Let \(S\) be a linear rank statistic of the form

\[S=\sum_{i=1}^{n} c(i) a\left(r_{i}ight)\]

If \(\mathbf{R}\) is a vector whose elements correspond to a random permutation of the integers in the set \(\{1, \ldots, n\}\) then prove that \(E(S)=n \bar{a} \bar{c}\) and

\[V(S)=(n-1)^{-1}\left\{\sum_{i=1}^{n}[a(i)-\bar{a}]^{2}ight\}\left\{\sum_{j=1}^{n}[c(j)-\bar{c}]^{2}ight\}\]

where

\[\bar{a}=n^{-1} \sum_{i=1}^{n} a(i)\]

and

\[\bar{c}=n^{-1} \sum_{i=1}^{n} c(i)\]