# Thread: Joint distribution, score function, likelihood

1. ## Joint distribution, score function, likelihood

Can anyone help me with this problem?

Consider the following statistical model

Yi=μ+(1+α xi)*εi, εi ~ iid N(0,1) i= 1,....,n
Where μ ∊ R and α ]-1,1[ are unkown parameters, while xi ∊ ]-1,1[ are known numbers.

1. Determine the joint distribution of Y1,....,Yn and find the distribution of the mean of the Y’s. Explain that the mean of the Y’s is a consistent estimator of μ.

2. Find the score function and the expected information. Give the approximate distribution of the maximum likelihood estimator of (α, δ) in large samples.

3. Set up the score statistic for the hypothesis H0: α = 0 and specify its approximate distribution under the hypothesis.