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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6010 • Linear Models

Spring 2015 • Question 5

Question

Math 6010 • Linear Models Spring 2015 • Question 5

Let \(X_1,\ldots ,X_{2n+1}\) be independent lognormal\((\mu ,\sigma ^2)\). That is, \(X_1=e^{Y_1},\ldots ,X_{2n+1}=e^{Y_{2n+1}}\) with \(Y_1,\ldots ,Y_{2n+1}\) being independent \(N(\mu ,\sigma ^2)\). Use a theorem to show that the sample median \(X_{n+1:2n+1}\) is asymptotically normal and find its asymptotic mean and asymptotic variance.

Raw LaTeX

Let \(X_1,\ldots,X_{2n+1}\) be independent
lognormal\((\mu,\sigma^2)\). That is,
\(X_1=e^{Y_1},\ldots,X_{2n+1}=e^{Y_{2n+1}}\) with
\(Y_1,\ldots,Y_{2n+1}\) being independent \(\N(\mu,\sigma^2)\). Use a theorem to
show that the sample median \(X_{n+1:2n+1}\) is asymptotically normal and find its
asymptotic mean and asymptotic variance.

Topic Tags

Analysis of Variance Covariance and Independence Uniform Integrability and Truncation Methods