A Starting Point for Navigating the World of Daily Fantasy Basketball

Fantasy sports, particularly the daily variety in which new lineups are selected each day, are a rapidly growing industry. The two largest companies in the daily fantasy business, DraftKings and Fanduel, have been valued as high as $2 billion. This research focuses on the development of a complete system for daily fantasy basketball, including both the prediction of player performance and the construction of a team. First, a Bayesian random effects model is used to predict an aggregate measure of daily NBA player performance. The predictions are then used to construct teams under the constraints of the game, typically related to a fictional salary cap and player positions. Permutation based and K-nearest neighbors approaches are compared in terms of the identification of “successful” teams—those who would be competitive more often than not based on historical data. We demonstrate the efficacy of our system by comparing our predictions to those from a well-known analytics website, and by simulating daily competitions over the course of the 2015–2016 season. Our results show an expected profit of approximately $9,000 on an initial $500 investment using the K-nearest neighbors approach, a 36% increase relative to using the permutation-based approach alone. Supplementary materials for this article are available online.     

Dynamic Bayesian forecasting of AFL match results using the Skellam distribution

The scoring and defensive abilities of Australian Rules Football teams change over time as a result of evolving player rosters, tactics and other management factors. We develop a dynamic model based on the Poisson difference (Skellam) distribution which simultaneously models the two different point scoring mechanisms in Australian Rules Football, the motivation for which comes from work on predicting outcomes in soccer matches. Our model is developed in a Bayesian framework and is fitted using the Stan modelling language. Model validation is performed on the 2015 Australian Football league (AFL) home and away season.     

A new sports ratings system: The tiddlywinks world ratings

A system for calculating relative playing strengths of tiddlywinks players is described. The method can also be used for other sports. It is specifically designed to handle cases where the number of games played in a season varies greatly between players, and thus the confidence that one can have in an assigned rating also varies greatly between players. In addition, the method is designed to handle situations in which some games in the tournament are played as individuals ("singles'), while others are played with a partner ("pairs'). These factors make application of some statistical treatments, such as the Elo rating system used in chess, difficult to apply. The new method characterizes each player's ability by a numerical rating together with an associated uncertainty in that player's rating. After each tournament, a "tournament rating' is calculated for each player based on how many points the player achieved and the relative strength of partner(s) and opponent(s). Statistical analysis is then used to estimate the likely error in the calculated tournament rating. Both the tournament rating and its estimated error are used in the calculation of new ratings. The method has been applied to calculate tiddlywinks world ratings based on over 13 r 000 national tournament games in Britain and the USA going back to 1985.     

Predicting Home Run Production in Major League Baseball Using a Bayesian Semiparametric Model

This article attempts to predict home run hitting performance of Major League Baseball players using a Bayesian semiparametric model. Following Berry, Reese and Larkey we include in the model effects for era of birth, season of play, and home ball park. We estimate performance curves for each player using orthonormal quartic polynomials. We use a Dirichlet process prior on the unknown distribution for the coefficients of the polynomials, and parametric priors for the other effects. Dirichlet process priors are useful in prediction for two reasons: (1) an increased probability of obtaining more precise prediction comes with the increased flexibility of the prior specification, and (2) the clustering inherent in the Dirichlet process provides the means to share information across players. Data from 1871 to 2008 were used to fit the model. Data from 2009 to 2016 were used to test the predictive ability of the model. A parametric model was also fit to compare the predictive performance of the models. We used what we called “pure performance” curves to predict future performance for 22 players. The nonparametric method provided superior predictive performance.     

中国居民消费价格指数间的关系与选择

居民消费价格是宏观经济监测的重要内容之一,其指数间的关系与选择具有重要意义。通过与国外发达国家的比较,从我国目前价格指数编制和选择使用的现状及存在的问题出发,研究环比、同比和定基价格指数以及环比“折年率”之间的相互转化情况和它们各自在经济监测中的特征。重点研究了基于X-12-ARIMA模型的环比价格指数的季节调整和同比价格指数相对环比价格指数对物价监测的时滞性。同时验证了研究中同比向定基指数转化的合理性。由此对我国居民价格实时监测的指数选择提供政策建议,也为我国价格指数的编制提出建议。     

Bayesian locally optimal design of knockout tournaments

The elimination or knockout format is one of the most common designs for pairing competitors in tournaments and leagues. In each round of a knockout tournament, the losers are eliminated while the winners advance to the next round. Typically, the goal of such a design is to identify the overall best player. Using a common probability model for expressing relative player strengths, we develop an adaptive approach to pairing players each round in which the probability that the best player advances to the next round is maximized. We evaluate our method using simulated game outcomes under several data-generating mechanisms, and compare it to random pairings, to the standard knockout format, and to two variants of the standard format.     

Reliability of a soccer player based on the bivariate Rayleigh distribution with right censored and ignorable missing data

In this paper, we study the performance of a soccer player based on analysing an incomplete data set. To achieve this aim, we fit the bivariate Rayleigh distribution to the soccer dataset by the maximum likelihood method. In this way, the missing data and right censoring problems, that usually happen in such studies, are considered. Our aim is to inference about the performance of a soccer player by considering the stress and strength components. The first goal of the player of interest in a match is assumed as the stress component and the second goal of the match is assumed as the strength component. We propose some methods to overcome incomplete data problem and we use these methods to inference about the performance of a soccer player.     

The Role of the Statistician: Scientist or Shoe Clerk

A carnival game is described in which a player rolls eight dice or eight marbles into holes. To win, the player must obtain a favorable outcome on a number of plays. The probability distribution for the outcome of a single play is discussed. The distribution of a winning combination of outcomes is discussed and approximated. These probabilities show that the game is essentially unbeatable.     

Ranking paired contestants

The ranking of paired contestants (players) after a series of contests is difficult when every player does not play every other player. In the 1975 JASA Mark Thompson presented a maximum likelihood solution based on the assumption that the probability of any one player defeating any other is a function only of the difference in their ranks. Here the linear approximation to that likelihood is shown to lead to a nonparametric measure of the efficacy of the ranking, called the net difference in ranks (NDR) , which is the sum of the differences in ranks of the paired players in the observed contests that agree with the ranking minus the sum of the differences in ranks in the observed contests that disagree with the ranking (upsets) . The subject is part of a large literature that has been consolidated by H.A. David in The Method of Paired Comparisons (1963, 1988). The method was introduced by the psychophysicist Fechner in 1860 and has been widely applied to sensory testing,     

A Bayesian Analysis of a Poisson Random Effects Model for Home Run Hitters

The problem of interest is to estimate the home run ability of 12 great major league players. The usual career home run statistics are the total number of home runs hit and the overall rate at which the players hit them. The observed rate provides a point estimate for a player's “true” rate of hitting a home run. However, this point estimate is incomplete in that it ignores sampling errors, it includes seasons where the player has unusually good or poor performances, and it ignores the general pattern of performance of a player over his career. The observed rate statistic also does not distinguish between the peak and career performance of a given player. Given the random effects model of West (1985), one can detect aberrant seasons and estimate parameters of interest by the inspection of various posterior distributions. Posterior moments of interest are easily computed by the application of the Gibbs sampling algorithm (Gelfand and Smith 1990). A player's career performance is modeled using a log-linear model, and peak and career home run measures for the 12 players are estimated.     

Randomization. pro and contra 1

We discuss a stochastic urn model in which there are two urns A and B. B is originally empty and A contains some fixed number of white and black balls. A player selects integers n>O and b>O. Balls are drawn with replacement in A and balls of the same color are put in B as long as the number of white balls in B exceeds (b-1) times the number of black balls in B. Under this condition, the player stops after drawing n+bx balls and is declared to be a winnear if urn B has x black balls. This number of black balls, x, is shown to have the generalized negative binomial distribution     

Quarterback evaluation in the national football league using tracking data

AStA Advances in Statistical Analysis - This paper evaluates quarterback performance in the National Football League. With the availability of player tracking data, there exists the capability to...     

Bayesian binary segmentation procedure for detecting streakiness in sports

Summary.  When an individual player or team enjoys periods of good form, and when these occur, is a widely observed phenomenon typically called 'streakiness'. It is interesting to assess which team is a streaky team, or who is a streaky player in sports. Such competitors might have a large number of successes during some periods and few or no successes during other periods. Thus, their success rate is not constant over time. We provide a Bayesian binary segmentation procedure for locating changepoints and the associated success rates simultaneously for these competitors. The procedure is based on a series of nested hypothesis tests each using the Bayes factor or the Bayesian information criterion. At each stage, we only need to compare a model with one changepoint with a model based on a constant success rate. Thus, the method circumvents the computational complexity that we would normally face in problems with an unknown number of changepoints. We apply the procedure to data corresponding to sports teams and players from basketball, golf and baseball.     

Action rate models for predicting actions in soccer

AStA Advances in Statistical Analysis - We present a data-driven approach to predict the next action in soccer. We focus on passing actions of the ball possessing player and aim to forecast the...     

Two stochastic models for the geeta distribution

The first stochastic model is based upon two urns A and B, where A contains a fixed number of white and black balls and B is empty. The player selects an integer β ≥, 2 and draws the balls one by one (with replacement) from urn A and balls of the same colour are put in urn B. The process is continued as long as the number of white balls in B exceeds (β-1) times the number of black balls in B. The player stops after drawing β(x-1) balls and is declared to be a winner if urn B has (x-1) black balls. It is shown that x has the Geeta distribution.

Assuming that the mean μ is a function of two parameters θ and β it has been shown that for small changes inthe value of θ there exists a difference-differential equation which leads to the Geeta distribution.     

Best-Ball Events in Golf: An Application of the Multinomial Distribution

Probability models are developed for two-man teams in best-ball events in golf. The multinomial distribution is used to develop the theoretical probabilities for possible team scores. Here each individual player is assigned a set of probabilities for making a certain score on each hole of an 18-hole round of golf. The two-man team results are compared with the 1980-1982 Bing Crosby Pro-Ams held in Monterey, California.     

行业投资者情绪的测量及其对收益的影响效应研究

投资者情绪对行业收益影响研究的关键在于行业情绪的测量。基于封闭式基金折价、交易量、波动率、行业beta值和每股流通市值这5个单项情绪指标提取主成分,构建了23个申万一级行业的投资者综合情绪指数(IICSI),并通过面板数据模型分析了各行业的投资者情绪对收益的影响效应。实证结果表明,投资者情绪对当期收益具有短期、中期和长期的正向激励作用,但对预期收益则只在短期和中期表现出正向效应,而在长期反转为负向的抑制效应,并且情绪对当期收益和预期收益的影响都具有显著的行业差异性,最后基于情绪对预期收益的行业影响差异构建了不同投资期限下的套利策略。     

Hazards regression for freemium products and services: a competing risks approach

We propose a competing risks approach to analyse customer behaviours in freemium products and services. The event of interest is when a customer starts to pay for additional features or functionalities. The observation of such an event may be preempted by an event where the customer quits using the product before paying and consuming the additional features or functionalities. One such freemium service is the online game category. The Fine-Gray regression model was implemented for an online game player activity data to study how covariates affect the paying hazard. Some covariates are hypothesized to have different discrete effects at multiple change points. We extend the model to allow for possible change points in the analysis.     

Survey of the major world sports rating systems

SUMMARY Using a common framework, this paper presents a survey of the major world sports rating systems (WSRSs) in skiing (sponsored by the International Skiing Federation (FIS)), men's tennis (Association of Tennis Professionals (ATP)), women's tennis (Women's Tennis Association (WTA)), soccer (Federation of International Football Associations (FIFA)) and golf (Royal and Ancient Golf Club of St Andrews). These systems are not otherwise available in the literature. Each of the WSRSs has three phases: first, the observed results are weighted to provide points for each competition; second, these points are combined to provide a seasonal value; third, the seasonal values are combined to provide a rating. The final result or placement (and not the score or time) is the most important factor in determining points for a given competition. In skiing, men's tennis and women's tennis, the rating is calculated from results over one season, while three seasons are used in golf and six seasons are used in soccer. In cross-country skiing and men's tennis, the seasonal value is calculated from the sum of the best values from that season's competitions. In alpine skiing and women's tennis, the sum of all values from that season's competitions is used. In golf and soccer, an averaging process is used. Besides potentially encouraging more entries, a 'best' system and one using all values also generates simple integer ratings rather than decimal ratings as are obtained with an averaging system. The simplest system is that of FIS in skiing, where one table of points is used for all alpine and cross-country disciplines. In contrast, considering that soccer (as a sport) prides itself on the simplicity of the game, it is surprising that FIFA's system is so complex, It is also surprising in soccer that a 'friendly' (often a pick-up exhibition used for player development) counts two-thirds as much as does a World Cup final played before a worldwide TV audience. It is hoped that this survey will serve as a valuable resource for those studying sports rating systems.     

AN ANALYSIS OF CLASSICAL AND TIE-BREAKER TENNIS

Expressions for the probability that a player wins a game, set or match of classical or tie-breaker tennis are obtained. Also, expressions for the distribution, the mean and the variance of the number of points in a game, set or match of classical or tie-breaker tennis are obtained. These results explain the long matches often previously observed in classical tennis between two players each with very effective serves on fast grass-court surfaces. The methodology used to derive these expressions could be used to obtain corresponding expressions for other "nested-type" scoring systems. For example, tennis is 3-nested (game, set, match) whereas squash is only 2-nested (game, match).