Reference no: EM132889805

Question: Two independent random samples X₁,......X_n and Y₁......Y_m follow Poisson distributions: X_i ~ Poi(λ) and Y ~ Poi(Yλ), where λ ≥ 0 and Y ≥ 0 are two unknown parameters. Suppose we wish to test the hypothesis

H_o : λ = λ₀,_Y Y = Y₀ v.s. H₁ : not H₀,

where λ₀ and Y₉ are given values. Let l(λ, Y) denote the log-likelihood function from these two samples.

(a) Show that the log-likelihood function is given by

l(λ, Y) = -nλ - mYλ + log λ (∑_i=1ⁿX_i + ∑_j=1^mY_j;) + log Y ∑_j=1^mY_j

(b) Hence show that the maximum likelihood estimators of λ and Y are

^λ = ∑_i=1ⁿX_i/n

^y = ∑_j=1^mY_j/m;/∑_i=1ⁿX_i/n

(c) Derive the likelihood ratio test statistic for testing this hypothesis.

(d) Specify the rejection region given by the likelihood ratio test.