We have already seen the usefulness of the LRT in dealing with problems with nuisance parameters. We now look at some other nuisance parameter problems.
(a) Find the LRT of
H0: θ ‰¤ 0 versus H1: θ > 0
based on a sample X1,..., Xn from a population with probability density function

where both 6 and A are unknown.
(b) We have previously seen that the exponential pdf is a special case of a gamma pdf. Generalizing in another way, the exponential pdf can be considered as a special case of the Weibull(γ, β). The Weibull pdf, which reduces to the exponential if γ = 1, is very important in modeling reliability of systems. Suppose that X1,..., Xn is a random sample from a Weibull population with both γ and β unknown. Find the LRT of H0: γ = 1 versus H1: γ ‰  1.

Transcribed Image Text:

(2|0, X) = te (-0/A Le, (1),