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On the joint asymptotic distribution of extreme midranges. (English) Zbl 0538.62013
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 M. B. Wilk and R. Gnanadesikan, Probability plotting methods for the analysis of data. Biometrika 55, 1-17 (1968), are discussed.
Reviewer: B.Rauhut
62E20 Asymptotic distribution theory in statistics
62F12 Asymptotic properties of parametric estimators
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