## 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

### Keywords:

majorization; empirical distributions; weak consistency; median
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### References:

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