or invariance, or
we can define uniformity in terms of some global risk. Because hypothesis
testing is essentially a binary decision problem, a minimax criterion usually
is not relevant, but use of global averaging may be appropriate. (This is done
in the Bayesian approaches described in Section 4.5, and we will not pursue
it further in this chapter.)
If we impose restrictions on certain properties of the acceptable tests, we
then proceed to find uniformly most powerful tests under those restrictions.
We discuss unbiasedness of tests in Section 7.2.3, and we discuss uniformly
most powerful unbiased tests in Section 7.2.4. In Section 7.3, we discuss general
methods for constructing tests based on asymptotic distributions. Next
we consider additional topics in testing statistical hypotheses, such as nonparametric
tests, multiple tests, and sequential tests.
Confidence sets are closely related to hypothesis testing. In general, rejection
of an hypothesis is equivalent to the hypothesis corresponding to a set
of parameters or of distributions outside of a confidence set constructed at
Theory of Statistics c○2000–2013 James E. Gentle