Negative Binomial-Crack (Nb-Cr) Distribution

Abstract

The objective of this paper is to provide an alternative distribution for modeling overdispersed count data. We propose a negative binomial-Crack (NB-CR) distribution which is obtained by mixing the NB distribution with a CR distribution. This new formulation distribution contains as special cases three parameter distribution, namely, negative binomial-inverse Gaussian (NB-IG), negative binomial-Birnbaum-Saunders (NB-BS) and negative binomial-length biased inverse Gaussian (NB-LBIG). In addition, we present some properties of the new distribution such as the factorial moments, the first four moments, variance, skewness and kurtosis. Parameters estimation are also implemented using maximum likelihood method and the application of NB-CR distribution is carried out on a sample of count data. The results show that the NB-CR provides a better fit compared to the Poisson and the NB distribution.

Investigation of the Factors Influencing the Crash Frequency in Expressway Tunnels: Considering Excess Zero Observations and Unobserved Heterogeneity

“…that make the mixture distributions more closely match the dataset of excess zero observations. The results of studies using this type of approach, with models mainly including the negative binomial Lindley model (NB-L) [49], [50], negative binomial-crack model (NB-CR) [51], negative binomial-generalized index model (NB-GE) [52], Poisson generalized Gaussian [53], Poisson weighted exponential [54], [55], Poisson inverse Gaussian (PIG) [56], and negative binomial with Dirichlet process [57], have verified that the introduction of the new distributions not only explains the logic of crash occurrence but also achieves a better goodness of fit compared to the traditional crash model. For example, Geedipally et al [49] and Lord and Geedipally [50] and verified that the NB-L model has better goodness of fit than the NB model and the ZINB model.…”

Applying a Correlated Random Parameters Negative Binomial Lindley Model to Examine Crash Frequency Along Highway Tunnels in China

“…Modeling efforts to deal with crash data with excess zeros have continued with the introduction of new distributions that are capable of handling observations with small counts and combining them with the parent distributions capturing the crash data generating process (NB distributions). These kinds of models mainly include the negative binomial Lindley (NB-L) model [40], [41], the negative binomial crack (NB-CR) model [42], the negative binomial generalized index (NB-GE) model [43], etc. Moreover, these studies also verify that the improvements of negative binomial distributions are more appropriate for actual crash frequency distributions.…”

Exploring the application of the Negative Binomial–Generalized Exponential model for analyzing traffic crash data with excess zeros

“…To overcome this problem, researchers have proposed different tools for analyzing datasets with a large number of zeros and long tails. They include the zero-inflated (ZI) models (Shankar et al, 1997;Shankar et al, 2003), the Negative Binomial-Lindley (NB-L) model (Geedipally et al, 2012;Hallmark et al, 2013;Xu and Sun, 2015), the Poisson-weighted exponential model (Zamani et al, 2014), the Poisson Inverse Gaussian (PIG) (Zha et al, 2015), the Negative Binomial-Crack (NB-CR) distribution (Saengthong and Bodhisuwan, 2013), and the Sichel (SI) model (Zou et al, 2013;2015). Lord et al (2005Lord et al ( , 2007 and Lord and Geedipally (2011; provide discussions about the advantages and limitations of these distributions and models.…”