Studentized residual

In statistics, a Studentized residual, named in honor of William Sealey Gosset, who wrote under the pseudonym Student, is a residual adjusted by dividing it by an estimate of its standard deviation. Studentization of residuals is an important technique in the detection of outliers.

How to Studentize

For this simple model, the design matrix is

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:X=left

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and the "hat matrix" H is the matrix of the orthogonal projection onto the column space of the design matrix:

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:H=X(X^T X)^{-1}X^T.

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The "leverage" hii is the ith diagonal entry in the hat matrix. The variance of the ith residual is

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:mbox{var}(widehat{arepsilon}_i)=sigma^2(1-h_{ii}).

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The corresponding Studentized residual is then

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:{widehat{arepsilon}_iover widehat{sigma} sqrt{1-h_{ii} }}

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where widehat{sigma} is an appropriate estimate of σ.

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