Feb 18, 2016

# Student Puzzle Corner 13: deadline now April 15

Bulletin Editor Anirban DasGupta sets this problem. Student members of the IMS are invited to submit solutions (to bulletin@imstat.org with subject “Student Puzzle Corner”). The deadline is now April 15, 2016.

We consider a problem on Gaussian extreme values. It comes across as a difficult calculation, but when looked at the right way, it is actually not at all difficult. Here is the exact problem.

Consider a sequence of iid random variables $X_1, X_2, \cdots \sim N(\mu , \sigma ^2)$. For any given $n \geq 1$, suppose $\bar{X} = \bar{X}_n$ denotes the mean and $X_{(n)}$ denotes the maximum of the first $n$ observations $X_1,\cdots , X_n$. Define $\mu_n(\bar{X}) = E(X_{(n)}|\bar{X})$, and $V_n(\bar{X}) = \mbox{Var}(X_{(n)}|\bar{X})$.

(a) Find explicit closed form deterministic sequences $a_n, b_n$ such that $b_n[\mu_n(\bar{X})-a_n] \stackrel {a.s.} {\rightarrow} 1$.

(b) Find explicit closed form deterministic sequences $c_n, d_n$ such that $d_n[V_n(\bar{X})-c_n] \stackrel {a.s.} {\rightarrow} 1$.

The solution to the previous puzzle is here.

## Welcome!

Welcome to the IMS Bulletin website! We are developing the way we communicate news and information more effectively with members. The print Bulletin is still with us (free with IMS membership), and still available as a PDF to download, but in addition, we are placing some of the news, columns and articles on this blog site, which will allow you the opportunity to interact more. We are always keen to hear from IMS members, and encourage you to write articles and reports that other IMS members would find interesting. Contact the IMS Bulletin at bulletin@imstat.org

## What is “Open Forum”?

In the Open Forum, any IMS member can propose a topic for discussion. Email your subject and an opening paragraph (to bulletin@imstat.org) and we'll post it to start off the discussion. Other readers can join in the debate by commenting on the post. Search other Open Forum posts by using the Open Forum category link below. Start a discussion today!