Theory of Statistics - George Mason University

An Optimal Test in a Simple Situation
7.2 Optimal Tests 511
First, consider the problem of picking the optimal critical region C in a problem
of testing the hypothesis that a discrete random variable has the probability
mass function p0(x) versus the alternative that it has the probability
mass function p1(x). We will develop an optimal test for any given significance
level based on one observation.
For x ∋ p0(x) > 0, let
r(x) = p1(x)
, (7.11)
p0(x)
and label the values of x for which r is defined so that
r(xr1) ≥ r(xr2) ≥ · · · .
Let N be the set of x for which p0(x) = 0 and p1(x) > 0. Assume that
there exists a j such that
j
p0(xri) = α.
i=1
If S is the set of x for which we reject the test, we see that the significance
level is
p0(x).
x∈S
and the power over the region of the alternative hypothesis is
p1(x).
x∈S
Then it is clear that if C = {xr1, . . ., xrj} ∪N, then
x∈S p1(x) is maximized
over all sets C subject to the restriction on the size of the test.
If there does not exist a j such that j i=1 p0(xri) = α, the rule is to put
xr1, . . ., xrj in C so long as
j
p0(xri) = α ∗ < α.
i=1
We then define a randomized auxiliary test R
Pr(R = d1) = δR(xrj+1)
= (α − α ∗ )/p0(xrj+1)
It is clear in this way that
x∈S p1(x) is maximized subject to the restriction
on the size of the test.
Theory of Statistics c○2000–2013 James E. Gentle