If the covariance
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we may as "variance estimator"
consider
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where
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This alternative "variance estimator"
can be shown to be uncorrelated with the point estimator of 
and to have a lower variance than the standard variance estimator. Now, by
substituting K by
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we arrive at the approximately t-distributed pivotal
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Consider then a two-sided t
test for a specific value (under the null hypothesis) of ,
using P as test statistic and significance level 
.
A confidence region with approximate confidence level 1 –
therefore consists of all values of
which is not rejected by this test, i.e., the values of
for which |P| < t. This is equivalent to
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We observe that the values of
for which this inequality
holds, constitute an interval with endpoints given by the solutions to
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We thus yield the confidence interval
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