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Computer based statistical treatment in models with incidental parameters: inspired by car crash data
Linköpings universitet, Matematisk statistik.ORCID iD: 0000-0002-9164-9221
2003 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Bootstrap and Markov chain Monte Carlo methods have received much attention in recent years. We study computer intensive methods that can be used in complex situations where it is not possible to express the likelihood estimates or the posterior analytically. The work is inspired by a set of car crash data from real traffic.

We formulate and develop a model for car crash data that aims to estimate and compare the relative collision safety among different car models. This model works sufficiently well, although complications arise due to a growing vector of incidental parameters. The bootstrap is shown to be a useful tool for studying uncertainties of the estimates of the structural parameters. This model is further extended to include driver characteristics. In a Poisson model with similar, but simpler structure, estimates of the structural parameter in the presence of incidental parameters are studied. The profile likelihood, bootstrap and the delta method are compared for deterministic and random incidental parameters. The same asymptotic properties, up to first order, are seen for deterministic as well as random incidental parameters.

The search for suitable methods that work in complex model structures leads us to consider Markov chain Monte Carlo (MCMC) methods. In the area of MCMC, we consider particularly the question of how and when to claim convergence of the MCMC run in situations where it is only possible to analyse the output values of the run and also how to compare different MCMC modellings. In Metropolis-Hastings algorithm, different proposal functions lead to different realisations. We develop a new convergence diagnostic, based on the Kullback-Leibler distance, which is shown to be particularly useful when comparing different runs. Comparisons with established methods turn out favourably for the KL.

In both models, a Bayesian analysis is made where the posterior distribution is obtained by MCMC methods. The credible intervals are compared to the corresponding confidence intervals from the bootstrap analysis and are shown to give the same qualitative conclusions.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet , 2003. , p. 34
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 814
Keywords [en]
Mathematical model, Accident, Statistics, Analysis, Driver, Characteristics
National Category
Mathematics
Research subject
80 Road: Traffic safety and accidents, 812 Road: Collation of accident statistics
Identifiers
URN: urn:nbn:se:vti:diva-7106Local ID: 1377ISBN: 91-7373-625-2 (print)OAI: oai:DiVA.org:vti-7106DiVA, id: diva2:747450
Public defence
2003-05-09, Sal Visionen, Hus B, Linköpings Universitet, 13:15 (Swedish)
Opponent
Available from: 2009-10-07 Created: 2014-09-16 Last updated: 2016-02-22Bibliographically approved
List of papers
1. Modelling and inference of relative collision safety in cars
Open this publication in new window or tab >>Modelling and inference of relative collision safety in cars
1998 (English)Licentiate thesis, monograph (Other academic)
Abstract [en]

We propose a new mathematical model for relative collision safety in cars. Our present research is restricted to head-on crashes between two cars and we try to determine how much of the injury risk in a crash that depends on car model. The relative risks include the driver populations of the different car models. When two cars crash they are exposed to the same force, but the damage severity is different depending on various factors such as car mass, change of speed and design of the car. To explore the relative risks between different car models, we build a model where we let car mass, change of speed and design of the car explain the injury outcome in the crashes. The mathematical model we use is a birth process where we let the states correspond to the injury classes. A data base containing police reported traffic accidents and hospital information is used to explore the relationships in our model.

A bootstrap analysis is made to produce a picture of the uncertainty of the estimates. The uncertainty from the bootstrap analysis is compared to the asymptotic estimate of the uncertainty given by the inverse of an information sub-matrix.

Place, publisher, year, edition, pages
Linköping: Linköpings Universitet, 1998. p. 51
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 685
Keywords
Collision, Mathematical model, Head on collision, Injury, Risk, Car
National Category
Mathematical Analysis
Research subject
90 Road: Vehicles and vehicle technology, 91 Road: Vehicle design and construction; 80 Road: Traffic safety and accidents, 85 Road: Personal injuries
Identifiers
urn:nbn:se:vti:diva-7103 (URN)91-7219-200-3 (ISBN)
Available from: 2013-01-07 Created: 2014-09-16 Last updated: 2016-02-22Bibliographically approved
2. The empirical KL-measure of MCMC convergence
Open this publication in new window or tab >>The empirical KL-measure of MCMC convergence
(English)Manuscript (preprint) (Other academic)
Abstract [en]

A new measure based on comparison of empirical distributions for sub sequences or parallel runs and the full sequence of Markov chain Monte Carlo simulation, is proposed as a criterion of stability or convergence. The measure is also put forward as a loss function when the design, including the proposal function, of a Markov chain is optimised. The comparison of empirical distributions is based on a Kullback-Leibler (KL) type distance over value sets defined by the output data. The singularity problem for such a measure is removed in a simple way.

The leading term in a series expansion of the measure gives an interpretation in terms of the relative uncertainty of cell frequencies measured by their average coefficient of variation. The validity of the leading term is studied by simulation in two analytically tractable cases with Markov dependency and selected acceptance rates. The agreement between the leading term and the KL-measure is close, in particular when the simulations are extensive enough for stable results. Comparisons with established criteria turn out favourably in examples studied.

Keywords
Calculation, Simulatin, Mathematical model
National Category
Natural Sciences
Research subject
SAB, T Mathematics
Identifiers
urn:nbn:se:vti:diva-7102 (URN)
Available from: 2013-01-07 Created: 2014-09-16 Last updated: 2016-02-22Bibliographically approved
3. On Gibbs sampler and Metrolopolis-Hastings applied to pairwise poisson and car crash data
Open this publication in new window or tab >>On Gibbs sampler and Metrolopolis-Hastings applied to pairwise poisson and car crash data
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We address the use of MCMC methods in the presence of incidental parameters. Two different models are studied, one pairwise Poisson model and a model for car crash data. In both models, a Bayesian analysis is made where the posterior density is obtained by MCMC methods. The empirical KL-measure is used to diagnose convergence. Credible intervals for structural parameters are computed in both models and compared to the corresponding confidence intervals obtained by frequentist analysis. Although some differences are observed, the methods give qualitatively the same conclusions.

Keywords
Accident rate, Statistics, Analysis (math), Method
National Category
Probability Theory and Statistics
Research subject
80 Road: Traffic safety and accidents, 812 Road: Collation of accident statistics
Identifiers
urn:nbn:se:vti:diva-7101 (URN)
Available from: 2013-01-07 Created: 2014-09-16 Last updated: 2016-02-22Bibliographically approved
4. Estimation in a model with incidental parameters
Open this publication in new window or tab >>Estimation in a model with incidental parameters
2002 (English)Report (Other academic)
Abstract [en]

We study the maximum likelihood method together with bootstrap analysis and other uncertainty measures in a situation with both structural and incidental parameters and a rather simple parametric setting. Our purpose is to study methods that can be generalised to and used in more complicated situations of similar nature.

Two different approaches to the incidental parameters are treated, one deterministic and one random. Both approaches are shown to give similar results. When analysing the asymptotic properties of the estimator of the structural parameter, the profile likelihood, the delta method and the bootstrap analysis seem to be equally good in the deterministic case. The bootstrap also works well with a random interpretation of the incidental parameters.

Publisher
p. 18
Series
LiTH-MAT-R, ISSN 0348-2960 ; 02
Keywords
Statistics, Calculation
National Category
Probability Theory and Statistics
Research subject
SAB, T Mathematics
Identifiers
urn:nbn:se:vti:diva-8302 (URN)
Available from: 2013-01-07 Created: 2016-02-03 Last updated: 2016-02-22Bibliographically approved
5. Including driver characteristics in a model of relative collision safety
Open this publication in new window or tab >>Including driver characteristics in a model of relative collision safety
2000 (English)Report (Other academic)
Abstract [en]

When the relative collision safety between different car makes has been estimated, it has been shown that a person's age and sex influence the injury risk in accidents that are otherwise similar. In an earlier work, relative collisionsafety in cars are studied. That model is now expanded by introducing parameters related to the driver's age and sex. Different models are compared and the "best" model is chosen by a likelihood-ratio-test. The estimated relative risks compensated for the driver's age and sex are compared to the relative risks without such information. The uncertainties of the different estimates are studied by a bootstrap analysis.

Publisher
p. 18
Series
LiTH-MAT-R, ISSN 0348-2960 ; 19
Keywords
Driver, Age, Man, Woman, Properties, Mathematical model, Accident, Risk
National Category
Probability Theory and Statistics
Research subject
80 Road: Traffic safety and accidents, 812 Road: Collation of accident statistics; 80 Road: Traffic safety and accidents, 85 Road: Personal injuries
Identifiers
urn:nbn:se:vti:diva-8303 (URN)
Available from: 2013-01-07 Created: 2016-02-03 Last updated: 2016-02-22Bibliographically approved

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Citation style
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