Discussion
Suppose that X1, X2, ........, Xn are i.i.d. random variables such that X1 has uniform distribution over (0, \(\theta\)), \(\theta \gt 0\). Then, which of the following statements is true ?
(I) The conditional density \(f_{y}(\cdot)\) of \(X_{(1)} \mid X_{(n)} = y\) is given by :
\[ f_{y}(x) = \begin{cases} (n-1) \cdot \frac{1}{y}\left(1 - \frac{x}{y}\right)^{n-2}, & \text{if } 0 \lt x \lt y \\ 0, & \text{otherwise} \end{cases} \]
(II) The conditional distribution of \(X_{(1)} \mid X_{(n)} = y\) does not depend on \(\theta\).
Answer: B
Both (I) and (II) are true.
Given the maximum \(X_{(n)} = y\), the remaining n − 1 observations are independent and uniform on (0, y). The minimum of n − 1 such variables has
\[ \begin{aligned} &P(X_{(1)} \gt x) \\ &= \left(1 - \frac{x}{y}\right)^{n-1},\quad 0 \lt x \lt y \end{aligned} \]
Differentiating, its density is \((n - 1)\frac{1}{y}\left(1 - \frac{x}{y}\right)^{n-2}\), which is the density in (I).
This density involves only y, not \(\theta\), which is (II). It reflects the fact that \(X_{(n)}\) is sufficient for \(\theta\), so the conditional distribution of any other statistic given it is free of \(\theta\).
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