Theory and Methods of Statistics

8.2.5 Specific Rank Tests Using Approximate Scores

First, note that the scores $a_N\left(i,f\right),a_{1N}(i,f),\dots$ can only be obtained from tables (if available) due to their complicated expressions. For example, there are tables for $\text{E}\left[{\Phi }^{-1}(U_{N:i})\right]$. However, since $\text{E}\left[U_{N:i}\right]=i/(N+1)$ and $\text{Var}\left[U_{N:i}\right]=i(N-i+1)/\left\{{(N+1)}^2(N+2)\right\}\to 0$ as $N\to \infty$, the distribution of U_N:i is concentrated near i