Two independent random samples X₁,..., Xn and Y₁,..., Ym follow Poisson distributions:
X₁
Poi(y), where A 20 and 20 are two unknown parameters.
Poi(X) and Y;
Suppose we wish to test the hypothesis
Ho: A=X0,7 = 70 v.s. H₁: not Ho,
where Ao and 79 are given values. Let 1(A, 7) denote the log-likelihood function from these
two samples.
(a) Show that the log-likelihood function is given by
m
1(x, y) = −nλ - myλ + log A (Xi+Y; +logy[Yj.
–
j=1
j=1
(b) Hence show that the maximum likelihood estimators of A and y are
 = ₁=1 X₁₁
ΣΧ
n
n
Σj=1Yj/m
ΣX₁/n
(c) Derive the likelihood ratio test statistic for testing this hypothesis.
7=
=
(d) Specify the rejection region given by the likelihood ratio test.