Question

X	H0	H1
x1	0.2	0.1
x2	0.3	0.4
x3	0.3	0.1
x4	0.2	0.4

What is the likelihood ratio test of H0 versus HA at level α = .2? What is the test at level α = .5?

Please show work or formulas on how to solve for values.

Answer 1

Compare the likelihood ratio, Λ for each possible value X and
order the xi according to Λ.
X H0 HA Λ
x1 .2 .1 2
x2 .3 .4 3/4
x3 .3 .1 3
x4 .2 .4 2/4
Sorting the xi from highest to lowest a : x3, x1, x2, x4
What is the likelihood ratio test of H0 versus HA at level α = 0.2?
We rewrite the table, ordering the rows by decreasing Λ. We add a
column corresponding to P(Λ < Λ(xi) | H0) this is the level of the test
that rejects H0 if Λ < Λ(xi)
X H0 HA Λ P(Λ(X) < Λ(xi) | H0)
x3 .3 .1 3 0.7
x1 .2 .1 2. 0.5
x2 .3 .4 3/4 0.2
x4 .2 .4 2/4 0.
The test that rejects H0 if Λ(X) < Λ(x2) has level 0.2 and the test
that rejects H0 if Λ(X) < Λ(x1) has level 0.5.

If the prior probabilities are P(H0) = P(HA) then the out-
comes favoring H0 are those for which the posterior probability ratio
is greater than 1:
1< [P(H0|X)/P(HA|X)] = {[P(H0)P(X|H0)]/[P(HA)P(X|HA)]}
= [P(H0)/P(HA)] . Λ(X)

= Λ(X)

For equal prior probabilities, the posterior odds is greater than 1 if the

likeihood ratio is greater than 1. This is true for outcomes x3 and x1.
(d). What prior probabilities correspond to the decision rules with
α = 0.2 and α = 0.5
For α = 0.2, the rejection region of the test is {x4} = {x : Λ < 3/4}.
For the posterior odds to be less than one, the prior odds must be less
than 4/3, which corresponds to
P(H0) = 1 − P(HA) < 4/7.
For α = 0.5, the rejection region of the test is {x4, x2} = {x : Λ < 2}.
For the posterior odds to be less than one, the prior odds must be less
than 1/2, which corresponds to
P(H0) = 1 − P(HA) < 1/3.
To include {x2} in the rejection region it must be that P(H0) > 4/7.
So for α = 0.5, the prior probabilities are such that
4/7 < P(H0) = 1 − P(HA) < 1/3.
Note that higher prior probability of H0 leads to requiring stronger
evidence (lower Λ) to reject H0.