On generalized geometric distributions and improved estimation of batting average in cricket

Batting average is the most popular way of measuring a batsman's performance in cricket. However, in light of scores from not-out innings, the conventional way of computing the batting average is unsatisfactory from theoretical statistical perspective, as well as from intuitive and practitioner's point of view. We adopt alternative methods of calculating batting average, treating not-outs as right-censored data and using generalized class of geometric distributions (GGD) as models for the runs scored. In the proposed family of GGD, the generalization lies in the hazard of getting out possibly changing from one score to another. Each postulated structure of the hazards leads to a different member of the GGD family. Selection of appropriate member from the GGD family and maximum likelihood estimation of the hazard parameters in the model are discussed theoretically with illustrations. The proposed method subsumes the traditional average and product limit (Kaplan-Meier) estimate as the two extreme scenarios within this structure. We also discuss two alternative methods of estimating the true mean under the proposed framework and deliberate on issues while adopting these practices in practice.     

Closed‐form likelihoods for stochastic differential equation growth models

The authors derive closed‐form expressions for the full, profile, conditional and modified profile likelihood functions for a class of random growth parameter models they develop as well as Garcia's additive model. These expressions facilitate the determination of parameter estimates for both types of models. The profile, conditional and modified profile likelihood functions are maximized over few parameters to yield a complete set of parameter estimates. In the development of their random growth parameter models the authors specify the drift and diffusion coefficients of the growth parameter process in a natural way which gives interpretive meaning to these coefficients while yielding highly tractable models. They fit several of their random growth parameter models and Garcia's additive model to stock market data, and discuss the results. The Canadian Journal of Statistics 38: 474–487; 2010 © 2010 Statistical Society of Canada     

Do Baseball Players Regress toward the Mean?

Baseball performances are an imperfect measure of baseball abilities, and consequently exaggerate differences in abilities. Predictions of relative batting averages and earned run averages can be improved substantially by using correlation coefficients estimated from earlier seasons to shrink performances toward the mean.     

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.     

Assessing the market values of soccer players – a robust analysis of data from German 1. and 2. Bundesliga

The paper examines to what extent a player's market value depends on his skills. Therefore, a data set covering 28 performance measures and the market values of about 493 players from 1. and 2. German Bundesliga is analysed. Applying robust analysis techniques, we are able to robustly estimate market values of soccer players. The results show (1) that there are significantly underrated and overrated players and (2) that a player's affiliation to a certain team may contribute to his market value. We conclude that a club's reputation affects the market values of its players and that star players are in tendency overrated.     

Asymptotically Optimal Nonparametric Empirical Bayes Via Predictive Recursion

An empirical Bayes problem has an unknown prior to be estimated from data. The predictive recursion (PR) algorithm provides fast nonparametric estimation of mixing distributions and is ideally suited for empirical Bayes applications. This article presents a general notion of empirical Bayes asymptotic optimality, and it is shown that PR-based procedures satisfy this property under certain conditions. As an application, the problem of in-season prediction of baseball batting averages is considered. There the PR-based empirical Bayes rule performs well in terms of prediction error and ability to capture the distribution of the latent features.     

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.     

An Application of Response Surface Methodology to the Atari® Miniature Golf Video Game

Response surface methodology is used to optimize a player's performance on the Atari® Miniature Golf computer video game. Because of the nature of the fitted response surface, a direct optimization procedure is used to estimate the optimum operating conditions.     

Results concerning the generalized partially linear single-index model

The central topic of this article is the estimation of parameters of the generalized partially linear single-index model (GPLSIM). Two numerical optimization procedures are presented and an S-plus program based on these procedures is compared to a program by Wand in a simulation setting. The results from these simulations indicate that the estimates for the new procedures are as good, if not better, than Wand's. Also, this program is much more flexible than Wand's since it can handle more general models. Other simulations are also conducted. The first compares the effects of using linear interpolation versus spline interpolation in an optimization procedure. The results indicate that by using spline interpolation one gets more stable estimates at a cost of increased computational time. A second simulation was conducted to assess the performance of a method for estimating the variance of alpha. A third set of simulations is carried out to determine the best criterion for testing that one of the elements of alpha is equal to zero. The GPLSIM is applied to a water quality data set and the results indicate an interesting relationship between gastrointestinal illness and turbidity (cloudiness) of drinking water.     

Likelihood estimation for exchangeable multinomial data

In this article, maximum likelihood estimates of an exchangeable multinomial distribution using a parametric form to model the parameters as functions of covariates are derived. The non linearity of the exchangeable multinomial distribution and the parametric model make direct application of Newton Rahpson and Fisher's scoring algorithms computationally infeasible. Instead parameter estimates are obtained as solutions to an iterative weighted least-squares algorithm. A completely monotonic parametric form is proposed for defining the marginal probabilities that results in a valid probability model.     

Efficiency of schemes of sampling from the inverse gaussian-weibull mixture model

The objective of this paper is to study the efficiency of sampling schemes suggested by Hosmer(1973), termed models Ml and M2, relative to the regular random sampling, termed model MO, when samples are drawn from a population having the Inverse Gaussian-Weibull (IG-W) mixture distribution.

It has been shown that whether the efficiency is based on relative variances of the maximum likelihood estimates (ML,E's) of the components of the vector of parameters or on the generalized variances of the MLE's of that vector, Hosmer's models Ml or M2 perform better than model MO.     

Combining non-linear regressions that have unequal error variances and some parameters in common

Methods of estimation and inference are presented for the situation where two non-linear regression models with unequal error variances contain some parameters in common. Such a situation arises in structural chemistry, when bond lengths are available for three nearly collinear atoms in crystals and a model is required to quantify the extent and form of the relationship between the longer and the shorter bond. Some atomic triples are symmetric and require a different model and error variance from those required by the asymmetric triples. The profile likelihood for the regression parameters is a weighted sum of the logarithms of the sums-of-squares functions from each model, and the estimates can be obtained by using a simple modification to a standard non-linear least squares program. A likelihood ratio test for assessing whether the parameters in common are equal is described. When these techniques are applied to two data sets consisting of bond lengths for bromine–tellurium–bromine and sulphur–tellurium–sulphur triples, there is no evidence against the equality hypothesis. An extension to the model to allow for a non-constant variance is required for proper analysis of the sulphur–tellurium–sulphur data.     

Logistic regression analysis of non‐randomized response data collected by the parallel model in sensitive surveys

To study the relationship between a sensitive binary response variable and a set of non‐sensitive covariates, this paper develops a hidden logistic regression to analyse non‐randomized response data collected via the parallel model originally proposed by Tian (2014). This is the first paper to employ the logistic regression analysis in the field of non‐randomized response techniques. Both the Newton–Raphson algorithm and a monotone quadratic lower bound algorithm are developed to derive the maximum likelihood estimates of the parameters of interest. In particular, the proposed logistic parallel model can be used to study the association between a sensitive binary variable and another non‐sensitive binary variable via the measure of odds ratio. Simulations are performed and a study on people's sexual practice data in the United States is used to illustrate the proposed methods.     

Approximate bayes estimation for mixtures of two weibull distributions under type-2 censoring

The main object of this paper is the approximate Bayes estimation of the five dimensional vector of parameters and the reliability function of a mixture of two Weibull distributions under Type-2 censoring. Under Type-2 censoring, the posterior distribution is complicated, and the integrals involved cannot be obtained in a simple closed form. In this work, Lindley's (1980) approximate form of Bayes estimation is used in the case of a mixture of two Weibull distributions under Type-2 censoring. Through Monte Carlo simulation, the root mean squared errors (RMSE's) of the Bayes estimates are computed and compared with the corresponding estimated RMSE's of the maximum likelihood estimates.     

Bayesian inference for the randomly censored Weibull distribution

In this paper, we consider the Bayesian inference of the unknown parameters of the randomly censored Weibull distribution. A joint conjugate prior on the model parameters does not exist; we assume that the parameters have independent gamma priors. Since closed-form expressions for the Bayes estimators cannot be obtained, we use Lindley's approximation, importance sampling and Gibbs sampling techniques to obtain the approximate Bayes estimates and the corresponding credible intervals. A simulation study is performed to observe the behaviour of the proposed estimators. A real data analysis is presented for illustrative purposes.     

Bayesian inference for the randomly censored Burr-type XII distribution

The article presents the Bayesian inference for the parameters of randomly censored Burr-type XII distribution with proportional hazards. The joint conjugate prior of the proposed model parameters does not exist; we consider two different systems of priors for Bayesian estimation. The explicit forms of the Bayes estimators are not possible; we use Lindley's method to obtain the Bayes estimates. However, it is not possible to obtain the Bayesian credible intervals with Lindley's method; we suggest the Gibbs sampling procedure for this purpose. Numerical experiments are performed to check the properties of the different estimators. The proposed methodology is applied to a real-life data for illustrative purposes. The Bayes estimators are compared with the Maximum likelihood estimators via numerical experiments and real data analysis. The model is validated using posterior predictive simulation in order to ascertain its appropriateness.     

On progressively censored generalized inverted exponential distribution

A generalized version of inverted exponential distribution (IED) is considered in this paper. This lifetime distribution is capable of modeling various shapes of failure rates, and hence various shapes of aging criteria. The model can be considered as another useful two-parameter generalization of the IED. Maximum likelihood and Bayes estimates for two parameters of the generalized inverted exponential distribution (GIED) are obtained on the basis of a progressively type-II censored sample. We also showed the existence, uniqueness and finiteness of the maximum likelihood estimates of the parameters of GIED based on progressively type-II censored data. Bayesian estimates are obtained using squared error loss function. These Bayesian estimates are evaluated by applying the Lindley's approximation method and via importance sampling technique. The importance sampling technique is used to compute the Bayes estimates and the associated credible intervals. We further consider the Bayes prediction problem based on the observed samples, and provide the appropriate predictive intervals. Monte Carlo simulations are performed to compare the performances of the proposed methods and a data set has been analyzed for illustrative purposes.     

Accelerated life tests under competing weibull causes of failure

Accelerated life testing of a product under more severe than normal conditions is cawiionly used to reduce test time and cost. Data collected at such accelerated conditions is used to obtain estimates of parameters of a stress translation function which is then used to make inference about the product's, per" formance under normal conditions. This problem is considered when the product is a p component series system with WeibuH distributed component lifetimes liaving a caimon shape parameter. A general stress translation function is used and estimates of model parameters are obtained for various censoring schemes.     

Interval estimates for diatom-inferred lake pH histories

We investigate ordinary least-squares and Bayesian methods for constructing interval estimates for historical lake pH's inferred from diatom sediments. The Bayesian method explicitly models several forms of variability, including the sampling and classification variability of the diatom records, estimation variability, and measurement error in observed pH's. The two methods produce similar interval estimates, but the Bayesian model allows design recommendations to be made.     

Effects of a single outlier on arma identification

Fox (1972), Box and Tiao (1975), and Abraham and Box (1979) have proposed methods for detecting outliers in time series whose ARMA form is known (or identified). We show that the existence of a single aberrant observation, innovation, or intervention causes an ARMA model to be misidentified using unadjusted autocorrelation (acf) and partial autocorrelation estimates. The magnitude, location, type of outlier, and in some cases the ARMA's parameters, affect the identification outcome. We use variance inflation, signal-to-noise ratios, and acf critical values to determine an ARMA model's susceptibility to misidentifi-cation. Numerical and simulation examples suggest how to iteratively use the outlier detection methods in practice.