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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
##### MSC:
 62E20 Asymptotic distribution theory in statistics 62F12 Asymptotic properties of parametric estimators
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