Intuition on simple linear regression signal plus noise model

Question

I'm currently studying linear regression on this book "F.M. Dekking - A Modern Introduction to Probability and Statistics: Understanding Why and How" where the signal+noise model is presented:

$Y_i = \alpha\ + \beta x_i\ + U_i$

where $x_i$ are non-random, $U_i$ are independent r.v.s with $E[U_i] = 0$ and $Var(U_i) = \sigma^2$ and $Y_i$ are, therefore, independent r.v.s with $E[Y_i] = \alpha\ + \beta x_i$ and $Var(Y_i) = \sigma^2$

He writes that "$\{Y_1, ..., Y_n \}$ do not form a random sample because given that each $Y_i$ have different expectation, they have different distributions" and he writes also that "if we assume that $U_i$ are normally distributed we can use MLE to estimate $\alpha,\beta$".

Now, given this two facts, how could we use MLE on something that is not a sample? Maybe the correct interpretation is that $\{Y_1, ..., Y_n \}$ is i.n.i.d. sample, but, if it is the case, would it be ok to use MLE?

Answer 1

Check the related question asking about logistic regression. The quotes you posted sound confusing, but what they are saying is that $Y_1,Y_2,\dots,Y_n$ are independent, but not identically distributed, since they have different means

$$ Y_i \sim \mathcal{N}(\mu_i, \sigma^2) $$

where $\mu_i$ is defined in terms of functional relationship $\mu_i = \alpha + \beta X_i$, what is equivalent of saying that

$$ \begin{align} \varepsilon_i &\sim \mathcal{N}(0, \sigma^2) \\ Y_i &= \alpha + \beta X_i + \varepsilon_i \end{align} $$

so $Y_1,Y_2,\dots,Y_n$ are random samples, but they are not i.i.d. Since they are random variables, we can define a likelihood function for the model describing their joint distribution

$$ L(\alpha,\beta,\sigma^2 \mid Y) = \prod_{i=1}^n \mathcal{N}(y_i; \alpha + \beta x_i, \; \sigma^2) $$

so we can estimate the parameters based on it (using MLE, or Bayesian estimation). You can also check the answers and comments in this thread for extended discussion and this thread for more formal discussion of somehow similar question.