So I have to solve the following question. My understanding and solution approach is at the bottom.
We observe $(X_1, Y_1) \dots (X_n, Y_n)$ that are iid from some unknown distribution.
Now we consider a likelihood model that $Y_i = α + \beta X_i + \epsilon_i$ where $\epsilon_i$'s are independent of $X_i$ and $\epsilon_1, \dots \epsilon_n$ ~ $N(0,\sigma ^2)$ for some unknown parameter $\sigma^2$ .
We further assume that $X_1, \dots ,X_n$ ~ $N(0, \tau^2)$
a)What is the likelihood function of $L(\alpha, \beta, \sigma^2, \tau^2|X_1, Y_1, \dots, X_n, Y_n)$
b) Suppose $\sigma^2, \tau^2$ are known, what are the MLE estimates of $\alpha, \beta$.
c) Suppose now $X_i$ ~ $U[0, \tau^2]$, are the MLE estimates different from the last part?
So I'm having trouble forming the likelihood function. I've seen this problem before but it doesn't mention anything about the distribution of $X_i$ like this problem does. It usually assumes that $\epsilon$ ~ $N(0, \sigma^2)$ and then it's easy from there.
So my question is, in this case, can I also just use $\epsilon_i$ to form the likelihood function and just ignore the distribution of $X_i$? Or I can use $Y_i$ to form $L$ where I can say $Y_i$ ~ $N(\alpha, \beta^2\tau^2 + \sigma^2)$
Even if I do this then how do I approach the part c)?
In the previous problem you saw, the $X_i$ are considered as fixed, so the likelihood function was the joint distribution of the $Y_1, \ldots, Y_n$. In this problem, the likelihood function is the joint density of $(X_1, Y_1), \ldots, (X_n, Y_n)$.
$$L(\alpha, \beta, \sigma^2, \tau^2 \mid x_1, y_1, \ldots, x_n, y_n) = \prod_{i=1}^n f_{X_i, Y_i}(x_i, y_i) = \prod_{i=1}^n f_{X_i}(x_i) f_{Y_i \mid X_i=x_i}(y_i).$$ You know $f_{X_i}$ is the density of $N(0, \tau^2)$ and $f_{Y_i \mid X_i = x_i}$ is the density of $N(\alpha + \beta x_i, \sigma^2)$.
In (c) the only difference is that $f_{X_i}$ is instead the density of $U[0, \tau^2]$. In both (b) and (c), you find the MLE by maximizing the likelihood in the usual way.