# Compute the profile log likelihood for $\alpha$ giving your answer in terms of $\hat{\alpha}_{\beta}$.

We have $$X_{1}, X_{2},...,X_{n}$$ IID random variables from a Poisson distribution with mean $$\mu_{i}=\exp{(\alpha + \beta z_{i})}$$.

i) For fixed $$\beta$$, find $$\hat{\alpha}_{\beta}$$, the maximum likelihood estimate for $$\alpha$$ assuming $$\beta$$ is known.

ii) Hence compute the profile log likelihood for $$\alpha$$ giving your answer in terms of $$\hat{\alpha}_{\beta}$$.

I'm confused with this question and I wonder if someone could clarify it for me. I had thought that in order to find the profile log likelihood for $$\alpha$$ we would find the maximum likelihood estimate for $$\beta$$ assuming $$\alpha$$ is known, $$\hat{\beta}_{\alpha}$$, and then $$Pl(\alpha)=l(\alpha, \hat{\beta}_{\alpha}$$). But the question asks for the profile log likelihood of $$\alpha$$ in terms of $$\hat{\alpha}_{\beta}$$, and I don't understand why this is.

Thanks for any help

• You are right. Probably they whant you to find the prpfile likelihood of $\beta$ in terms of $\hat \alpha_{\beta}$. – Riccardo Sven Risuleo May 11 at 11:54