On the joint asymptotic distribution of extreme midranges (Q793456)

Let \(X_ 1,...,X_ n\) be i.i.d. r.v.'s with c.d.f. F and \(Z_ 1,...,Z_ n\) be the order statistics of the \(X_ i\), that is \(Z_ 1\leq Z_ 2\leq...\leq Z_ n\). Let further denote \(M_ i=2^{-1}(Z_ i+Z_{n-i+1})\) the i-th midrange, \({\tilde M}_ i=2^{-1}(U_ i+V_ i)=(2b_ n)^{-1}(Z_ i+Z_{n-i+1})\) the i-th normalized midrange (this means \(U_ i=b_ n^{-1} (Z_ i+a_ n)\) and \(V_ i=b_ n^{-1} (Z_{n-i+1}-a_ n)\) such that \(F_{b_ n\!^{-1}(Z_ n- a_ n)}(x)\to^{n\to \infty}\exp(-\exp(-x)))\) and \(T_ k=\max_{1\leq i\leq k}\tilde M_ i\) with \(k=[(n+1)/2].\) For distributions F with \(f(x)=F'(x)>0\), F''(x) exists for all \(x\geq x_ 1\in {\mathbb{R}}\) and \(\lim_{x\to \infty}(d/dx)[f^{-1}(x)(1-F(x))]=0\) the joint asymptotic distribution of \({\tilde M}_ 1,...,\tilde M_ k\) is given. Also, the asymptotic distribution of \(T_ k\) with respect to n and k is derived. These results are used to show that the length of the interval which contains the maximum likelihood estimation of the location parameter of distributions with p.d.f. \(f_ g(x,\mu)=c_ g \exp(-g(x-\mu))\), \(x\in {\mathbb{R}}\), with \[ g\in G=\{g| \quad g:{\mathbb{R}}\to {\mathbb{R}},\quad g(o)=0,\quad g(x)=g(-x),\quad g''\quad exists \] \[ and\quad g''>0\quad for\quad all\quad x\neq 0,\quad \lim_{x\to \infty}g''(x)/[g'(x)]^ 2=0\} \] does not tend to zero stochastically. This means that different maximum likelihood estimators for \(\mu\) are possible. Finally, the consequences of this for the test of symmetry of \textit{M. B. Wilk} and \textit{R. Gnanadesikan}, Probability plotting methods for the analysis of data. Biometrika 55, 1-17 (1968), are discussed.