Friday, March 25, 2016

Round of 16: What's Sweet about It?

The Statistics Department is running a March Madness contest, and I couldn't resist entering.
It is organized a little differently than the usual bracket picking, which made it more fun to
prepare an entry.  You are given a budget of 100 units,  and you must pick a subset of teams
as many as you want subject to the budget constraint:  Teams seeded 1 cost 25, 2 cost 19, ...
16 seeds cost 1.  I simulated 10,000 brackets, and recorded the survival probabilities as in
the earlier survival plot on this blog, and then computed the expected number of wins for
each team, normalized by their cost, ordered the teams and produced the following list of
teams.  The winner is the entry whose teams accumulate the largest number of wins.

                 EWins Seeds Cost      Bang CumCost CumEWins
Gonzaga         1.1619    11    4 0.2904750       4   1.1619
Pittsburgh      0.9041    10    4 0.2260250       8   2.0660
Cincinnati      1.0757     9    5 0.2151400      13   3.1417
Iowa            1.6385     7    8 0.2048125      21   4.7802
Syracuse        0.8112    10    4 0.2028000      25   5.5914
VA Commonwealth 0.7855    10    4 0.1963750      29   6.3769
West Virginia   2.4625     3   13 0.1894231      42   8.8394
Duke            2.2013     4   12 0.1834417      54  11.0407
Purdue          1.9888     5   11 0.1808000      65  13.0295
Connecticut     0.9040     9    5 0.1808000      70  13.9335
Butler          0.8602     9    5 0.1720400      75  14.7937
Indiana         1.8908     5   11 0.1718909      86  16.6845

Texas A&M       1.8507     3   13 0.1423615      99  18.5352

Monday, March 14, 2016

Bracketology 2016

Again, it is time to fill the brackets.  This year, again, I'm planning to join the Kaggle gaggle, but
I thought I would report here what this years survival probabilities look like.  The situation is quite
different than last year, when Kentucky was the overwhelming favorite.  This year there is a quite
close race with three teams:  Kansas, MSU, and UNC all above 0.10 probability of winning it all,
with 0.16, 0.12, 0.11 respectively.  This is based on my 1000 simulations of the tournament with
my standard QR model, just like last year when Kentucky was at 0.40.  Here is a Tufte type
spark lines graphic with this years survival functions.

This year we have also posted a new bracket generator at

http://www.econ.uiuc.edu:8080/QBracketology/

that can be used to generate a random bracket according to this year's fitted model.  Thanks to
Ignacio Sarmiento Barbieri for help with the R shiny implementation of this.  Further details
about the methods underlying all of this are available here:

http://www.econ.uiuc.edu/~roger/research/bracketology/MM.html

Sunday, January 3, 2016

Edgeworth and the Newsvendor

Several months ago I received a new paper by Mukherjee, Brown and Rusmevichientong on
empirical Bayes methods for the "newsvendor problem."  It is a matter of considerable personal
embarrassment that I hadn't ever heard about the newsvendor problem.  In its simplest form we
have a newspaper vendor (remember them?) who must decide how many papers of various sorts
he should stock.  He sacrifices profits if he stocks too many or too few, the loss is linear in both
directions, but asymmetric.  He knows the distribution, F, of the (random) demand for each
paper, and of course his solution is to stock the a/(a+b) quantile of F, where a is the marginal
cost of stocking too few papers, and b is the marginal cost of stock too many.  I learned this
bit of basic decision theory wisdom from Tom Ferguson's wonderful textbook in a course from
Bruce Hill eons ago, and I was aware that it appears in Raiffa and Schlaifer's text as well, but
I was surprised to learn that there was an extensive operations research literature going back
at least to the seminal paper of Arrow, Harris and Marschak in 1951.  Why I was surprised,
since this is a rather fundamental problem in inventory policy, is another question, but I won't
delve into that. *  Instead, I decided to look into the earlier history of this idea and began to see
quite a lot of references to Edgeworth's (1888) paper "The mathematical theory of banking."
More embarrassment ensued when I realized that I'd never read this paper.  Reading it revealed
a very clear formulation of the newsvendor problem applied to banking, unfortunately Edgeworth's
solution seemed a little murky.  I've tried to untangle all this in a couple of pages below, but as
you will see, it resists a fully satisfactory untangling.

Edgeworth.pdf

* I would like to stipulate that the simple static model underlying the newsvendor problem
is only a small part of the accomplishment of the AHM paper that introduced dynamic sS
rules as well.

Thursday, December 3, 2015

The Pianists' Birthday Problem

We are used to the idea that common birthdays are not that unusual.  I recently attended a
chamber music concert involving the young German pianist Markus Groh.  A quick look at
his website revealed a rather extreme version of birthday matching:  Groh was born January 5th,
the same day as Alfred Brendel, Maurizo Pollini, and Arturo Benedetti Michelangeli!

Sunday, November 15, 2015

Requirements for Academic Work

Inquiring minds often wanted know:  What sort of qualifications were necessary for academic
positions?  Finally we have an answer from the University of Arkansas.  In addition to passing
background checks for prior criminal records one must be capable of:

Sedentary Work - Exerting 10 pounds: Occasionally, Light Work - Exerting up to 20 pounds: Occasionally, Kneeling: Occasionally, Climbing (Stairs, Ladders, etc.): Occasionally, Lifting 5 -10 lbs: Occasionally, Lifting 10-25 lbs: Occasionally, Carrying 5-10 lbs: Occasionally, Pushing/pulling 10-25 lbs: Occasionally, Sitting for long periods of time: Occasionally, Standing for long periods of time: Occasionally, Speaking; Essential, Hearing: Essential, Vision: Ability to distinguish similar colors, depth perception, close vision: Essential, Walking - Short Distances: Frequently, Walking - Moderate Distances; Occasionally.

Friday, November 6, 2015

Ab Uno Disce Omnes

Virgil's dictum:  "From one example, all is revealed,"  would seem to be the very antithesis of
statistical thinking, so it seems surprising to discover that a single observation is enough to
construct a confidence interval for the mean in the standard Gaussian model with unknown
mean and variance.  The construction is described below and comes from class notes of Charles Stein,related to me by Steve Portnoy.  Any further information about the origins of this problem
or its solution would be most welcome.

Details in html
Details in pdf