Shubhabratha Das

Inconsistency in Response When Questions are Repeated with the Number of Options Changing

Discarding haphazard and insincere respondents can improve the quality of data resulting in more efficient survey analysis. This may be achieved by repeating a question with varied scale and then checking the consistency of relevant responses. A formal measure of inconsistency of a respondent is formulated in this work on the basis of his/her response to the same question repeated in multiple scales. The measure can be alternatively viewed as a measure of fuzziness attributed to respondent or attribute depending on its formulation. The probability distribution of this measure is obtained if the respondent marks completely at random. Undesirable respondents may be screened using the above mentioned probability distribution in the framework of statistical testing of hypothesis. The application extends to identifying fuzzy or unclear attributes along the similar lines. The paper also proposes a model based approach as well as another heuristic that can deal with screening inconsistent respondents and fuzzy attributes simultaneously.

Rank Consistent Bradley-Terry Models for Repeated Tournaments

The primary objective of this paper is to model the win-loss records of matches in a repeated tournament using the ranks of the teams. The work proposes modifications of Bradley-Terry (BT) model to make the estimation consistent with the ranks of the participating teams. The BT model with restricted maximum likelihood estimation involves too many parameters and the estimates typically lack strict monotonicity. A proposed class of rank-percentile BT models based on different parametric distribution resolves both the issues. Parameter estimation, goodness-of-fit using suitably framed test statistic and its null distribution, change point analysis in a nested model framework, as well as other estimation aspects are discussed in this article. Adaptive variations of the model that allow estimates to alter are also discussed. For demonstration, National Collegiate Athletic Association (NCAA) men and women basketball tournament data are considered. The discussed models provide excellent fit to the historical data using only a few parameters. The fit validates the ranking procedure implemented by the NCAA. The models can be extended in more general tournament structures, as shown through an analysis of results from the Indian Premiere League. The work done has potential for application in the wider domain of paired comparisons.

On Generalized Geometric Distributions: Application to Modeling Scores in Cricket and Improved Estimation of Batting Average in Light of Notout Innings

In the game of cricket, batting average is the most common and basic measure of a batsman’s performance during a short duration, like a series or calendar year, as well over a longer span like the career. Batting average is considered in isolation or in combination with other measures like strike rate, at times depending on the form of the game. However, in either case, treatment of runs scores from notout innings throws particular challenge in adopting batting average as a measure of true performance. The conventional way of computing batting average enjoys favour as well as criticism from intuitive standpoint — but it can be justified as the maximum likelihood estimate if the scores come from an Exponential or Geometric distribution. Either of these distributions is quite unreasonable in modeling cricket scores of a batsman because of obviously non-constant hazard or propensity to get out after scoring different runs. Towards this, we discuss the role of the Kaplan Meir estimator treating the scores from the notout innings as right censored data. We show that while it provides a vast conceptual improvement over the traditional average, there are some associated some problems as well. The first of these is because of its nonparametric nature, specially in the context of reflecting true average performance in a short duration like a tournament or a series — the other because of its inability to produce a finite-valued estimate when the largest score is from a notout innings. To address these concerns, we propose a generalized class of Geometric distributions (GGD) as model for the runs scored by individual batsmen. The generalization comes in the form of hazard of getting out changing from one score to another. We consider the change points as the known or specified parameters and derive the general expressions for the restricted maximum likelihood estimators of the hazard rates under the generalized structure considered. Given the domain context, we propose and test ten different variations of the GGD model and carry out the test across the nested models using the asymptotic distribution of the likelihood ratio statistic to determine the best possible model. This family of GGD subsumes the traditional average as well as the Kaplan-Meir based estimate, as the parameter GGD is the simple Geometric distribution, while the infinite order GGD corresponds to the non-parametric Kaplan-Meir based survival function. Finally to estimate the true batting average, we propose two methods: first being the simple mean of the fitted GGD and in the second case the notout scores are replaced by conditional mean of the fitted GGD, before averaging out. We show that while the two methods coincide for the two extreme GGD (simple Geometric and nonparametric) it is not so in general. We also discuss how different approaches for estimating average over a short or long time horizon. Finally we compute batting averages by the different methods for all top players, in both forms of the game and study the rank correlation. We also present results from numerical computation is carried out using scores of all opening batsmen as well as No 11 batsmen in one day cricket matches, to illustrate model selection procedures. This also establishes that any model in the family need not be appropriate for all situation. We also focus on Batting average of two players. In particular, we show that quite possibly Bradman’s true average was greater than 100, while Bevan have been distinctly beneficiary of prevalent way of computing average as his 1-day average seems to be an overestimate by fair degree.

0-1 Knapsack Problems with Random Budgets

Given a set of elements, each having a profit and cost associated with it, and a budget, the 0-1 knapsack problem finds a subset of the elements with maximum possible combined profit subject to the combined cost not exceeding the budget. In this paper we study a stochastic version of the problem in which the budget is random. We propose two different formulations of this problem, based on different ways of handling infeasibilities, and propose exact and heuristic algorithms to solve the problems represented by these formulations. We also present the results from some computational experiments.

Solving Discrete Optimization Problems When Element Costs are Random

In a general class of discrete optimization problems with min-sum objective function, some of the elements may have random costs associated with them. In such a situation, the notion of optimality needs to be suitably modified. We define an optimal solution to be a feasible solution with the minimum risk. It is shown that the knowledge of the means of these random costs is enough to reduce such a problem into one with no random costs.