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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Fall 2015 • Question 11

Question

Math 6010 • Linear Models Fall 2015 • Question 11

Let \(U_1,U_2,\ldots ,U_n\) be independent and 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.

(a) Compute the limit distribution of \(Z=n(U_{3,n}-U_{2,n})\).

(b) Show that \(Z_1=n(U_{3,n}-U_{2,n})\) and \(Z_2=n(U_{4,n}-U_{3,n})\) are asymptotically independent.

Raw LaTeX

Let \(U_1,U_2,\ldots,U_n\) be independent and 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.

\textbf{(a)} Compute the limit distribution of \(Z=n(U_{3,n}-U_{2,n})\).

\textbf{(b)} Show that \(Z_1=n(U_{3,n}-U_{2,n})\) and
\(Z_2=n(U_{4,n}-U_{3,n})\) are asymptotically independent.

Topic Tags

Covariance and Independence Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods