Math 6010 • Linear Models

Let \(U_1,U_2,\ldots,U_n\) be independent identically distributed random variables, uniform on \([0,1]\). Let \(U_{1,n}\leq U_{2,n}\leq\cdots\leq U_{n,n}\) denote the order statistics. Let \(X_1,X_2,\ldots,X_{n+1}\) be independent identically distributed exponential random variables with \(EX_1=1\). Define \(S(i)=X_1+\cdots+X_i\).
\begin{parts}
\item Prove that the random vectors \(\{U_{1,n},\ldots,U_{n,n}\}\) and
\[
\left\{\frac{S(1)}{S(n+1)},\ldots,\frac{S(n)}{S(n+1)}\right\}
\]
have the same distribution.
\item Compute the asymptotic distribution of \(n(U_{i+3,n}-U_{i,n})\), as \(n\to\infty\) when \(i\) is fixed.
\end{parts}