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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2016 • Question 9

Question

Math 6010 • Linear Models Spring 2016 • Question 9

Let \(X_1,X_2,\ldots ,X_n\) be iid from a distribution with CDF \(F(x)=1-1/x\) for \(1\leq x<\infty \), and zero otherwise.

(a) Derive the CDF of the smallest order statistic \(X_{(1)}\).

(b) Find the limiting distribution of \(X_{(1)}\).

(c) Find the limiting distribution of \(X_{(1)}^n\).

Raw LaTeX

Let \(X_1,X_2,\ldots,X_n\) be iid from a distribution with CDF
\(F(x)=1-1/x\) for \(1\leq x<\infty\), and zero otherwise.

\textbf{(a)} Derive the CDF of the smallest order statistic \(X_{(1)}\).

\textbf{(b)} Find the limiting distribution of \(X_{(1)}\).

\textbf{(c)} Find the limiting distribution of \(X_{(1)}^n\).

Topic Tags

Estimator Distributions Multivariate Normal Distributions Uniform Integrability and Truncation Methods Weak Convergence and Distributional Limits