Theory of Statistics - George Mason University

7.7 The Likelihood Principle and Tests of Hypotheses 535
In a sequential testing procedure, at any point, the question of whether
or not to continue observing random variables depends on the inference that
could be made at that point. If the hypothesis can be rejected or if it is
very unlikely that it can be rejected, the decision is made, and the test is
terminated; otherwise, the test continues. When we refer to a “sequential
test”, this is the type of situation we have in mind.
7.6.1 Sequential Probability Ratio Tests
Let us again consider the test of a simple null hypothesis against a simple
alternative. Thinking of the hypotheses in terms of a parameter θ that indexes
these two PDFs by θ0 and θ1, for a sample of size n, we have the likelihoods
associated with the two hypotheses as Ln(θ0; x) and Ln(θ1; x). The best test
indicates that we should reject if
for some appropriately chosen k.
define and show optimality
7.6.2 Sequential Reliability Tests
Ln(θ0; x)
≤ k,
Ln(θ1; x)
7.7 The Likelihood Principle and Tests of Hypotheses
*** introduce; refer to likelihood in N-P
Tests of Hypotheses that Depend on the Data-Generating Process
***
Example 7.12 Sampling in a Bernoulli distribution; p-values and
the likelihood principle revisited
In Examples 3.12 and 6.1, we have considered the family of Bernoulli distributions
that is formed from the class of the probability measures Pπ({1}) = π
and Pπ({0}) = 1 − π on the measurable space (Ω = {0, 1}, F = 2 Ω ). Suppose
now we wish to test
H0 : π ≥ 0.5 versus H1 : π < 0.5.
As we indicated in Example 3.12 there are two ways we could set up an
experiment to make inferences on π. One approach is to take a random sample
of size n, X1, . . ., Xn from the Bernoulli(π), and then use some function of
that sample as an estimator. An obvious statistic to use is the number of 1’s
in the sample, that is, T = Xi. To assess the performance of an estimator
Theory of Statistics c○2000–2013 James E. Gentle