Jun 27, 2018

# Solution to Student Puzzle 20

Congratulations to Mirza Uzair Baig [pictured below], at the University of Hawai’i at Mānoa, who wrote an excellent solution to the problem.

Note that the statistic Tn may be represented as

$T_n = I_{Y_{(1)} < X_{(1)}, Y_{(n)} < X_{(n)}}\, \bigg [\sum_{i = 1}^n I_{Y_i < X_{(1)}} + \sum_{i = 1}^n I_{X_i > Y_{(n)}}\bigg ]$
$+ \,I_{X_{(1)} < Y_{(1)}, X_{(n)} < Y_{(n)}}\, \bigg [\sum_{ i = 1}^n I_{X_i < Y_{(1)}} + \sum_{i = 1}^n I_{Y_i > X_{(n)}}\bigg ].$

Denote the empirical CDF of $X_1, \cdots , X_n$ by $F_n$ and that of $Y_1, \cdots , Y_n$ by $G_n$. Then, this above representation yields

$T_n = n\,I_{Y_{(1)} < X_{(1)}, Y_{(n)} < X_{(n)}}\, \bigg [G_n(X_{(1)}) +1-F_n(Y_{(n)})\bigg ]$ $+\,n\,I_{X_{(1)} < Y_{(1)}, X_{(n)} < Y_{(n)}}\, \bigg [F_n(Y_{(1)}) +1-G_n(X_{(n)})\bigg ].$ Use the fact that for given $u, v, nF_n(u)$ and $nG_n(v)$ are binomial random variables with success probabilities $F(u)$ and $G(v)$. Now use the iterated expectation formula by conditioning on the minima and the maxima to get the mean, and similarly, but with a longer calculation, the variance. It is useful to think of $T_n$ as approximately a sum of two geometrics. Suppose $W$ is a negative binomial with parameters $r = 2, p = \frac{1}{2}$. Then for $n$ not too small, $T_n$ would have a point mass at zero mixed with the negative binomial. That is, write down a Bernoulii variable $Z$ with parameter $\frac{1}{2}$; then $T_n$ (in law) is approximately $Z\,(W+2)$. This gives a quick explanation for why the mean and the variance under the null of $T_n$ should be about $2$ and $6$. You can see a plot below of the null distribution of $T_n$ below when $n = 300$; it is distribution-free in its usual sense.

Under specified alternatives, the negative binomial would be replaced by a sum of two geometrics, approximately independent, but not i.i.d.

The next puzzle, number 21, is here. Can you solve it? Send us your answer by September 7.

## 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!