Editing Studentized residual (section)

==Background==

For this simple model, the [[design matrix]] is

:<math>X=\left[\begin{matrix}1 & x_1 \\  \vdots & \vdots \\ 1 & x_n \end{matrix}\right]</math>

and the [[hat matrix]] ''H'' is the matrix of the [[orthogonal projection]] onto the column space of the design matrix:

:<math>H=X(X^T X)^{-1}X^T.\,</math>

The [[Leverage (statistics)|leverage]] ''h''<sub>''ii''</sub> is the ''i''th diagonal entry in the hat matrix.  The variance of the ''i''th residual is

:<math>\operatorname{var}(\widehat{\varepsilon\,}_i)=\sigma^2(1-h_{ii}).</math>

In case the design matrix ''X'' has only two columns (as in the example above), this is equal to

:<math>\operatorname{var}(\widehat{\varepsilon\,}_i)=\sigma^2\left( 1 - \frac1n -\frac{(x_i-\bar x)^2}{\sum_{j=1}^n (x_j - \bar x)^2 }  \right). </math>

In the case of an [[arithmetic mean]], the design matrix ''X'' has only one column (a [[vector of ones]]), and this is simply:

:<math>\operatorname{var}(\widehat{\varepsilon\,}_i)=\sigma^2\left( 1 - \frac1n \right). </math>