(Invariance) Suppose that R 2 and consider testing the restriction 1 = 2 in the form 1 / 2 =

(Invariance) Suppose that θ ∈ R² and consider testing the restriction θ₁ = θ₂ in the form θ₁/θ₂ = 1.

(a) Reparameterize the likelihood function in terms of θ₁ and y = θ₁/θ₂.

(b) Find the score, Hessian, and information matrix for the reparameterization in terms of the original score, Hessian, and information matrix.

(c) Show that the score test is invariant to the reparameterization if the information matrix is used to estimate the variance of the score.

(d) Show that the score test is not invariant if the Hessian matrix is used to estimate the variance of the score. What term can be dropped from the Hessian to make the test invariant? Is this score test statistic always positive?