A special case of a normal family is one in which the mean and the variance are related, the n(θ, αθ) family. If we are interested in testing this relationship, regardless of the value of θ, we are again faced with a nuisance parameter problem.
(a) Find the LRT of H0: a = 1 versus H1: a =≠1 based on a sample X1,..., Xn from a n(θ, aθ) family, where θ is unknown.
(b) A similar question can be asked about a related family, the n(θ, aθ2) family. Thus, if X1,... ,Xn are iid n(θ,aθ2), where θ is unknown, find the LRT of H0 : a = 1 versus H1:α ≠ 1.