ECE 830 Homework 6: Composite Hypothesis Testing

ECE 830
Homework 6: Composite Hypothesis Testing
1. Suppose we observe two iid samples from an exponential distribution of the form
{ λ e
p(x) =
−λx , x > 0
0, x ≤ 0
and consider the hypothesis testing problem
H 0 : λ = λ 0
H 1 : λ > λ 0
for a known λ 0 > 0. Propose a one-dimensional test statistic and P FA
threshold. Does a UMP test exist? Explain your answer.
as a function of the
2. Suppose we observe two iid samples from an Laplacian distribution of the form
p(x) = λ 2 e−λ|x| ,
x ∈ R
and consider the hypothesis testing problem
H 0 : λ = λ 0
H 1 : λ > λ 0
for a known λ 0 > 0. Propose a one-dimensional test statistic and P FA
threshold. Does a UMP test exist? Explain your answer.
as a function of the
3. Suppose you have started a new job at a wireless communications company and your first
assignement is to design a test for detecting high bit error rates. For the test, a known n-
bit sequence is transmitted on a wireless channel and the received bit sequence is recorded.
Let x 1 , . . . , x n be binary variables that indicate the bit errors; i.e., x i = 1 if the ith bit is
incorrectly received and 0 otherwise. Assume that the bit errors are iid.
a. Design a test to distinguish the following hypotheses:
H 0 : bit error rate = p 0
H 1 : bit error rate > p 0
where p 0 is the acceptable bit error rate for the channel.
b. Does a UMP test exist for this problem? Explain your answer.
c. Determine the false alarm probability if n = 10 6 and p 0 = 10 4 .
4. In class we discussed the following composite hypothesis testing problem.
H 0 :
H 1 :
x 1 , . . . , x n
iid
∼ N (0, 1)
x 1 , . . . , x n
iid
∼ N (0, σ 2 )
where σ 2 > 0 is an unknown parameter.

a. Design a GLRT for this problem.
b. Apply Wilks’ Theorem to choose a threshold for a test with false-alarm probability of
0.05.
c. Simulate the detection problem with n = 30 in Matlab. Generate T = 1000 independent
trials of the problem with H 0 true in half of the trials. Check whether the empircal falsealarm
rate of the GLRT is close to the desired level of 0.05. If not, try a larger value
for n in your simulation. Plot the (empirical) distribution of the log GLR test statistic
when H 0 is true. Compare it to the asymptotic distribution of the statistic (according to
Wilks Theorem). Is the asymptotic distribution close to the observed distribution (i.e.,
is n = 30 large enough)?
d. Design a GLRT with P F A = 0.05 for a slightly different set of hypotheses:
H 0 :
H 1 :
x 1 , . . . , x n
iid
∼ N (0, σ 2 )
x 1 , . . . , x n
iid
∼ N (µ, σ 2 )
where both σ 2 and µ are unknown.