# Bounds on CDF for the median of samples from an exchangeable distribution

Suppose $x_1,\dotsc, x_n$ are $n = 2k-1$ samples from an EXCHANGEABLE sequence, where the common marginal distribution is assumed UNIFORM on $[0,1]$. Let $x_{(1;n)} \le \dotsc \le x_{(n;n)}$ be the corresponding order statistics, for which $G_{k;n}(x)$ is the CDF for the median $x_{(k:n)}$, and denote by $H_{k;n}(x)$ the CDF for the median in the case of $n$ IID uniform samples (which is a known polynomial).

Under what additional assumptions on the exchangeable distribution for $x_1, \dotsc, x_n$ can we conclude that $G_{k}(x)$ is bounded between $x$ and $H_{k}(x)$ inclusive, i.e., to ensure that:

$H_{k;n}(x) \le G_{k;n}(x) \le x$ for $0 \le x \le \tfrac{1}{2}$

and

$x \le G_{k;n}(x) \le H_{k;n}(x)$ for $\tfrac{1}{2} \le x \le 1$ ?

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