Multiple comparison

The problem of testing hypotheses with respect to the values of scalar products $ \pmb\theta ^ {T} \cdot \mathbf c $ of a vector $ \pmb\theta = ( \theta _ {1}, \dots, \theta _ {k} ) ^ {T} $, the coordinates of which are unknown parameters, with a number of given vectors $ \mathbf c = ( c _ {1}, \dots, c _ {k} ) ^ {T} $. In statistical research the multiple comparison problem often arises in dispersion analysis where, as a rule, the vectors $ \mathbf c $ are chosen so that $ c _ {1} + \dots + c _ {k} = 0 $, and the scalar product $ \pmb\theta ^ {T} \cdot \mathbf c $ itself, in this case, is called a contrast. On the assumption that $ \theta _ {1}, \dots, \theta _ {k} $ are unknown mathematical expectations of one-dimensional normal laws, J.W. Tukey and H. Scheffé proposed the $ T $-method and the $ S $-method, respectively, for the simultaneous estimation of contrasts, which are the fundamental methods in the problem of constructing confidence intervals for contrasts.

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