Theory and Methods of Statistics

Finally, we consider the two-sample statistics D_mn and $W_{mn}^2$ given by Eq. (14) under the null hypothesis F₁ = F₂ = F (unknown).

For $m,n\to \infty$ in such a way that $m/(m+n)=m/N\to \lambda$ and $n/(m+n)=n/N\to 1-\lambda$ for 0 < λ < 1, write

$\begin{array}{ll}\hfill D_{mn}& =\sqrt{\frac{mn}{m+n}}\underset{x}{sup}|\left\{F_{1m}(x)-F(x)\right\}-\left\{F_{2n}(x)-F(x)\right\}|\hfill \\ =\sqrt{\frac{mn}{m+n}}\underset{0\le t\le 1}{sup}|m^{-1/2}\left[\sqrt{m}\left\{G_{1m}(t)-t\right\}\right]-n^{-1/2}\left[\sqrt{n}\left\{G_{2n}(t)-t\right\}\right]|\hfill \\ \stackrel{\mathcal{L}}{\to }\sqrt{\lambda \left(1-\lambda \right)}\underset{0\le t\le 1}{sup}|{\lambda }^{-1/2}{Y}_1(t)-{\left(1-\lambda \right)}^{-1/2}{Y}_2(t)|\hfill \\ =\underset{0\le t\le 1}{sup}|\sqrt{1-\lambda }\hfill \end{array}$