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It looks like weighted least squares in which you divide by variance is the same as minimizing the norm of the Studentized residual. Is that right? —Ben FrantzDale (talk) 15:24, 26 November 2008 (UTC)[reply]

Almost. It is minimizing the sum (over all data points) of the squares of the Studentized residuals. Physchim62 (talk) 12:58, 3 May 2009 (UTC)[reply]

The way I read this, it is suggested that the Studentized residuals are defined by:

${\displaystyle r_{i}={\frac {\varepsilon _{i))((\sqrt {\frac {RSS-\varepsilon _{i}^{2)){df-1))}{\sqrt {1-h_{i,i))))))$

However I believe the correct definition is:

${\displaystyle r_{i}={\frac {\varepsilon _{i))((\sqrt {\frac {RSS-\varepsilon _{i}^{2}/(1-h_{i,i})}{df-1))}{\sqrt {1-h_{i,i))))))$

(I looked at equation 8.1.17 of Draper and Smith, Applied Regression Analysis and at how this is implemented in ls.diag in R).

Erwin.kalvelagen (talk) 12:34, 1 October 2009 (UTC)[reply]

It is claimed at User_talk:Michael_Hardy/Archive7#Studentized_residuals that a correction is needed in a formula in this article. I will look at that shortly. In the mean time, I've put a "factual accuracy" tag atop the article. Michael Hardy (talk) 19:09, 18 February 2014 (UTC)[reply]

In Kendall's Advanced Theory of Statistics, 5th Ed., by Alan Stuart and J. Keith Ord, page 1044, the variance of the residuals is given as

var (y[n+1] - y-hat[n+1] = sigma^2 * (1 + x_0' (X'X)^-1 x_0)

and Judge, et al., in Introduction to the Theory and Practice of Econometrics, 2nd Ed., page 210 we have

E[(y-hat_0 - y_0)(y-hat_0 - y_0)'] = sigma^2 [X_0 (X'X)^-1 X_0' + I_T0],

and noting that for the simple regression case, (Kendall, page 1045),

x_0' (X'X)^-1 x_0 = 1/n + (x_0 - x-bar)^2 / Sum(x - x-bar)^2

Thus the formula for the variance of the residuals has 1 + 1/n + (x_0 - x-bar)^2 / Sum(x - x-bar)^2, i.e., 1 + h_ii rather than 1 - h_ii

Did I miss anything? — Preceding unsigned comment added by 108.18.33.210 (talk) 19:58, 21 April 2015 (UTC)[reply]

Yes, this is the formula for the variance of a new observation; the formula for the variance for the residuals uses 1 - h_ii — Preceding unsigned comment added by 204.188.186.4 (talk) 00:55, 23 April 2015 (UTC)[reply]

"Typically the standard deviations of residuals in a sample vary greatly from one data point to another"

Can someone explain how a single residual can have its own standard deviation? Isn't it a single number? — Preceding unsigned comment added by 2601:641:200:1356:25C7:46B:898F:63B8 (talk) 06:55, 12 November 2015 (UTC)[reply]

That would be the a priori/assumed/expected standard deviation; it can be corrected or scaled a posteriori estimating a global variance factor for all residuals, or variance factors for groups of residuals. fgnievinski (talk) 01:43, 13 November 2015 (UTC)[reply]

wouldn't that be a standard error?

The introduction is unclear. If by a Studentized residual we mean the distance from zero to that residual divided by the standard deviation of all the other residuals, excluding that residual, then we would have a more general definition that is not limited to the assumption of ordinary least squares (OLS) in y. That is, such a more general definition would work as well for a minimized proportional norm type error, or any other more general than OLS residual set, be a lot clearer and simpler to define, and likely more generally useful. CarlWesolowski (talk) 00:42, 2 September 2022 (UTC)[reply]