conditional distribution of sample given maximum

Question

Given an i.i.d. sample $X_1,\dots, X_n$ from the uniform distribution on $[0,\theta]$, and denoting their order statistics by $X_{(1)} < X_{(2)} < \cdots < X_{(n)}$, it is easy to show that $X_{(1)}, \dots, X_{(n-1)}| X_{(n)}$ is equal in distribution to the order statistics of an i.i.d. sample from the uniform distribution on $[0,X_{(n)}]$. My question is this: what does the previous result tell us about the distribution of $X_1,\dots,X_{i-1},X_{i+1},\dots,X_n| X_{(n)}=X_i =t$ for some fixed $i$?

Answer 1

I’m not sure whether it’s the previous result that tells us this, but that distribution is just the distribution of $n-1$ samples independently uniformly drawn from $[0,t]$. There’s no reason why some particular one of the $X_k$ being the maximum should break the independence or uniformity of the others; the only information gained about them is that they’re $\le t$.