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Necessary and sufficient conditions for weak consistency of the median of independent but not identically distributed random variables. (English) Zbl 0934.62052

For each \(n=1,2,\dots\), suppose that \(X_{n1}\), \(X_{n2},\dots, X_{nn}\) are independent random variables with distribution functions \(F_{n1}\), \(F_{n2},\dots,F_{nn}\). Let \(\mathbb{F}_n\) denote the empirical distribution function of the \(X_{ni}\)’s: \[ \mathbb{F}_n(x)\equiv n^{-1}\sum^n_{i=1}1_{(-\infty, x]}(X_{ni}), \] and let \(\overline F_n\) be the average distribution function \[ \overline F_n\equiv n^{-1}\sum^n_{i=1}F_{ni}. \] For any distribution function \(G\), let \(G^{-1}\) be the left-continuous inverse of \(G\) defined by \(G^{-1}(u) \equiv\inf\{x:G(x)\geq u\}\), \(0<u<1\). Throughout this paper, unless otherwise noted, we call \(G^{-1}(1/2)\) the median of \(G\), even when there is a nondegenerate interval \([m_0,m_1]\) of median points in the sense that \(P_G(Y\leq m)\geq 1/2\) and \(P_G(Y\geq m)\geq 1/2\) for \(m\in [m_0,m_1]\). The problem in this paper is to give necessary and sufficient conditions for weak consistency of the sample median \(\mathbb{F}_n^{-1}(1/2)\): under what conditions on the \(F_{ni}\)’s does it hold that \(\mathbb{F}_n^{-1} (1/2)-\overline F_n^{-1}(1/2)\to_P0\)?

MSC:

62G20 Asymptotic properties of nonparametric inference
62G30 Order statistics; empirical distribution functions
60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
62F12 Asymptotic properties of parametric estimators
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