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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

Â = 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.