A winning strategy for lotto games?

In lotto games, the distribution of k-tuples chosen by participants is not uniform, but the chance of any k-tuple being the winner is the same. The winning categories consist of matching exactly k — i numbers from the winning k-tuple for i = 0, 1, …, m for some m. The total prize pool for a category is divided equally among all the winning tickets in the category. Therefore the strategy of buying a ticket with a k-tuple consisting of unpopular numbers will increase the expected amount of the prize if this k-tuple is a winner in some category, because the prize pool is shared among fewer tickets. By modelling the distribution of 6-tuples chosen by participants of Lotto 6/49 in Canada, the expected return and standard deviation of return can be computed. It is shown that the expected return can be more than the amount spent when the carryover is large, but the large standard deviation means that it would take tens of thousands of years to millions of years for the strategy to have a high probability of yielding a profit.     

Assessing risk for rare events

This paper develops a method for assessing the risk for rare events based on the following scenario. There exists a large population with an unknown percentage p of defects. A sample of size N is drawn from the population and, in the sample, 0 defects are drawn. Given these data, we want to determine the probability that no more than n defects will be found in another random sample of N drawn from the population. Estimates on the range of p and n are calculated from a derived joint distribution which depends on p, n and N. Asymptotic risk results based on an infinite sample are then developed. It is shown that these results are applicable even with relatively small sample spaces.     

Optimized Sampling Strategies for Identifying Modes in Length-frequency Distributions

An approach to evaluate sampling strategies to detect modes in length-distributions is presented. Distributions based on various numbers of samples (S) and sample sizes (n) were simulated from the original cod and capelin data in Icelandic waters, incorporating within sample correlations. A peak was discerned if the difference between any simulated and original distribution did not exceed a pre-specified Δ, given a probability. This was achieved with numerous combinations of S and n, and the optimal choice will depend on the sampling costs. Variance-equivalence curves also illustrate the difference between demanding precision of mean lengths versus precision of length distributions.     

Who wins the football?

Portsmouth have dumped their manager—will it change their results? Chelsea are on a winning streak—does it mean anything? Media pundits and sports fans—and punters—devote time and energy to speculation about who will win Saturday's beautiful game. The discussions—in pubs, newspapers and dressing rooms—are usually based on subjective opinion, or evidence that is at best partial or impressionistic. Can statisticians do better? John Goddard offers a large-scale analysis of patterns in English league results from the past 35 football seasons—new managers, winning and losing streaks, home advantages and all.     

The case of the random telephone numbers

An unusual pattern of telephone numbers recorded on suspect credit-card purchases led to the development of a generalized multinomial distribution. One usage of credit cards included 26 telephone numbers with a pattern of 7 unique numbers: 16 of one kind, 4 of another, 2 of another, and 4 different ones. With the assumption that the probability of any digit appearing in a telephone number is 1/10, the pattern of the 26 telephone numbers is found to have a probability of occurrence on the order of 1/10⁶⁵. That is, the chance of such a pattern occurring at random is roughly equivalent to winning nine times in a row the popular ottery, LOTO 6/49.     

Experimental designs for detecting synergy and antagonism between two drugs in a pre‐clinical study

The identification of synergistic interactions between combinations of drugs is an important area within drug discovery and development. Pre‐clinically, large numbers of screening studies to identify synergistic pairs of compounds can often be ran, necessitating efficient and robust experimental designs. We consider experimental designs for detecting interaction between two drugs in a pre‐clinical in vitro assay in the presence of uncertainty of the monotherapy response. The monotherapies are assumed to follow the Hill equation with common lower and upper asymptotes, and a common variance. The optimality criterion used is the variance of the interaction parameter. We focus on ray designs and investigate two algorithms for selecting the optimum set of dose combinations. The first is a forward algorithm in which design points are added sequentially. This is found to give useful solutions in simple cases but can lack robustness when knowledge about the monotherapy parameters is insufficient. The second algorithm is a more pragmatic approach where the design points are constrained to be distributed log‐normally along the rays and monotherapy doses. We find that the pragmatic algorithm is more stable than the forward algorithm, and even when the forward algorithm has converged, the pragmatic algorithm can still out‐perform it. Practically, we find that good designs for detecting an interaction have equal numbers of points on monotherapies and combination therapies, with those points typically placed in positions where a 50% response is expected. More uncertainty in monotherapy parameters leads to an optimal design with design points that are more spread out. Copyright © 2015 John Wiley & Sons, Ltd.     

Reflections on the UK National Lottery

Gambling in Britain has been transformed by the National Lottery. The small pink ticket from the newsagent's electronic outlet offers temptation of wealth beyond the dreams of avarice. John Haigh examines a bet that, statistically, not one of our readers will live to win.     

The negative contagion reflection of the polya-eggenberger distribution

The Polya-Eggenberger distribution Involves drawing a ball from an urn containing black and white balls and, after each drawing, returning the ball together with s balls of the same color, The model represents positive contagion since the added balls are the same color as the one drawn, See Johnson and Kotz, (1977),

This paper derives and examines the probability distribution which results from the Polya-Eggenberger model with only one change namely, the s additional balls added after each drawing are of the opposite color, producing a negative contagion model.

Formulas in closed form are presented for the probability distribution function, the mean and variance, all binomial moments and, where s is greater than or equal to the number of balls in the urn at start, the mode, A formula for the mode is conjectured where s is less than the number of balls in the urn at start.

Finally, the probability of obtaining k black balls in n drawings is shown in certain instances to be equal to A_nk/n!

where A_nk are the Eulerian numbers.     

Certain Inverse Multivariate Hypergeometric and Multinomial Distributions

A sample is drawn from a population in such a way that it contains at least certain numbers from certain of its subpopulations. This paper obtains the frequency function of the number of units in the sample and its distribution over the subpopulations, and suggests some applications.     

Interregional Price Difference in the New Orleans Auctions Market for Slaves

Our article investigates the variation of winning bids in slave auctions held in New Orleans from 1804 to 1862. Specifically, we measure the variation in the price of slaves conditional on their geographical origin. Previous work using a regression framework ignored the auction mechanism used to sell slaves. This introduced a bias in the conditional mean of the winning bid because it depended on the number of bidders participating in the auction. Unfortunately, the number of bidders is unobserved by the econometrician.We adopt the standard framework of a symmetric independent private value auction and propose an estimation strategy to attempt to deal with this omitted variable bias. Our estimate of the mean number of bidders doubled from 1804 to 1862. We find the number of bidders had a significant positive effect on the average winning bid. An increase from 20 to 30 bidders in an auction raised the average winning bid by around 10%%. The price variation according to the geographical origin of slaves found in earlier work continues to persist after accounting for the omitted variable. We also find a new result that a considerable premium is paid for slaves originating from New Orleans. However, this price variation disappears once we account for regional productivity differences.     

A new rank test for the K-Sample problem

The treatment sum of squares in the one-way analysis of variance can be expressed in two different ways: as a sum of comparisons between each treatment and the remaining treatments combined, or as a sum of comparisons between the treatments two at a time. When comparisons between treatments are made with the Wilcoxon rank sum statistic, these two expressions lead to two different tests; namely, that of Kruskal and Wallis and one which is essentially the same as that proposed by Crouse (1961,1966). The latter statistic is known to be asymptotically distributed as a chi-squared variable when the numbers of replicates are large. Here it is shown to be asymptotically normal when the replicates are few but the number of treatments is large. For all combinations of numbers of replicates and treatments its empirical distribution is well approximated by a beta distribution     

Rating teams and analysing outcomes in one-day and test cricket

Summary.  Multiple linear regression techniques are applied to determine the relative batting and bowling strengths and a common home advantage for teams playing both innings of international one-day cricket and the first innings of a test-match. It is established that in both forms of the game Australia and South Africa were rated substantially above the other teams. It is also shown that home teams generally enjoyed a significant advantage. Using the relative batting and bowling strengths of teams, together with parameters that are associated with common home advantage, winning the toss and the establishment of a first-innings lead, multinomial logistic regression techniques are applied to explore further how these factors critically affect outcomes of test-matches. It is established that in test cricket a team's first-innings batting and bowling strength, first-innings lead, batting order and home advantage are strong predictors of a winning match outcome. Contrary to popular opinion, it is found that the team batting second in a test enjoys a significant advantage. Notably, the relative superiority of teams during the fourth innings of a test-match, but not the third innings, is a strong predictor of a winning outcome. There is no evidence to suggest that teams generally gained a winning advantage as a result of winning the toss.     

The Promise of Prediction Contests

This article examines the prediction contest as a vehicle for aggregating the opinions of a crowd of experts. After proposing a general definition distinguishing prediction contests from other mechanisms for harnessing the wisdom of crowds, we focus on point-forecasting contests—contests in which forecasters submit point forecasts with a prize going to the entry closest to the quantity of interest. We first illustrate the incentive for forecasters to submit reports that exaggerate in the direction of their private information. Whereas this exaggeration raises a forecaster's mean squared error, it increases his or her chances of winning the contest. And in contrast to conventional wisdom, this nontruthful reporting usually improves the accuracy of the resulting crowd forecast. The source of this improvement is that exaggeration shifts weight away from public information (information known to all forecasters) and by so doing helps alleviate public knowledge bias. In the context of a simple theoretical model of overlapping information and forecaster behaviors, we present closed-form expressions for the mean squared error of the crowd forecasts which will help identify the situations in which point forecasting contests will be most useful.     

Can Business Owners Form Accurate Counterfactuals? Eliciting Treatment and Control Beliefs About Their Outcomes in the Alternative Treatment Status

A survey of participants in a large-scale business plan competition experiment, in which winners received an average of U.S. $50,000 each, is used to elicit ex-post beliefs about what the outcomes would have been in the alternative treatment status. Participants are asked the percent chance they would be operating a firm, and the number of employees and monthly sales they would have, had their treatment status been reversed. The study finds the control group to have reasonably accurate expectations of the large treatment effect they would experience on the likelihood of operating a firm, although this may reflect the treatment effect being close to an upper bound. The control group dramatically overestimates how much winning would help them grow the size of their firm. The treatment group overestimates how much winning helps their chance of their business surviving and also overestimates how much winning helps them grow their firms. In addition, these counterfactual expectations appear unable to generate accurate relative rankings of which groups of participants benefit most from treatment.     

Large sample interval estimation for the logarithmic series distribution

It is assumed that a large random sample of fixed size n is drawn from a logarithmic series distribution with parameter $tH and that it is desired to estimate e by means of two-sided confidence interval. Using asymptotic results, charts of confidence intervals are prepared for n = 50, 100, 200, 500 and 1,000, and confidence coefficients 0.90 and 0.95.     

Construction of a semi-Markov model for the performance of a football team in the presence of missing data

Using play-by-play data from the very beginning of the professional football league in Turkey, a semi-Markov model is presented for describing the performance of football teams. The official match results of the selected teams during 55 football seasons are used and winning, drawing and losing are considered as Markov states. The semi-Markov model is constructed with transition rates inferred from the official match results. The duration between the last match of a season and the very first match of the following season is much longer than any other duration during the season. Therefore these values are considered as missing values and estimated by using expectation–maximization algorithm. The effect of the sojourn time in a state to the performance of a team is discussed as well as mean sojourn times after losing/winning are estimated. The limiting probabilities of winning, drawing and losing are calculated. Some insights about the performance of the selected teams are presented.     

Why a Bayesian approach to safety analysis in pharmacovigilance is important

Large databases of routinely collected data are a valuable source of information for detecting potential associations between drugs and adverse events (AE). A pharmacovigilance system starts with a scan of these databases for potential signals of drug-AE associations that will subsequently be examined by experts to aid in regulatory decision-making. The signal generation process faces some key challenges: (1) an enormous volume of drug-AE combinations need to be tested (i.e. the problem of multiple testing); (2) the results are not in a format that allows the incorporation of accumulated experience and knowledge for future signal generation; and (3) the signal generation process ignores information captured from other processes in the pharmacovigilance system and does not allow feedback. Bayesian methods have been developed for signal generation in pharmacovigilance, although the full potential of these methods has not been realised. For instance, Bayesian hierarchical models will allow the incorporation of established medical and epidemiological knowledge into the priors for each drug-AE combination. Moreover, the outputs from this analysis can be incorporated into decision-making tools to help in signal validation and posterior actions to be taken by the regulators and companies. We discuss in this paper the apparent advantage of the Bayesian methods used in safety signal generation and the similarities and differences between the two widely used Bayesian methods. We will also propose the use of Bayesian hierarchical models to address the three key challenges and discuss the reasons why Bayesian methodology still have not been fully utilised in pharmacovigilance activities.     

Lower bounds on expectations of positive L-statistics from without-replacement models

We consider samples drawn without replacement from finite populations. We establish optimal lower non-negative and upper non-positive bounds on the expectations of linear combinations of order statistics centered about the population mean in units generated by the population central absolute moments of various orders. We also specify the general results for important examples of sample extremes, Gini mean differences and sample range. The paper completes the results of Papadatos and Rychlik [2004. Bounds on expectations of L-statistics from without replacement samples. J. Statist. Plann. Inference 124, 317–336], where sharp negative lower and positive upper bounds on the expectations of the combinations were presented for the without-replacement samples.     

Extremely efficient acceptance-rejection method for simulating uncorrelated Nakagami fading channels

Multipath fading is one of the most common distortions in wireless communications. The simulation of a fading channel typically requires drawing samples from a Rayleigh, Rice or Nakagami distribution. The Nakagami-m distribution is particularly important due to its good agreement with empirical channel measurements, as well as its ability to generalize the well-known Rayleigh and Rice distributions. In this paper, a simple and extremely efficient rejection sampling (RS) algorithm for generating independent samples from a Nakagami-m distribution is proposed. This RS approach is based on a novel proposal density composed of three pieces of well-known densities from which samples can be drawn easily and efficiently. The proposed method is valid for any combination of parameters of the Nakagami distribution, without any restriction in the domain and without requiring any adjustment from the final user. Simulations for several parameter combinations show that the proposed approach attains acceptance rates above 90% in all cases, outperforming all the RS techniques currently available in the literature.     

Applying discrete choice models to predict Academy Award winners

Summary.  Every year since 1928, the Academy of Motion Picture Arts and Sciences has recognized outstanding achievement in film with their prestigious Academy Award, or Oscar. Before the winners in various categories are announced, there is intense media and public interest in predicting who will come away from the awards ceremony with an Oscar statuette. There are no end of theories about which nominees are most likely to win, yet despite this there continue to be major surprises when the winners are announced. The paper frames the question of predicting the four major awards—picture, director, actor in a leading role and actress in a leading role—as a discrete choice problem. It is then possible to predict the winners in these four categories with a reasonable degree of success. The analysis also reveals which past results might be considered truly surprising—nominees with low estimated probability of winning who have overcome nominees who were strongly favoured to win.