Exercise4.13 (F-test fortheequalityofvariances) Consider twoindependentnormalsamples X1, ...,Xn N (1,2 1 ) and Y1, ...,Ym N (2,2 2

Exercise4.13 (F-test fortheequalityofvariances) Consider twoindependentnormalsamples X1, ...,Xn ∼N (μ1,σ2 1 ) and Y1, ...,Ym ∼N (μ2,σ2 2 ), wherealltheparametersareunknown.

1. Derivethegeneralizedlikelihoodratiofortestingtheequalityofvariances H0 : σ2 1 = σ2 2 vs.

H1 : σ2 1 ̸= σ2 2 and showthatitisafunctionofthesampledvariancesratio F = s2y

/s2x

, where s2y

= 1 m−1 Σmj

=1(Yj −¯Y )2 and s2x

= 1 n−1 Σn i=1(Xi− ¯X )2.

2. Showthatunderthenullhypothesis, F H0∼ Fm−1,n−1.

3. DerivethecorrespondingapproximateGLRTatlevel α.

4. In particular,suchatestisrelevantfortestingtheassumptionoftheequalityofvariancesrequired for thetwo-sample t-test (seeExercise4.12).Checkthisassumptionfortheratssurvivaldatain