Adjustment for Maximum Likelihood Estimate of Negative Binomial Dispersion Parameter

2008 ◽  
Vol 2061 (1) ◽  
pp. 9-19 ◽  
Author(s):  
Byung-Jung Park ◽  
Dominique Lord
Keyword(s):  
Maximum Likelihood ◽  
Sample Size ◽  
Maximum Likelihood Estimate ◽  
Negative Binomial ◽  
Highway Safety ◽  
Likelihood Estimate ◽  
Dispersion Parameter ◽  
Dispersion Parameters ◽  
Sample Mean ◽  
Crash Data

The negative binomial (NB) (or Poisson–gamma) model has been used extensively by highway safety analysts because it can accommodate the overdispersion often exhibited in crash data. However, it has been reported in the literature that the maximum likelihood estimate of the dispersion parameter of NB models can be significantly affected when the data are characterized by small sample size and low sample mean. Given the important roles of the dispersion parameter in various types of highway safety analyses, there is a need to determine whether the bias could be potentially corrected or minimized. The objectives of this study are to explore whether a systematic relationship exists between the estimated and true dispersion parameters, determine the bias as a function of the sample size and sample mean, and develop a procedure for correcting the bias caused by these two conditions. For this purpose, simulated data were used to derive the relationship under the various combinations of sample mean, dispersion parameter, and sample size, which encompass all simulation conditions performed in previous research. The dispersion parameter was estimated by using the maximum likelihood method. The results confirmed previous studies and developed a reasonable relationship between the estimated and true dispersion parameters for reducing the bias. Details for the application of the correction procedure were also provided by using the crash data collected at 458 three-leg unsignalized intersections in California. Finally, the study provided several discussion points for further work.

Recurrence formula and the maximum likelihood estimation of the age in a simple branching process

1982 ◽  
Vol 19 (4) ◽  
pp. 776-784 ◽  
Author(s):  
M. Adès ◽  
J.-P. Dion ◽  
G. Labelle ◽  
K. Nanthi
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Maximum Likelihood Estimate ◽  
Branching Process ◽  
Negative Binomial ◽  
Recurrence Formula ◽  
Likelihood Estimation ◽  
Likelihood Estimate ◽  
Offspring Distribution ◽  
Watson Process

In this paper, we consider a Bienaymé– Galton–Watson process {Xn; n ≧ 0; Xn = 1} and develop a recurrence formula for P(Xn = k), k = 1, 2, ···. The problem of obtaining the maximum likelihood estimate of the age of the process when p0 = 0 is discussed. Furthermore the maximum likelihood estimate of the age of the process when the offspring distribution is negative binomial (p0 ≠ 0) is obtained, and a comparison with Stigler's estimator (1970) of the age of the process is made.

Recurrence formula and the maximum likelihood estimation of the age in a simple branching process

1982 ◽  
Vol 19 (04) ◽  
pp. 776-784 ◽  
Author(s):  
M. Adès ◽  
J.-P. Dion ◽  
G. Labelle ◽  
K. Nanthi
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Maximum Likelihood Estimate ◽  
Branching Process ◽  
Negative Binomial ◽  
Recurrence Formula ◽  
Likelihood Estimation ◽  
Likelihood Estimate ◽  
Offspring Distribution ◽  
Watson Process

In this paper, we consider a Bienaymé– Galton–Watson process {Xn ; n ≧ 0; Xn = 1} and develop a recurrence formula for P(Xn = k), k = 1, 2, ···. The problem of obtaining the maximum likelihood estimate of the age of the process when p 0 = 0 is discussed. Furthermore the maximum likelihood estimate of the age of the process when the offspring distribution is negative binomial (p 0 ≠ 0) is obtained, and a comparison with Stigler's estimator (1970) of the age of the process is made.

Effects of Varying Dispersion Parameter of Poisson–Gamma Models on Estimation of Confidence Intervals of Crash Prediction Models

2008 ◽  
Vol 2061 (1) ◽  
pp. 46-54 ◽  
Author(s):  
Srinivas Reddy Geedipally ◽  
Dominique Lord
Keyword(s):  
Confidence Intervals ◽  
Negative Binomial ◽  
Prediction Models ◽  
Motor Vehicle ◽  
Dispersion Parameter ◽  
Transportation Safety ◽  
Dispersion Parameters ◽  
Crash Data ◽  
Varying Dispersion ◽  
Using Data

In estimating safety performance, the most common probabilistic structures of the popular statistical models used by transportation safety analysts for modeling motor vehicle crashes are the traditional Poisson and Poisson–gamma (or negative binomial) distributions. Because crash data often exhibit overdispersion, Poisson–gamma models are usually the preferred model. The dispersion parameter of Poisson–gamma models had been assumed to be fixed, but recent research in highway safety has shown that the parameter can potentially be dependent on the covari-ates, especially for flow-only models. Given that the dispersion parameter is a key variable for computing confidence intervals, there is reason to believe that a varying dispersion parameter could affect the computation of confidence intervals compared with confidence intervals produced from Poisson–gamma models with a fixed dispersion parameter. This study evaluates whether the varying dispersion parameter affects the computation of the confidence intervals for the gamma mean (m) and predicted response (y) on sites that have not been used for estimating the predictive model. To accomplish that objective, predictive models with fixed and varying dispersion parameters were estimated by using data collected in California at 537 three-leg rural unsignalized intersections. The study shows that models developed with a varying dispersion parameter greatly influence the confidence intervals of the gamma mean and predictive response. More specifically, models with a varying dispersion parameter usually produce smaller confidence intervals, and hence more precise estimates, than models with a fixed dispersion parameter, both for the gamma mean and for the predicted response. Therefore, it is recommended to develop models with a varying dispersion whenever possible, especially if they are used for screening purposes.

Box-Cox Transformations

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Maximum Likelihood ◽  
Normal Form ◽  
Maximum Likelihood Estimate ◽  
Likelihood Estimate ◽  
Grouped Data ◽  
Numerical Example ◽  
Alternative Method ◽  
Modified Likelihood

This chapter examines the well-known Box-Cox method, which transforms a sample of non-normal observations into approximately normal form. Two non-standard aspects are highlighted. First, the likelihood of the transformed sample has an unbounded maximum, so that the maximum likelihood estimate is not consistent. The usually suggested remedy is to assume grouped data so that the sample becomes multinomial. An alternative method is described that uses a modified likelihood similar to the spacings function. This eliminates the infinite likelihood problem. The second problem is that the power transform used in the Box-Cox method is left-bounded so that the transformed observations cannot be exactly normal. This biases estimates of observational probabilities in an uncertain way. Moreover, the distributions fitted to the observations are not necessarily unimodal. A simple remedy is to assume the transformed observations have a left-bounded distribution, like the exponential; this is discussed in detail, and a numerical example given.

scholarly journals GLM-Based Flexible Monitoring Methods: An Application to Real-Time Highway Safety Surveillance

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 362
Author(s):  
Arshad Jamal ◽  
Tahir Mahmood ◽  
Muhamad Riaz ◽  
Hassan M. Al-Ahmadi
Keyword(s):  
Real Time ◽  
Control Charts ◽  
Negative Binomial ◽  
Highway Safety ◽  
Monitoring Methods ◽  
Count Data Models ◽  
Crash Data ◽  
Safety Surveillance ◽  
Study Results ◽  
Best Fit

Statistical modeling of historical crash data can provide essential insights to safety managers for proactive highway safety management. While numerous studies have contributed to the advancement from the statistical methodological front, minimal research efforts have been dedicated to real-time monitoring of highway safety situations. This study advocates the use of statistical monitoring methods for real-time highway safety surveillance using three years of crash data for rural highways in Saudi Arabia. First, three well-known count data models (Poisson, negative binomial, and Conway–Maxwell–Poisson) are applied to identify the best fit model for the number of crashes. Conway–Maxwell–Poisson was identified as the best fit model, which was used to find the significant explanatory variables for the number of crashes. The results revealed that the road type and road surface conditions significantly contribute to the number of crashes. From the perspective of real-time highway safety monitoring, generalized linear model (GLM)-based exponentially weighted moving average (EWMA) and cumulative sum (CUSUM) control charts are proposed using the randomized quantile residuals and deviance residuals of Conway–Maxwell (COM)–Poisson regression. A detailed simulation-based study is designed for predictive performance evaluation of the proposed control charts with existing counterparts (i.e., Shewhart charts) in terms of the run-length properties. The study results showed that the EWMA type control charts have better detection ability compared with the CUSUM type and Shewhart control charts under small and/or moderate shift sizes. Finally, the proposed monitoring methods are successfully implemented on actual traffic crash data to highlight the efficacy of the proposed methods. The outcome of this study could provide the analysts with insights to plan sound policy recommendations for achieving desired safety goals.

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