Normal circle model: Suppose Y is a bivariate normal random vector with mean ( cos(), sin())

Normal circle model: Suppose Y is a bivariate normal random vector with mean (ρ cos(θ), ρ sin(θ)) and covariance matrix n−1/2

. Let ρ = ρ0 be given.

(i) Find the maximum likelihood estimate of θ.

(ii) Derive the likelihood ratio statistic for testing the hypothesis Η0

:θ = 0.

(iii) Interpret the first two log likelihood derivatives and the likelihood ratio statistic geometrically.

(iv) Show that the Bartlett correction for testing the hypothesis in (ii) is b (θ) = 1/ (4ρ

2 0).

(v) Show that the first derivative of the log likelihood function is normally distributed: find its variance.