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.

Internal and external Studentization

The estimate of σ2 may be

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:widehat{sigma}^2={1 over n-2}sum_{j=1}^n widehat{arepsilon}_j^2.

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But it is desirable to exclude the ith observation from the process of estimating the variance when one is considering whether the ith case may be an outlier. Consequently one may use the estimate

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:widehat{sigma}_{(i)}^2={1 over n-3}sum_{j=1}^n widehat{arepsilon}_j^2,

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based on all but the ith case. If the latter estimate is used, excluding the ith case, then the residual is said to be externally Studentized; if the former is used, including the ith case, then it is internally Studentized.

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If the errors are independent and normally distributed with expected value 0 and variance σ2, then the probability distribution of the ith externally Studentized residual is a Student's t-distribution with n − 3 degrees of freedom.

Related Topics:
Normally distributed - Expected value - Probability distribution - Student's t-distribution - Degrees of freedom

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