dians: Let \(F\) and \(G\) be the CDFs corresponding to \(P\) and \(Q\), and denote \(\theta(F)\) the median of \(F\) i.e. \(\theta(F)=\inf\{x:F(x)\ge1/2\}\). Assume that \(F\) is continuously differentiable at \(\theta(P)\) with derivative \(F'\) (and the same with \(F\) replaced by \(G\)). Here, the null hypothesis is of the form  \(H_0: \theta(P)-\theta(Q)=0\), and the corresponding test statistic is given by
$$T_{m,n}=\frac{N^{1/2}\left(\theta(\hat{P}_m)-\theta(\hat{Q})\right)}{\hat{\upsilon}_{m,n}}$$
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