Linear Models, Statistics

Question

For part (a), do I need to calculate the MLE of the betas?

For the second part, I'm not really sure how to approach this question.

Any help would be greatly appreciated!

Thanks

Answer 1

1. Not necessarily, because you can view this model as $$ Y=\mu + \epsilon, $$ thus $Y_i \sim N(\mu, \sigma^2)$, hence $\hat{\mu}_n = \bar{Y}_n$, where $\bar{Y}_n \sim N(\mu, \sigma^2/n)$.

Clearly, if you are interested in the MLE for each of the coefficients, you will have to find the MLE using $Y_i \sim N(\beta_0 + \beta_1x_i+ \beta_2z_i, \sigma^2)$, i.e., take derivatives w.r.t all the three parameters.

2. I would try $Y_i = \beta_0 + \beta_1 (x_{t_i} - \bar{x}_{t_n})+\epsilon_i$, where $x_{t_i} = e^{-t_i}$ and $\epsilon_i \sim N(0, \sigma^2)$.