Estimating the Dispersion Parameter of the Negative Binomial Distribution for Analyzing Crash Data Using a Bootstrapped Maximum Likelihood Method

The objective of this study is to improve the estimation of the dispersion parameter of the negative binomial distribution for modeling motor vehicle collisions. The negative binomial distribution is widely used to model count data such as traffic crash data, which often exhibit low sample mean values and small sample sizes. Under such situations, the most commonly used methods for estimating the dispersion parameter, the method of moment and the maximum likelihood estimate, may become inaccurate and unstable. A bootstrapped maximum likelihood estimate is proposed to improve the estimation of the dispersion parameter. The proposed method combines the technique of bootstrap resampling with the maximum likelihood estimation method to obtain better estimates of the dispersion parameter. The performance of the bootstrapped maximum likelihood estimate is compared with the method of moment and the maximum likelihood estimates through Monte Carlo simulations. To validate the simulation results, the methods are applied to observed data collected at 4-legged unsignalized intersections in Toronto, Ont. Overall, the results show that the proposed bootstrap maximum likelihood method produces smaller biases and more stable estimates. The improvements are more pronounced with small samples and low sample means.