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EXHIBIT
D
FILED: NEW YORK COUNTY CLERK 11/24/2015 06:39 PM INDEX NO. 453054/2015
NYSCEF DOC. NO. 103 RECEIVED NYSCEF: 11/24/2015
SUPREME COURT OF THE STATE OF NEW YORK COUNTY OF NEW YORK --- ------------------- -------- -------x THE PEOPLE OF THE STATE OF NEW YORK, by ERIC T. SCHNEIDERMAN, Attorney General of the State ofNew York, Index No. 453054/2015
Plaintiff, IAS Part 13
Justice Manuel Mendez
- against-AFFIDAVIT OF DR. ABRAHAM J. WYNER
DraftKings, Inc.,
Defendant. --- -------- -------------- ------ -- - -- x
COMMONWEAL TH OF PENNSYLVANIA ) ) ss.:
PHILADELPHIA COUNTY )
DR. ABRAHAM J. WYNER. being duly sworn, deposes and says:
1. I am a tenured Professor of Statistics at the University of Pennsylvania's Wharton
School of Business, where I also serve as Chair of the Undergraduate Program in Statistics. In
1988 I completed my own undergraduate education magna cum laude at Yale University with a
BS in mathematics. Then, in 1993, I earned my Ph.D. in Statistics from Stanford University.
My expertise is in probability theory and statistics, topics on which I publish scholarly works and
present at conferences and universities. I currently also host a weekly ractio show on Sirius XM
called "Wharton MoneyBall" where fantasy sports games are frequently discussed.
2. I submit this affidavit in opposition to the State's motion for a pre liminary
injunction against DraftKings. Inc. ("DraftKings"). I provided the analysis below at the request
of Boies, Schiller & Flexner LLP, attorneys for DraftKings.
3. The documents that I have reviewed are shown in Exhibit 1. A current copy of
my curriculum vitae can be found in Exhibit 2.
Background&Scope
4. The Attorney General of the State of New York has filed a complaint on behalf of
the people of New York against DraftK.ings, a company that offers daily fantasy sports ("DFS")
competitions. The complaint asserts that DFS is illegal gambling because participants allegedly
wager on "contests of chance" or on "future contingent events" not under the participant's
"control or influence."
5. There are games in which chance has a sufficiently low influence on the outcome
so as to be immaterial. This determination is difficult to make because there is no well-defined
principle in mathematics or statistics that can be used to measure the precise influence that
chance has over the outcome of a contest or game.
6. Nonetheless, in my research and writings, I have worked to understand the
interplay between chance and skill, so that an informed determination can be made. My analysis
applies this knowledge to determine whether chance plays a material role in DFS contests.
Guiding Examples and Basic Principles
7. First, consider a few examples of chance- and skill-based games to help
understand the basic principles involved:
8. Consider games that are entirely chance-based (e.g., casino games like slot
machines, craps, or roulette). The key feature in games of this type is that there is little or
nothing to learn other than the rules. There are no strategies. There are no decisions that can
alter the odds of winning. Every player is the equal of every other.
2
9. Now consider the opposite end of the spectrum: games that are virtually entirely
skill-based (e.g., chess). For these contests, the difference between players' abilities, talents, and
skills is nearly always the determining factor. The key feature is the nearly unlimited levels of
ability that differentiates participants. Players can improve their skills with effort and learning.
These games have large numbers of choices, decisions, and strategies that have a material impact
on the outcome. In contests that pit two players at different levels of mastery, the stronger player
wiJI usually defeat the weaker player.
I 0. There are games at other positions along the spectrum of skill and chance that
involve both-for example, Scrabble. The popular word game Scrabble involves both skill and
chance. The skill component involves strategies and talents that introduce considerable
differentiation among players. But chance still plays a role; for example, letter tiles are drawn at
random. Nevertheless, it is generally acknowledged that skill dominates: a more skilled player
will usually defeat a weaker player. To master skills in Scrabble takes a great deal of effort and
learning.
11. Poker is at a different place along the spectrum. Poker involves considerable skill
and in the long run better players will earn more money than weaker players. But in a given
hand of play, the game is chance-dominated. Since a player's options are quite )jmited, it often
happens that a weak player will defeat a stronger player on a given band, but not in the long run
over the course of a sufficiently large number of hands.
12. Statistically, an event is "random" if it is unpredictable. There are different types
of randomness-some of these are internal to the game and other external. Obvious examples of
internal randomness are cards or dice-the stock features of traditional forms of gambling.
Another form of internal randomness is introduced by unpredictable variation in a player's
3
performance (e.g., a soccer goalie's successful block of a penalty kick) or caused by a referee's
inconsistent and sometimes erroneous decision making.
13. External randomness has nothing to do with the game itself but is sometimes an
important consideration. External randomness is introduced when a player' s opponent is
assigned using an unpredictable or random mechanism. If an opponent is disqualified (perhaps
by injury) then victory can be secured by a mechanism having nothing to do with the game itself.
These forms of randomness are external but still important, since they can have an enormous and
often dominant impact on a players "chance of winning" while having nothing to do with a
game's relative balance of skill and chance.
Requirements of Skill-Dominant Competitions
14. Skill-dominant games (i.e., games in which chance does not play a material role
in the outcome) possess three properties: Depth, Complexity, and Differential Impact. These
properties are necessary and sufficient.
a. DeQth: The game must have depth with difficult material to study, learn, and
practice. A game that is skill-dominant can be mastered, but only with a great
deal of effort, dedication, and talent.
b. Complexity: During gameplay, there must be many decisions, strategies, and
choices with material impact on the outcome. That is, the game must be complex
enough so that novices cannot stumble accidently into a winning position with
any frequency .
c. Differential Impact: There must be many levels of proficiency. Furthermore,
when two players with distinct levels of mastery (i.e., skill level) enter into a head
to head contest, the more proficient player should usually win.
4
15. Note that it is not sufficient for an expert to usually defeat a novice. The "expert"
needs to be able to consistently defeat the "very good" player, too, and so forth down the ladder
of levels (e.g., the "merely good" needs to consistently defeat the "average").
16. Next, consider a head-to-head competition involving two players of nearly equal
skill level. Since the skill levels are balanced, if the contest has any internal chance component
at all, then the specific outcome of the contest will be determined predominantly by a
combination of internal chance and external randomness. This may appear to be a paradox, but it
is nevertheless to be expected even in games that are almost entirely skill-based. Even in a game
as skill-dominated as chess, for example, an equally skiUed opponent's choice of opening moves
may affect the ultimate outcome of the game. Likewise, an internal chance mechanism for
equally skilled chess players comes from the simple selection of which player moves first.
17. A similar situation arises when contestants are randomly selected among a large
pool of competitors of varied quality. In such situations, an unskilled player's likelihood of
winning the contest depends predominantly on external randomness- the choice of their
opponent-even when the game itself is skill dominant. This is especially true when the field
contains many unskilled players and a relatively small proportion of skilled players, then an
unskilled player's "chance of winning" will depend significantly on the quality of the opponent;
a chance determination that has nothing to do with the game itself.
18. Returning to the previous example of Scrabble, it is easy to see that the game
possesses all of the three properties mentioned above. First, there are many complex strategies
and difficult tasks; this satisfies the qualification for depth. Second, in any one Scrabble game
there are innumerable choices and decisions; this satisfies the qualification for complexity.
Finally, better players will usually defeat weaker players across the entire ladder of skill; this
5
satisfies the qualification for differential impact. Note that there is nevertheless a component of
chance: When players of nearly equal ability play each other, the random tiles chosen will
determine the outcome, not skill.
19. In contrast, poker fails to satisfy two oftbe three qualifications. Poker is certainly
a deep game with much to study and many skills to acquire. Nevertheless, in a single hand of
poker there is not enough decision complexity; players have only a limited palette of choices
(fold, call, or raise) and thus even a novice will frequently play "properly" just by chance. Also,
talent and skill do not have a large enough impact on a round. A great player will not beat a
good player with regularity, although he will have an edge- the cards have too great an impact.
In a hand of poker, differences in skill do not have a large impact on the outcome.
Daily Fantasy Sports
20. As explained below, daily fantasy sports contests satisfy all three requirements to
be skill-dominant competitions.
Depth
21. To play DFS successfully, contestants need to have deep knowledge in three
difforent subject areas: (i) sports, (ii) statistics, and (iii) economics. Since advanced knowledge
in all three areas is necessary to be extremely skilled at DFS, it follows that highly successful
players should be exceptional, which is indeed the case.
a. Selecting a fantasy roster requires an understanding of the sport and its rules. It is
particularly important to understand that fantasy teams are largely not evaluated
on the performance of teams in real-world games- for example, a real-world
team could lose a game, while the DFS player who selects that team's athletes for
his fantasy team could still win the DFS competition. Therefore DFS players
6
need to understand the real-world athletes and how they function in the different
player positions which affect the fanta!,y score, since that impact follows its own
set of patterns.
b. Statistical knowledge is essential for forecasting player performance. In fact, a
successful team is built on high quality prediction systems.
c. Similarly, knowledge and proficiency with economic concepts is necessary
because a fantasy team is constructed using a salary-based system, whereby
players assemble a roster by recruiting athletes at set prices (i.e., based on the
"salaries" ascribed to the athletes). A quality player must learn to select high
impact athletes at bargain prices in order to consistently defeat opponents. In
essence, a player must learn to play a high quality Moneyball.
Complexity
22. In DFS, a pfayer has to construct a fantasy roster with multiple player choices at
multiple positions. These decisions are the determining factor in the outcome (since they
determine how points are ultimately attributed to the team).
23. The player also must abide by budget constraints (i.e., a total "salary cap" on the
team), which make strategy very important and challenging. For this reason, it is extremely
unlikely that a novice accidentally assembles a high-performing team.
Differential Impact
24. To establish differential impact of skill level on the outcomes in DFS I refer to
two sets of research reports that address the question of how players' different skill levels impact
the outcome of play.
7
a. The report prepared by Professor Zvi Gilula, former Chair of the Department of
Statistics at Hebrew University and current Visiting Professor of Statistics and
Econometrics at the University of Chicago Booth School of Business.
b. Four analyses undertaken by Gaming Laboratories International, LLC (GLI).
25. The GLI analyses show, in four different professional sports (MLB, NFL, NHL,
and NBA), that highly skilled players will defeat complete novices at DFS (represented by
"unskilled" computer simulations selecting randomly-generated teams) anywhere from 80 to 90
percent of the time, depending on the sport. They also show that the most skilled players will
typically defeat skilled computer-based strategies.
26. Prof. Gilula's report shows that there are players whose successes cannot be
explained by chance. From Prof. Gilula's data, I can also deduce that there are differing skill
levels among the top-performing players: Table 5 of the report shows the win ratio obtained by
top performing players. There are players who have won every single one of the more than
approximately 75 entries submitted, players who have won approximately 80%, and others who
have won 67% of their entries. What this means is that not only are there highly skilled players
for whom chance plays a non-material role, but that there are many levels of skill among players,
where the best win most, the second-best win less often, and so on down the ladder of skill
levels.
Conclusion
27. DFS is deep and complex, and players with the most skill will usually and
consistently defeat players with less skill. Although there is a chance component in certain DFS
contests, DFS satisfies all the necessary and sufficient requirements for skill-based games in
which the outcome does not depend in a material degree on chance.
8
28. I reserve my right to revise my opinion as additional information becomes
available to me.
DATED: Philadelphia, Pennsylvania November 23, 2015
Sworn to before me this 23rd day of November, 2015
NOTARIAL SEAL . 1
GERALDINE PETTEY. Notary Pubhc I Citv o· p, i1ad,'!o;. ia, Phila. County
My Cc1'..rni""''0'1 ~: ,.~ . i"~!J?.:Y 14, 2017 ...._.,___·_·_• ··"'''"':-.: ...... : ... :.J::·_. _4..-.--
Dr. Abraham J. Wyner
9
CERTIFICATE OF CONFORMITY
The undersigned does hereby certify that he is an attorney at law duly admitted to
practice in the State of Pennsylvania and is a resident of the State of Pennsylvania; that he makes
this affidavit in accordance with the requirements of the Clerk of the County of New York
pertaining to the acknowledgement of the proof of the Affidavit of Abraham J. Wyner, filed in
the Supreme Court of the State ofNew York, New York County; that the foregoing
acknowledgment of Abraham J. Wyner named in the foregoing instrument taken before
Geraldine Pettey, a Notary in the State of Pennsylvania, being the state in which it was taken,
and based upon my review thereof, appears to conform with the law of the State of Pennsylvania
as to the purpose for which it is submitted and filed.
COMMONWEALTH OF PENNSYLVANIA Notarial Seal
Rosemary L. Lavin, Notary Public City of Philadelphia, Philadelphia County My Commission. Expires Aug. 23, 2016
MEMP~fi. P~NNiiiYLVANtA M~ocn110N OF NOTARIES
Franco A. Corrado
EXHIBIT
1
Document Reviewed
• Memo to Tim Dent re Skill Simulation Analysis of DraftKings Daily Fantasy Hockey Contest by Gaming Laboratories International (April 15, 2015).
• Memo to Tim Dent re Skill Simulation Analysis of DraftKings Daily Fantasy Basketball Contest by Gaming Laboratories International (June 19, 2015).
• Memo to Tim Dent re Skill Simulation Analysis of DraftKings Daily Fantasy Hockey Contest by Gaming Laboratories International (July 9, 2015).
• Memo to Tim Dent re Skill Simulation Analysis of DraftKings Daily Fantasy Football Contest by Gaming Laboratories International (July 24, 2015).
• Expert Report of Professor Zvi Gilula, dated April 27, 2015.
EXHIBIT
2
Abraham J. Wyner Department of Statistics, The Wharton School, University of Pennsylvania
448 Jon M. Huntsman Hall, Philadelphia, PA 19104 [email protected]
Citizenship: United States
Education Stanford University, Stanford, CA. Ph.D., Statistics, June 1993. Advisor: Ani Adhikari. Yale University, New Haven, CT. BS, Mathematics, May 1988. Graduated magna cum laude, with distinction in the major.
Academic Employment Wharton School, University of Pennsylvania, Philadelphia, PA. Full Professor of Statistics, July 1, 2013. Chair, Undergraduate Program in Statistics. Wharton School, University of Pennsylvania, Philadelphia, PA. Associate Professor of Statistics, September 2005-2013. Chair, Undergraduate Program in Statistics. Wharton School, University of Pennsylvania, Philadelphia, PA. Assistant Professor of Statistics, September 1998-2005. University of California at Berkeley, Berkeley, CA. Visiting Assistant Professor of Statistics, September 1995- June 1998. Stanford University, Stanford, CA. Acting Assistant Professor of Statistics, September 1993-June 1995.
Non-Academic Employment IBM Thomas J. Watson Research Center, Yorktown Heights, NY. Research Associate, Speech Recognition Group. June 1989-August 1989. AT&T Bell Laboratories, Murray Hill, NJ. Consultant, Data Compression Lab. June 1987-present.
Statistical Consulting Services (2012-present)
• Blue Man Group Holdings. Provided statistical expertise on accounting matters and
survey design. May 2013. • Berger & Montague, P.C. Expert report and deposition. Miriam Haskins et. al. v.
First American Title Insurance Co. November 2012- May, 2013.
• DII Investment Trust. Statistical modeling and consulting. November 2011- Present. • NYC Police Department. Expert report. Carmichael v. City of New York et al. July,
2012. • Tycko & Zavareei LLP. Expert report. June, 2012. • Kramer, Levin, Frankel and Naftalis LLP. Expert services. U.S. v. Chad Elie & John
Campos, et al. May, 2012. • Paul, Weiss, Rifkind, Wharton & Garrison LLP. Expert report and deposition,
Bermanger et. Al. vs. CGM et al. April 2012. Client: Citigroup. • Paul, Weiss, Rifkind, Wharton & Garrison LLP. Expert consulting for advertising
arbitration involving Castrol. January, 2012.
Awards
• National Science Foundation (NSF) Post-Doctoral Fellowship in the Mathematical Sciences, 1995-1998
• National Science Foundation Fellowship, 1989-1991 • Hertz Foundation Award, 1988 • Abrams Award, Stanford Statistics Department, 1988 • Anthony D. Stanley Prize for Excellence in Mathematics, 1988
Professional Activities
Member of NSF statistics expert panel 2003. Referee for NSF grant proposals, Israel-US Bi-National Science Foundation. Referee for numerous articles in many journals including the Annals of Statistics, IEEE Transactions on Information Theory, Machine Learning, Annals of Probability, Annals of Applied Probability, Journal of Computational Biology, Computer Journal, Journal of Information Retrieval, Statistical Science, Annals of Applied Statistics, Geographical Review Letters, NIPS, AI-Stats, Proceedings of Information Theory Symposium (also on Programming Committee).
Refereed Publications1
(*denotes alphabetical ordering)
1. Mathieu E. Wimmer, Justin Rising, Raymond J. Galante, Abraham Wyner, Allan I. Pack, Ted Abel , “Aging in Mice Reduces the Ability to Sustain Sleep/Wake States”, PloS one 8 (12), e81880, December, 2013.
2. McShane, Blakely B.; Jensen, Shane T.; Pack, Allan I.; Wyner, Abraham J. “Modeling
Time Series Dependence for Scoring Sleep in Mice.” Journal of the American Statistical Association, 108 (504), 1147-1162, 2013.
3. McShane, Blakely B.; Jensen, Shane T.; Pack, Allan I.; Wyner, Abraham J. “Rejoinder:
Modeling Time Series Dependence for Scoring Sleep in Mice.” Journal of the American Statistical Association, 108 (504), 2013.
4. Driver, R. J., Lamb, A. L., Wyner, A. J., & Raizen, D. M. "DAF-16/FOXO Regulates
Homeostasis of Essential Sleep-like Behavior during Larval Transitions in C. elegans." Current Biology (2013).
5. Hu FY, Hanna GM, Han W, Mardini F, Thomas SA, Wyner AJ, Kelz MB, “Hypnotic Hypersensitivity to Volatile Anesthetics and Dexmedetomidine in Dopamine-Hydroxylase Knockout Mice.” Anesthesiology. Accepted for Publication, November 2012.
6. Hu FY, Hanna GM, Han W, Mardini F, Thomas SA, Wyner AJ, Kelz MB, “Hypnotic
Hypersensitivity to Volatile Anesthetics and Dexmedetomidine in Dopamine- Hydroxylase Knockout Mice.” Anesthesiology. Accepted for Publication, November 2012.
7. *Rising, Justin and Wyner, Abraham. “Partial Kelly Portfolios and Shrinkage
Estimators.” IEEE International Symposium on Information Theory Proceedings, July 2012.
1 Including articles, discussions and rejoinders in refereed journals and refereed conference proceedings.
8. Naidoo, N., Ferber M., Galante, R.J., McShane, B.B., Hu, J.H., Zimmerman, J., Maislin, G., Cater, J., Wyner, A. J., Worley, P., and Pack, A.I., “Role of Homer Proteins in the Maintenance of Sleep-Wake States.” PLoS ONE Genetics, 2012, 7(4): e35174, 2012.
9. McShane, BB; Galante, RJ; Biber, M; Jensen, ST; Wyner, AJ; Pack, AI. “Assessing
REM Sleep in Mice Using Video Data.” SLEEP; 35(3):433-442, 2012.
10. Pick, Jeremy; Chen, Yihan; Moore, Jason T; Sun, Yi; Wyner, Abraham J.; Friedman, Eliot B; Kelz, Max B. “Rapid Eye Movement Sleep Debt Accrues in Mice Exposed to Volatile Anesthetics.” Anesthesiology: Volume 115 - Issue 4 - p 702–712, 2011.
7. *McShane, B.B. and Wyner, A.J. “Rejoinder: A Statistical Analysis of Multiple
Temperature Proxies: Are Reconstructions of Surface Temperatures over the Last 1000 Years Reliable?” The Annals of Applied Statistics, Vol. 5, No. 1, 99–123, 2011.
8. *McShane, B.B. and Wyner, A.J. “A Statistical Analysis of Multiple Temperature
Proxies: Are Reconstructions of Surface Temperatures over the Last 1000 Years Reliable?” The Annals of Applied Statistics, Vol. 5, No. 1, 5–44, 2011.
9. Blakeley B. McShane, Raymond J. Galante, Shane T. Jensen, Nirinjini Naidoo, Allan I.
Pack*, and Abraham Wyner * (*Co-Senior authors), “Characterization of the Bout Durations of Sleep and Wakefulness.” Journal of Neuroscience Methods 193, 321–333, 2010.
10. *Jensen, S.T., McShane, B.B and Wyner, A.J. “Rejoinder: Hierarchical Bayesian
modeling of hitting performance in baseball.” Bayesian Analysis, pp. 669–674, 2009.
11. *Jensen, S.T., McShane, B.B., and Wyner, A.J. “Hierarchical Bayesian modeling of hitting performance in baseball.” Bayesian Analysis 4, pp. 631–652, 2009.
12. *Jensen, S.T, Shirley, K.E., and Wyner, A.J. “Bayesball: A Bayesian Hierarchical Model
for Evaluating Fielding in Major League Baseball.” The Annals of Applied Statistics, Vol. 3, No. 2, 491–520, 2009.
13. *Bradlow, E., Jensen, S., Wolfers, J. and Wyner, A. “A Statistical Look at Roger
Clemens’ Pitching Career.” Chance, Vol. 21, No. 3, 24–30, 2008.
14. *D. Mease, A. Wyner. “Evidence Contrary to the Statistical View of Boosting.” Journal of Machine Learning Research, 9 131-156, 2008.
15. *D. Mease, A. Wyner. “Evidence Contrary to the Statistical View of Boosting: A Rejoinder to Responses.” Journal of Machine Learning Research, 9 195-201, 2008.
16. Mease, D., Wyner, A.J. and Buja, A., “Boosted Classification Trees and Class
Probability/Quantile Estimation.” Journal of Machine Learning Research, 8 (Mar): 409--439, 2007.
17. *Buja, A., Mease, D. and Wyner, A.J. “Comment: Boosting Algorithms: Regularization,
Prediction and Model Fitting.” Statistical Science, Vol.22, No. 4, 506-512, 2007.
18. *Belentepe, C., Wyner, A.J. “A Statistical View of Universal Stock Market Portfolios.” Proceedings of the 2005 International Symposium on Information Theory, Adelaide, Australia, September, 2005.
19. *Ferrari, F. and Wyner, A.J., “Estimation of General Stationary Processes by Variable Length Markov Chains.” Scandinavian Journal of Statistics, Vol. 30, pp. 459-480, September, 2003.
20. Wyner A.J., “On Boosting and the Exponential Loss.” in C. M. Bishop and B. J. Frey
(eds), Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics, Jan 3-6, Key West, FL. 2003.
21. *Foster, D., Stine, B. and Wyner, A.J., “Universal Codes for Finite Sequences of Integers
Drawn from a Monotone Distribution.” IEEE Transactions on Information Theory, Volume: 48 Issue: 6, pp. 1713 -1720, June 2002.
22. *Krieger, A., Long, C., and Wyner, A.J., “Boosting Noisy Data.” C. Brodley, A. P.
Danyluk (Eds.): Proceedings of the Eighteenth International Conference on Machine Learning , Williams College, Williamstown, MA, USA, June 28 - July 1, 2001. Morgan Kaufmann, pp. 274-281, 2001.
23. *Friedman, N., Goldszmidt, M., and Wyner, A.J., “Data Analysis with Bayesian
Networks: A Bootstrap Approach.” Proceeding of the Fifteenth Conference on Uncertainty in Artificial Intelligence, pp. 196-205. Morgan Kaufmann, San Francisco, 1999.
24. *Friedman, N., Goldszmidt, M. and Wyner, A.J., “On the Application of the Bootstrap
for Computing Confidence Measures on Features of Induced Bayesian Networks.” Artificial Intelligence and Statistics: Proceeding of the Seventh International Workshop on Intelligence and Statistics, pp. 197-202. Morgan Kaufmann, San Francisco, 1999.
25. Wyner, A.J., “More on Recurrence and Waiting Times.” The Annals of Applied
Probability, Vol. 9, No. 3, pp. 780-796, 1999.
26. *Buhlmann, P. and Wyner, A.J., “Variable Length Markov Chains.” The Annals of Statistics, Vol. 27, No. 2, pp. 480-513, 1999.
27. Wyner, A.D., Ziv. J. and Wyner, A.J., “On the Role of Pattern Matching in Information
Theory.” IEEE Transactions on Information Theory, Vol. 44, no. 6, pp. 2045-2056, October, 1998.
28. Kontoyiannis, I., Algoet, P.H., Suhov, M. and Wyner, A.J., “Nonparametric Entropy
Estimation for Stationary Processes and Random Fields, with Applications to English Text.” IEEE Transactions Information Theory. Vol. IT-44, pp. 1319 - 1327, May, 1998.
29. Wyner, A.J., “The Redundancy and Distribution of the Phrase Lengths of the Fixed-
Database Lempel-Ziv Algorithm.” IEEE Transactions of Information Theory, Vol. IT-43, pp. 1452 - 1464, September 1997.
30. *Wyner, A.D., and Wyner, A.J., “Improved Redundancy of a Version of the Lempel-Ziv-
Algorithm.’’ IEEE Transactions on Information Theory, Vol. IT-41, pp. 723 – 731, May, 1995.
31. *Farach, M., Noordewier, N., Sevari, S., Shepp, L., Wyner, A.J., and Ziv, J., “On the
Entropy of DNA: Algorithms and Measurements based on Memory and Rapid Convergence.” Proc AC-SIAM, Symposium on Discrete Algorithms (SODA)}, Philadelphia, PA, pp. 48-57. 1995.
32. *Plotkin, N. and Wyner, A.J., “An Entropy Estimator Algorithm and
Telecommunications Applications.” G.R. Heidbreder (ed.), Maximum Entropy and Bayesian Methods, 351-363. Kluwer Academic Publishers. Santa Barbara, CA, 1993. The Netherlands, 1996.
Papers Submitted to Refereed Journals
33. Wyner, A.J. and Foster, D., “On the Lower Limits of Entropy Estimation.” Entropy, Re-Submitted October 2013.
Non-Refereed Publications (*denotes alphabetical ordering)
34. Wyner, Abraham. “A Statistician Reads the Sports Pages: Can the Skill Level of a Game
of Chance Be Measured?” Shane Jensen (column editor) Chance, Vol. 25.3, 2012.
35. *Bradlow, E., Jensen, S., Wolfers, J. and Wyner, A. “Keeping Score: Report Backing Clemens Chooses Its Facts Carefully.” New York Times, February. 10, 2008.
36. Wyner, Abraham. “Why Do Women’s Salaries Still Lag Behind? The Forward,
December 20, 2013.
Book Chapters
37. Wyner, A.D., J. Ziv and Wyner, A.J., “On the Role of Pattern Matching in Information Theory.” Information Theory: Fifty Years of Discovery, S. Verdu and S. McLaughlin, editors. IEEE Press, Piscataway, NJ. 2000. Originally appeared in [27].
Working Papers
38. Wei Han, Max Kelz and Abraham Wyner. “Continuous Segmental Best Fit Analysis using Approximate Entropy.”
39. Lawrence Shepp, Phillip Ernst and Abraham Wyner, “On the alleged correlation of
uncorrelated processes.”
Grants and Contracts 1. ESPN Baseball Project: Received a $ 50K grant from ESPN to study baseball. This grant
will fund a large collaboration with several statisticians in our department (Shane Jensen, Dylan Small), graduate students and professors from outside the department.
2. Mechanisms of Alterations in Sleep with Age. Multi-million dollar program project grant will be funded by National Institute on Aging. It will provide 5 years of summer support for a collaboration with the Sleep Center at the University of Pennsylvania. Approved June, 2006.
3. NIH PhD Training Grant: This grant will provide full time support for a statistics PhD student beginning from Fall 2007 through 2014.
4. SBIR Grant to NIH “High Throughput Phenotyping of Mouse using Video”. This grant proposal submitted in January and approved for funding in March will provide 40K of funding to develop a collaboration with small business partner NeuroCare to explore the commercialization our newly developed technology.
5. Mechanisms of Alerations in Sleep with Age (renewal). Mutlimillion program program projet grant. Approved Feb, 2013.
6. Patent Proposal: we are in the process of patenting this technology.
Doctoral Students: Thesis Advisees 1. Chuan Long, Ph.D. Statistics, 2001. 2. Cengiz Belentepe, Ph.D. Statistics, 2005. 3. Blakeley McShane, Ph.D. Statistics, 2010. 4. Justin Rising, PhD. Candidate Department of Statistics, 2013 5. Wei Han, PhD Candidate, Department of Applied Mathematics, 2013 (co-advisor with
Alexander Rakhlin). 6. Joshua Magarick, PhD. Candidate Department of Statistics, 2015
Wharton Service • Director of the Undergraduate Program in Statistics (2005-present). • MBA Faculty cohort Advisor (2003-2007). • Member of Executive Committee (2003-2004). • Faculty advisor to Wharton Gaming Club (2002-2003).
Invited Presentations Presented technical talks at conferences and universities around the world, including:
Universities: Stanford University, University of California Berkeley, University of Southern California, University of Chicago, Yale University, Columbia University (Statistics), Rutgers University, Hebrew University (CS), Carnegie-Mellon, Weizmann Institute, Technion, Tel Aviv University, ETH-Zurich. Conferences: Workshop on Information Theory and Applications, International Conference on Complexity, DIMACS, International conference on Neural Information and Processing Systems Workshop, IEEE International Workshop of Information Theory, International Conference on Mathematics and Information Theory, Conference on Information Science and Systems, International Symposium on Information Theory, Mid-West Statistical Society Conference, Harvard University Statistical Research Conference, Joint Statistical Meetings, Wharton Sports Business Initiative Conference.
Teaching Experience • Undergraduate: Intro. Statistics, Introduction to Statistics for Social Scientists and
Economics Majors, Introduction to Statistics to Wharton students, Probability Theory. • Undergraduate Preceptorial: Gambling and Probability (2001-2003).
• Graduate: Stochastic Processes with Applications (for OR Ph.D. students), Probability (statistics Ph.D. students), Stochastic Processes (for Ph.D. students), Seminar in Advanced Topics in Statistics and Information Theory. Statistics and the Law.
• MBA: Statistics for Managers (core courses: Stat 611, 612, 603, 621).
• MBA Independent Study: Vik Kakkar, Jackie Pangilinan, Brooks Powlen, Clint Siegfried, Timothy Skender (Sports Research).
• Undergraduate Independent Study and Research (since 2005): 1. Andrew Yang (Sequential Prediction). 2. Dan Short (Gambling/Basketball). 3. Matt Kozim, Matt Corruth, Elan Fuld, Charlie Boatwright, Julie Kandel-Krieger,
Sam Mondry-Cohen (Baseball Research). 4. Michael Olivares, Sydney Scott (Psychology). 5. Divya Krishnan Danish Munir (Financial Engineering). 6. Tamara Pier.