Testing the Gumbel hypothesis by Galton's ratio (Q5954059)

If \(X_{1,n}\leq\dots\leq X_{n,n}\) are order statistics of an i.i.d. sample, then the Galton ratio is defined as \[ G_n(s):=(X_{n-s+1,n}-X_{n-s-1,n})(X_{n-s,n}-X_{n-s-1,n})^{-1}. \] The authors derive the weak limit of \(G_n(S)\) as \(n\to\infty\). E.g., the d.f. of \(G_n(1)\) tends to \(F_\gamma\) (\(\gamma\) being the tail index of the sample distribution), where \(F_{\gamma}\) is concentrated on \((1,\infty)\) and for \[ x>1,\;\gamma>0:\quad 1-F_\gamma(x)=2\int_0^\infty(1+(z^\gamma-1)x)^{-1/\gamma}dz/z^2; \] \[ \gamma=0:\quad 1-F_0(x)=2/(1+x); \] \[ \gamma<0:\quad 1-F_\gamma(x)=2\int_0^{(1-1/x)^{-1/\gamma} (1+(z^\gamma-1)x)^{-1/\gamma}dz/z^2}. \] The authors use these relations to derive tests for the nonparametric hypothesis \(\gamma=0\) (i.e., the sample distribution belongs to the max-domain of attraction of the Gumbel distribution). Results of simulations are presented.