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Let $X_{(1)}\leq X_{(2)}\leq \cdots X_{(n)}$ be the order statistics for a random sample from a continuous distribution with c.d.f. $F(x)$ and density $f(x)$. Define $U_{i}$, $i=1,2\ldots,n,$ by $$U_{i}=\frac{F(X_{(i)})}{F(X_{(i+1)})}, \quad i=1,\ldots,n-1, \: \mbox{and }\: U_{n}=F(X_{(n)}).$$

Find the joint distribution of $U_{1},\ldots,U_{n}$.

Remark: I need a suggestion or someone tell me in which book I can find this exercise or related theory that allows me to solve it.

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    $\begingroup$ Where did this question come from? $\endgroup$
    – Yemon Choi
    Commented Feb 23, 2017 at 0:27
  • $\begingroup$ @YemonChoi I found it in "Introduction The Theory of Nonparametric Statistics, Randles & Wolfes " $\endgroup$
    – user70004
    Commented Feb 23, 2017 at 1:05

1 Answer 1

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See the discussion after Theorem 1.2.9 in the book you refer to (Introduction to the Theory of Nonparametric Statistics, Randles & Wolfe).

If we let $V_i=F(X_{(i)})$ then it is explained that, because (by Theorem 1.2.9) $F(X)\sim U(0,1)$, we have that $V_1\leq \cdots \leq V_n$ are distributed as the order statistics from a uniform distribution on $(0,1)$. That is to say, they have joint density $g(v_1,\ldots ,v_n)=n!$ if $0<v_1<\cdots <v_n<1$, or $0$ otherwise.

To obtain the joint distribution of $U_1,\ldots ,U_n$ we just need to calculate the Jacobian of the transformation between the $U_i$'s and the $V_i$'s. The Jacobian is triangular in this case, so is quite straightforward to compute, giving the joint density for the $U_i$'s as $f(u_1,\ldots ,u_n)=\prod_{i=1}^n iu_i^{i-1}$ if $0<u_i<1$ for all $i$, or $0$ otherwise.

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