Normal spherical model: Suppose Y is a trivariate normal random vector with mean ( cos() cos(), cos() sin(), sin()) and covariance matrix n1

Normal spherical model: Suppose Y is a trivariate normal random vector with mean (ρ cos(θ) cos(ϕ), ρ cos(θ) sin(ϕ), ρ sin(θ)) and covariance matrix n−1 I3

. Let ρ = ρ0 be given.

(i) Find the maximum likelihood estimate of (θ, ϕ).

(ii) Derive the likelihood ratio statistic for testing the hypothesis H0

: θ = 0,

ϕ = 0.

(iii) Show that the Bartlett correction for testing the hypothesis in (ii) is identically zero regardless of the value of ρ0.