Independence of studentised residuals in Q-Q plot (normal linear model) I'm working with a normal linear model and know that the normal Q-Q plot is used for a distributional check for the normality assumption in the model. I also know that studentised residuals are plotted against the quantiles.
Here's what I don't understand: the distributional check used says:
"If $Y_{1},...,Y_{n}$ are independent $N(\mu,\sigma^{2})$ distributed random variables, then $P(Y_{i} \leq y) = P(\frac{Y_{i}-\mu}{\sigma} \leq \frac{y-\mu}{\sigma}) = \Phi(\frac{y-\mu}{\sigma})$. Then the empirical cdf $F_{n}(x):=\frac{1}{n}\sum_{i=1}^{n}I(Y_{i}\leq x) \rightarrow \Phi(\frac{x-\mu}{\sigma})$ as $\text{n} \rightarrow \infty$. Therefore, $\Phi^{-1}(F_{n}(y_{i})) \approx \frac{y_{i}-\mu}{\sigma}$"
I'm pretty sure that there's no assumption that the studentised residuals $e_{i}$ are independent so how does the QQ plot work? 
 A: For a normal linear model with (matrix notation, assume $X$ have a column of 1's)
$$
   Y =X\beta+\epsilon
$$
where the error terms $\epsilon_i$ are iid $\text{N}(0,\sigma^2)$.  The least squares estimator is $\hat{\beta}= (X^TX)^{-1}X^TY$, the hat (projection) matrix is $H=X(X^TX)^{-1}X^T$ the predicted values is 
$\hat{Y}= HY$ and the residuals vector is $e=Y-\hat{Y}=(I-H)Y$. 
Then (assuming the model is correct) we find that the expected value of residuals is 0, and its  variance is $\DeclareMathOperator{\V}{\mathbb{V}} \V e =\sigma^2 (I-H)$. So the raw residuals are centered, but with unequal variances!  So, for residuals analysis, we should standardize the residuals, the standardized residuals is
$$
  r_i= \frac{e_i}{\sqrt{1-h_i}}
$$
where $h_i$ are the diagonal values of the hat matrix $H$.  One can show that $0\le h_i \le 1$. So the standardized residuals $r_i$ will all have the same marginal normal distribution, though they will be dependent. This should be enough for utility of QQplots, thoug if $p$ (the number of columns of $X$) is close to $n$, there might maybe be problems.  Simulated confidence bands on QQplot will usually  include this correlation in the simulations, so should be correct. 
However, if you want a formal hypothesis test of normality based on the residuals, we might have problems with the dependency between the different standardized residuals.  It would be better then to transform the residuals to some ($n-p$) independent residuals, which can be done in various ways. One idea is recursive residuals, see for instance  https://www.jstor.org/stable/2683242?seq=1#page_scan_tab_contents   (I will come back with more details)  
