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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

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

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