Let X and Y be independentandexponentiallydistributedrandom variableswith E(X) = 1 and E(Y ) = 2 where

Let X and Y be independentandexponentiallydistributedrandom variableswith E(X) = λ1 and E(Y ) = λ2 where λ = (λ1, λ2) ∈ R2

+. andlet

(X1, Y1), ..., (Xn, Yn) be asamplefromthisdistribution.Considerthehypothesis H0 : λ1λ2 = 1 as inExercise3.9,Exercise4.10,andExercise5.10.

a) Determinethelog-likelihoodratiostatisticforthecompositehypothesis H0.

b) DetermineaWaldteststatisticforthecompositehypothesisbasedonthefactthat the hypothesismaybeexpressedthroughtheparameterfunction

ϕ(λ) =log λ1 + log λ2 as H0 : ϕ(λ) =0.

c) Doesthefollowing(simulated)samplesupportthehypothesis H0? Useboththe likelihoodratioandWaldteststatisticsasderivedabove?

Nowweshallinvestigatethesimplehypothesis H1 : λ1 = λ2 = 1 within themodel determined by H0.

d) Derivethelikelihoodratiostatisticfor H1 under theassumptionof H0.

e) Derivethescoreteststatisticfor H1 under theassumptionof H0.

f) Arethedataunderc)compatiblewith H1?
g) DeterminetheMonteCarlo p-valuefortheabovetests.

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X 0.581 y 0.469 0.621 0.084 0.362 3.739 4.354 0.409 1.843 1.705 0.312 2.050 2.704 0.034 0.061 0.419