Estimating dispersion parameter of negative binomial distribution for analysis of crash data: Bootstrapped maximum likelihood method

Transportation Research Record

Volumn , Issue 2019, 2007, Pages 15-21

  Zhang, Yunlong ^a   Ye, Zhirui ^a   Lord, Dominique ^a  

^a Department of Electrical and Computer Engineering   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CRASHWORTHINESS; HIGHWAY ACCIDENTS; MAXIMUM LIKELIHOOD; PARAMETER ESTIMATION;

BINOMIAL DISTRIBUTION; DISPERSION PARAMETER;

DATA REDUCTION;

EID: 41549144969     PISSN: 03611981     EISSN: None     Source Type: Journal    
DOI: 10.3141/2019-03     Document Type: Article

References (44)

1. Nicholson, A., and Y. Wong, Are Accidents Poisson Distributed? A Statistical Test. Accident Analysis & Prevention, Vol. 25, No. 1, 1993, pp. 91-97.
2. Bliss, C. I., and R. A. Fisher. Fitting the Negative Binomial Distribution to Biological Data and Note on the Efficient Fitting of the Negative Binomial. Biometrics, Vol. 9, 1953, pp. 176-200.
3. Poch, M., and F. L. Mannering. Negative Binomial Analysis of Intersection Accident Frequency. Journal of Transportation Engineering, ASCE, Vol. 122, No. 2, 1996, pp. 105-113.
4. Lord, D. Modeling Motor Vehicle Crashes Using Poisson-Gamma Models: Examining the Effects of Low Sample Mean Values and Small Sample Size on the Estimation of the Fixed Dispersion Parameter. Accident Analysis & Prevention, Vol. 38, No. 4, 2006, pp. 751-766.
5. Hauer, E. Observational Before-After Studies in Road Safety: Estimating the Effect of Highway and Traffic Engineering Measures on Road Safety. Pergamon Press, Elsevier Science Ltd., Oxford, England, 1997.
6. Washington, S. P., M. G. Karlaftis, and F. L. Mannering. Statistical and Econometric Methods for Transportation Data Analysis. Chapman & Hall, CRC, London, 2003.
7. Taylor, L. R. Aggregation, Variance, and the Mean. Nature, Vol. 189, 1961, pp. 732-735.
8. Bliss, C. I., and A. R. G. Owen. Negative Binomial Distributions with a Common k. Biometrika, Vol. 45, No. 1/2, 1958, pp. 37-58.
9. Fisher, R. A. The Negative Binomial Distribution. Annals of Eugenics, Vol. 11,1941, pp. 182-187.
10. Neyman, J. On a New Class of 'Contagious' Distributions, Applicable in Entomology and Bacteriology. Annals of Mathematical Statistics, Vol. 10, 1939, pp. 35-57.
11. Willson, L. J., J. L. Folks, and J. H. Young. Multistage Compared with Fixed-Sample-Size Estimation of the Negative Binomial Parameter k. Biometrics, Vol. 40, 1984, pp. 109-117.
12. Bliss, C. I., and A. R. G. Owen. Negative Binomial Distributions with a Common k. Biometrika, Vol. 45, No. 1/2, 1958, pp. 37-58.
13. Anscombe, F. J. Sampling Theory of the Negative Binomial and Logarithmic Series Distributions. Biometrika, Vol. 37, 1950, pp. 358-382.
14. Wood, G. R. Confidence and Prediction Intervals for Generalized Linear Accident Models. Accident Analysis & Prevention, Vol. 37, No. 2, 2005, pp. 267-273.
15. Hauer, E. Overdispersion in Modelling Accidents on Road Sections and in Empirical Bayes Estimation. Accident Analysis & Prevention, Vol. 33, No. 6, 2001, pp. 799-808.
16. Persaud, B., and C. Lyon. Empirical Bayes Before-After Safety Studies: Lessons Learned from Two Decades of Experience and Future Directions. Accident Analysis & Prevention, Vol. 39, No. 3, May 2007, pp. 546-555.
17. Anscombe, F. J. The Statistical Analysis of Insect Counts Based on the Negative Binomial Distributions. Biometrics, Vol. 5, 1949, pp. 165-173.
18. Lawless, J. F. Negative Binomial and Mixed Poisson Regression. Canadian Journal of Statistics, Vol. 15, 1987, pp. 209-225.
19. Clark, S. J., and J.N. Perry. Estimation of the Negative Binomial Parameter it by Maximum Quasi-Likelihood. Biometrics, Vol. 45, 1989, pp. 309-316.
20. Maycock, G., and R. D. Hall. Accidents at 4-Arm Roundabouts. TRRL Laboratory Report 1129. U.K. Transport and Road Research Laboratory, Crowthorne, Berkshire, 1984.
21. Fridstrom, L. J. I., S. Ingebrigtsen, S. R. Kulmala, and L. K. Thomsen. Measuring the Contribution of Randomness, Exposure, Weather, and Daylight to the Variation in Road Accident Counts. Accident Analysis & Prevention, Vol. 27, No. 1, 1995, pp. 1-20.
22. Maher, M. J., and I. Summersgill. A Comprehensive Methodology for the Fitting Predictive Accident Models. Accident Analysis & Prevention, Vol. 28, No. 3, 1996, pp. 281-296.
23. Kumara, S. S. P., H. C. Chin, and W. M. S. B. Weerakoon. Identification of Accident Causal Factors and Prediction of Hazardousness of Intersection Approaches. In Transportation Research Record: Journal of the Transportation Research Board, No. 1840, Transportation Research Board of the National Academies, Washington, D.C., 2003, pp. 116-122.
24. Oh, J., C. Lyon, S. Washington, B. Persaud, and J. Bared. Validation of FHWA Crash Models for Rural Intersections: Lessons Learned. In Transportation Research Record: Journal of the Transportation Research Board, No. 1840, Transportation Research Board of the National Academies, Washington, D.C., 2003, pp. 41-49.
25. Lord, D., and J. A. Bonneson. Calibration of Predictive Models for Estimating Safety of Ramp Design Configurations. In Transportation Research Record: Journal of the Transportation Research Board, No. 1908, Transportation Research Board of the National Academies, Washington, D.C., 2005, pp. 88-95.
26. Piegorsch, W. W. Maximum Likelihood Estimation for the Negative Binomial Dispersion Parameter. Biometrics, Vol. 46, 1990, pp. 863-867.
27. Toft, N., G. T. Innocent, D. J. Mellor, and S. W. J. Reid. The GammaPoisson Model as a Statistical Method to Determine If Micro-Organisms Are Randomly Distributed in a Food Matrix, Food Microbiology, Vol. 23, No. 1, 2006, pp. 90-94.
28. Diaconis, P., and B. Efron. Computer-Intensive Methods in Statistics. Scientific American, 1983, pp. 116-130.
29. Efron, B. Bootstrap Methods: Another Look at the Jackknife. Annals of Statistics, Vol. 7, 1979, pp. 1-26.
30. Efron, B. Nonparametric Estimates of Standard Error: The Jackknife, the Bootstrap and Other Methods. Biometrika, Vol. 63, 1981, pp. 589-599.
31. Efron, B. The Jackknife, the Bootstrap, and Other Resampling Plans. CBMS-NSF Monograph 38. Society of Industrial and Applied Mathematics, Philadelphia, Pa., 1982.
32. Efron, B., and R. J. Tibshirani. An Introduction to the Bootstrap. Chapman & Hall, New York, 1993.
33. Zoubir, A. M., and D. R. Iskander. Bootstrap Techniques for Signal Processing. Cambridge University Press, New York, 2004.
34. Politis, D. N. Computer-Intensive Methods in Statistical Analysis. IEEE Signal Processing Magazine, Vol. 15, 1998, pp. 39-55.
35. Tamstorf, R., and H. W. Jensen. Adaptive Sampling and Bias Estimation in Path Tracing. In Rendering Techniques '97(J. Dorsey and P. Slusallek, eds.), Springer-Verlag, London, 1997, pp. 285-295.
36. Rosychuk, R., X. Sheng, and J. L. Stuber. Comparison of Variance Estimation Approaches in a Two-State Markov Model for Longitudinal Data with Misclassification. Statistics in Medicine, Vol. 25, No. 11, 2006, pp. 1906-1921.
37. Yeo, D., H. Mantel, and T. P. Liu. Bootstrap Variance Estimation for the National Population Health Survey. Proc., Annual Meeting of the American Statistical Association (Survey Research Methods Section), American Statistical Association, Baltimore, Md., 1999.
38. Zoubir, A. M. Multiple Bootstrap Tests and Their Application, Proc., IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) VI, Adelaide, Australia, April 1994, pp. 69-72.
39. Hall, P., and S. R. Wilson. Two Guidelines for Bootstrap Hypothesis Testing. Biometrics, Vol. 47, 1991, pp. 757-762.
40. Fisher, N. I., and P. Hall. Bootstrap Confidence Regions for Directional Data. Journal of the American Statistical Society, Vol. 84, 1989, pp. 996-1002.
41. Reid, D. C., A. M. Zoubir, and B. Boashash. The Bootstrap Applied to Passive Acoustic Aircraft Parameter Estimation. Proc., ICASSP '96, Atlanta, Ga., 1996, pp. 3154-3157.
42. Lord, D., and L. F. Miranda-Moreno. Effects of Low Sample Mean Values and Small Sample Size on Estimation of Fixed Dispersion Parameter of Poisson-Gamma Models for Motor Vehicle Crashes: Bayesian Perspective. Presented at 86th Annual Meeting of the Transportation Research Board, Washington, D.C., 2006.
43. Pieters, E. G., C. E. Gates, J. H. Matis, and W. L. Sterling. Small Sample Comparisons of Different Estimators of Negative Binomial Parameters, Biometrics, Vol. 33, 1977, pp. 718-723.
44. Lord, D. The Prediction of Accidents on Digital Networks: Characteristics and Issues Related to the Application of Accident Prediction Models. PhD dissertation. Department of Civil Engineering, University of Toronto, Toronto, Ontario, Canada, 2000.