arXiv subjects

Kevin Dayaratna

Publications and source records attributed to Kevin Dayaratna.

2 recordsLinked to original sources

arXiv2026

Applications of Improvements to the Pythagorean Won-Loss Expectation in Optimizing Rosters

Bill James' Pythagorean formula has for decades done an excellent job estimating a baseball team's winning percentage from very little data: if the average runs scored and allowed are denoted respectively by ${\rm RS}$ and ${\rm RA}$, there is some $γ\approx 2$ such that the winning percentage is approximately ${\rm RS}^γ/ ({\rm RS}^γ+ {\rm RA}^γ)$. One use case is to determine the value of potential signings to the team, as it allows us to estimate how many more wins one obtains over a season given an estimated change in run production and concession. We summarize earlier work on the subject, and extend the earlier theoretical model of Miller (who assumed the home and away teams' runs arise from independent Weibull distributions with the same shape parameter $γ$; this has been observed to describe the observed run data well and yields a win probability equivalent to that of James' formula). We extend this work to model runs scored and allowed as being drawn from independent Weibull distributions with different shape parameters, and then consider the first and second moments to solve a system of four equations in the four unknowns. Doing so fits the training data better, yielding a higher winning percentage over the last 30 MLB seasons (1994 to 2023). This comes at a small cost as we no longer have a closed form expression for the win probability, but must evaluate a two-dimensional integral of Weibull distributions and numerically estimate the solutions to the system of equations. These are trivial to do with simple computational programs.

stat.AP↗

arXiv2006

Closed-Form Bayesian Inferences for the Logit Model via Polynomial Expansions

Articles in Marketing and choice literatures have demonstrated the need for incorporating person-level heterogeneity into behavioral models (e.g., logit models for multiple binary outcomes as studied here). However, the logit likelihood extended with a population distribution of heterogeneity doesn't yield closed-form inferences, and therefore numerical integration techniques are relied upon (e.g., MCMC methods). We present here an alternative, closed-form Bayesian inferences for the logit model, which we obtain by approximating the logit likelihood via a polynomial expansion, and then positing a distribution of heterogeneity from a flexible family that is now conjugate and integrable. For problems where the response coefficients are independent, choosing the Gamma distribution leads to rapidly convergent closed-form expansions; if there are correlations among the coefficients one can still obtain rapidly convergent closed-form expansions by positing a distribution of heterogeneity from a Multivariate Gamma distribution. The solution then comes from the moment generating function of the Multivariate Gamma distribution or in general from the multivariate heterogeneity distribution assumed. Closed-form Bayesian inferences, derivatives (useful for elasticity calculations), population distribution parameter estimates (useful for summarization) and starting values (useful for complicated algorithms) are hence directly available. Two simulation studies demonstrate the efficacy of our approach.

math.ST↗