I am having trouble with the following question, particularly the first part. Doesn't least squares require that errors be the same across the RV drawn from a distribution. However the variance changes for each RV though it is some fixed value. Could someone show me how to do this problem:
Let $ X_j \sim N(j\mu, j^3\sigma^2) $ be independent random variables j = 1,...,n where $\mu $ is the unknown parameter and $\sigma^2 >0 $ is some known fixed value.
Compute:
(a) $\hat{\mu_1}$, the (unweighted) least squares estimator for $\mu$
(b) $\hat{\mu_2}$, the maximum likelihood estimator for $\mu$