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Suppose $\{X_i\}_{i\in 1\ldots n}$ are $n$ independent, non-identically distributed RV's. Let $X_i \sim f_i(x) \mathbf{1}_{[0,1]}$, where $f_i$ is the $i$-th parent supported on $[0,1]$.

I am confused between two notions: the joint CDF $F(X_1,\ldots, X_n)$ and the joint cdf of the order statistics $F(X_{(1)},\ldots,X_{(n)})$.

Are they related at all, as they do for iid RVs? If so, how?

It seems to me that over that subset of $\mathbb{R}^n$ on which the tuple $(x_1,\ldots,x_n)$ is nondecreasing (a simplex), the two cdfs would coincide. But then, $F(X_{1\ldots n})$ is a product of parent CDFs wherever it is defined, and is easy to compute. On the other hand, $F(X_{(1\ldots n)})$ has an expression driven by the permanent of an $n\times n$ matrix, and is thus intractable. What am I missing?

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  • $\begingroup$ Look at this books.google.com/… . $\endgroup$ – Mark L. Stone Jan 9 '16 at 1:22
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    $\begingroup$ I've been through that section, but my understanding of the question is still foggy. $\endgroup$ – PKG Jan 9 '16 at 1:28

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