Comparing the Adequacy of Some Non-Homogeneous Poisson Models to Estimate Ozone Exceedances in Mexico City

Affiliation(s)

Faculdade de Medicina de Ribeir?o Preto, Universidade de S?o Paulo, S?o Paulo, Brazil.

Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico.

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico.

Faculdade de Medicina de Ribeir?o Preto, Universidade de S?o Paulo, S?o Paulo, Brazil.

Facultad de Ciencias, Universidad Nacional Autónoma de México, Mexico City, Mexico.

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico.

ABSTRACT

We consider some non-homogeneous Poisson models to estimate the mean number of times that a given environmental threshold of interest is surpassed by a given pollutant. Seven different rate functions for the Poisson processes describing the models are taken into account. The rate functions considered are the Weibull, exponentiated-Weibull, and their generalisation the Beta-Weibull rate function. We also use the Musa-Okumoto, the Goel-Okumoto, a generalised Goel- Okumoto and the Weibull-geometric rate functions. Whenever thought justifiable, the model allowing the presence of change-points is also going to be considered. The different models are applied to the daily maximum ozone measurements data provided by the monitoring network of the Metropolitan Area of Mexico City. The aim is to compare the adjustment of different rate functions to the data. Even though, some of the rate functions have been considered before, now we are applying them to the same data set. In previous works they were used in different data sets and therefore a comparison of the adequacy of those models were not possible. The measurements considered here were obtained after a series of environmental measures were implemented in Mexico City. Hence, the data present a different behaviour from that of earlier studies.

We consider some non-homogeneous Poisson models to estimate the mean number of times that a given environmental threshold of interest is surpassed by a given pollutant. Seven different rate functions for the Poisson processes describing the models are taken into account. The rate functions considered are the Weibull, exponentiated-Weibull, and their generalisation the Beta-Weibull rate function. We also use the Musa-Okumoto, the Goel-Okumoto, a generalised Goel- Okumoto and the Weibull-geometric rate functions. Whenever thought justifiable, the model allowing the presence of change-points is also going to be considered. The different models are applied to the daily maximum ozone measurements data provided by the monitoring network of the Metropolitan Area of Mexico City. The aim is to compare the adjustment of different rate functions to the data. Even though, some of the rate functions have been considered before, now we are applying them to the same data set. In previous works they were used in different data sets and therefore a comparison of the adequacy of those models were not possible. The measurements considered here were obtained after a series of environmental measures were implemented in Mexico City. Hence, the data present a different behaviour from that of earlier studies.

KEYWORDS

MCMC Algorithms; Non-Homogeneous Poisson Models; Change-Points; Ozone Air Pollution; Mexico City

MCMC Algorithms; Non-Homogeneous Poisson Models; Change-Points; Ozone Air Pollution; Mexico City

Cite this paper

J. Achcar, J. Barrios and E. Rodrigues, "Comparing the Adequacy of Some Non-Homogeneous Poisson Models to Estimate Ozone Exceedances in Mexico City,"*Journal of Environmental Protection*, Vol. 3 No. 9, 2012, pp. 1213-1227. doi: 10.4236/jep.2012.329139.

J. Achcar, J. Barrios and E. Rodrigues, "Comparing the Adequacy of Some Non-Homogeneous Poisson Models to Estimate Ozone Exceedances in Mexico City,"

References

[1] M. L. Bell, A. McDermontt, S. L. Zeger, J. M. Samet and F. Dominici, “Ozone and Short-Term Mortality in 95 US Urban Communities, 1987-2000,” Journal of the American Medical Society, Vol. 292, 2004, pp. 2372-2378.

[2] M. L. Bell, R. Peng and F. Dominici, “The Exposure- Response Curve for Ozone and Risk of Mortality and the Adequacy of Current Ozone Regulations,” Environmental Health Perspectives, Vol. 114, 2005, pp. 532-536. doi:10.1289/ehp.8816

[3] M. L. Bell, R. Goldberg, C. Rogrefe, P. L. Kinney, K. Knowlton, B. Lynn, J. Rosenthal, C. Rosenzwei and J. A. Patz, “Climate Change, Ambient Ozone, and Health in 50 US Cities,” Climate Change, Vol. 82, No. 1-2, 2007, pp. 61-76. doi:10.1007/s10584-006-9166-7

[4] W. J. Gauderman, E. Avol, F. Gililand, H. Vora, D. Tho- mas, K. Berhane, R. McConnel, N. Kuenzli, F. Lurmman, E. Rappaport, H. Margolis, D. Bates and J. Peter, “The Effects of Air Pollution on Lung Development from 10 to 18 Years of Age,” New England Journal of Medicine, Vol. 351, 2004, pp. 1057-1067. doi:10.1056/NEJMoa040610

[5] G. Likens (Lead Author), W. Davis, L. Zaikowski and S. C. Nodvin (Topic Editors), “Acid Rain, 2011,” In: J. Cutler, Ed., Encyclopedia of Earth, Environmental Information Coalition, National Council for Science and the Environment, Cleveland, 2011. http://www.eoearth.org/article/Acid_rain

[6] D. Loomis, V. H. Borja-Arbuto, S. I. Bangdiwala and C. M. Shy, “Ozone Exposure and Daily Mortality in Mexico City: A Time Series Analysis,” Health Effects Institute Research Report, Vol. 75, 1996, pp. 1-46.

[7] M. R. O’Neill, D. Loomis and V. H. Borja-Aburto, “Ozone, Area Social Conditions and Mortality in Mexico City,” Environmental Research, Vol. 94, No. 3, 2004, pp. 234-242. doi:10.1016/j.envres.2003.07.002

[8] WHO (World Health Organization), “Air Quality Guidelines—2005. Particulate Matter, Ozone, Nitrogen Dioxide and Sulfur Dioxide,” World Health Organization Regional Office for Europe, 2006.

[9] NOM, “Modificación a la Norma Oficial Mexicana NOM-020-SSA1-1993 (In Spanish),” Diario Oficial de la Federación, 2002.

[10] J. Horowitz, “Extreme Values from a Nonstationary Stochastic Process: An Application to Air Quality Analysis,” Technometrics, Vol. 22, No. 4, 1980, pp. 469-482. doi:10.1080/00401706.1980.10486195

[11] E. M. Roberts, “Review of Statistics Extreme Values with Applications to Air Quality Data. Part I. Review,” Journal of the Air Pollution Control Association, Vol. 29, No. 7, 1979, pp. 632-637. doi:10.1080/00022470.1979.10470856

[12] E. M. Roberts, “Review of Statistics Extreme Values with Applications to Air Quality Data. Part II. Applications,” Journal of the Air Pollution Control Association, Vol. 29, No. 6, 1979, pp. 733-740. doi:10.1080/00022470.1979.10470835

[13] R. L. Smith, “Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone,” Statistical Sciences, Vol. 4, No. 4, 1989, pp. 367-393. doi:10.1214/ss/1177012400

[14] J. B. Flaum, S. T. Rao and I. G. Zurbenko, “Moderating Influence of Meteorological Conditions on Ambient Ozone Concentrations,” Journal of the Air & Waste Management Association, Vol. 46, No. 1, 1996, pp. 33-46. doi:10.1080/10473289.1996.10467439

[15] N. Gouveia and T. Fletcher, “Time Series Analysis of Air Pollution and Mortality: Effects by Cause, Age and Socio-Economics Status,” Journal of Epidemiology and Community Health, Vol. 54, No. 10, 2000, pp. 750-755. doi:10.1136/jech.54.10.750

[16] U. Kumar, A. Prakash and V. K. Jain, “A Multivariate Time Series Approach to Study the Interdependence among O3, NOx and VOCs in Ambient Urban Atmosphere,” Environmental Modeling & Assessment, Vol. 14, No. 5, 2010, pp. 631-643. doi:10.1007/s10666-008-9167-1

[17] M. Lanfredi and M. Macchiato, “Searching for Low Dimensionality in Air Pollution Time Series,” Europhysics Letters, Vol. 40, No. 6, 1997, pp. 589-594. doi:10.1209/epl/i1997-00504-y

[18] J.-N. Pan and S.-T. Chen, “Monitoring Long-Memory Air Quality Data Using ARFIMA Model,” Environmetrics, Vol. 19, No. 2, 2008, pp. 209-219. doi:10.1002/env.882

[19] L. J. álvarez, A. A. Fernández-Bremauntz, E. R. Rodrigues and G. Tzintzun, “Maximum a Posteriori Estimation of the Daily Ozone Peaks in Mexico City,” Journal of Agricultural, Biological Statistics, Vol. 10, No. 3, 2005, pp. 276-290. doi:10.1198/108571105X59017

[20] J. Austin and H. Tran, “A Characterization of the Weekday-Weekend Behavior of Ambient Ozone Concentrations in California,” In: Air Pollution VII, WIT Press, Southampton, 1999, pp. 645-661.

[21] J. A. Achcar, H. C. Zozolotto and E. R. Rodrigues, “Bivariate Volatility Models Applied to Air Pollution Data,” Revista Brasileira de Biologia, Vol. 26, 2008, pp. 67-81.

[22] J. A. Achcar, H. C. Zozolotto and E. R. Rodrigues, “Bivariate Stochastic Volatility Models Applied to Mexico City Ozone Pollution Data,” In: G. C. Romano and A. G. Conti, Eds., Air Quality in the 21st Century, Nova Publishers, New York, 2010, pp. 241-267.

[23] J. A. Achcar, E. R. Rodrigues and G. Tzintzun, “Using Stochastic Volatility Models to Analyse Weekly Ozone Averages in Mexico City,” Environmental and Ecological Statistics, Vol. 18, No. 2, 2011, pp. 271-290. doi:10.1007/s10651-010-0132-1

[24] G. Huerta and B. Sansó, “Time-Varying Models for Extreme Values,” Environmental and Ecological Statistics, Vol. 14, No. 3, 2007, pp. 285-299. doi:10.1007/s10651-007-0014-3

[25] L. W. Davis, “The Effect of Driving Restrictions on Air Quality in Mexico City,” Journal of Political Economy, Vol. 116, No. 1, 2008, pp. 39-81. doi:10.1086/529398

[26] M. Zavala, S. C. Herndon, E. C. Wood, T. B. Onasch, W. B. Knighton, L. C. Marr, C. E. Kolb and L. T. Molina, “Evaluation of Mobile Emissions Contributions to Mexico City’s Emissions Inventory Using on Road and Cross-Road Emission Measurements and Ambient Data,” Atmospheric Chemistry and Physics, Vol. 9, No. 17, 2009, pp. 6305-6317. doi:10.5194/acp-9-6305-2009

[27] J. S. Javits, “Statistical Interdependencies in the Ozone National Ambient Air Quality Standard,” Journal of the Air Pollution Control Association, Vol. 30, No. 1, 1980, pp. 58-59. doi:10.1080/00022470.1980.10465918

[28] A. E. Raftery, “Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone,” Statistical Sciences, Vol. 4, No. 4, 1989, pp. 378-381. doi:10.1214/ss/1177012401

[29] J. A. Achcar, A. A. Fernández-Bremauntz, E. R. Rodrigues and G. Tzintzun, “Estimating the Number of Ozone Peaks in Mexico City Using a Non-Homogeneous Poisson Model,” Environmetrics, Vol. 19, No. 5, 2008, pp. 469-485. doi:10.1002/env.890

[30] J. A. Achcar, E. R. Rodrigues, C. D. Paulino and P. Soares, “Non-Homogeneous Poisson Processes with a Change-Point: An Application to Ozone Exceedances in Mexico City,” Environmental and Ecological Statistics, Vol. 17, No. 4, 2010, pp. 521-541. doi:10.1007/s10651-009-0114-3

[31] J. A. Achcar, E. R. Rodrigues and G. Tzintzun, “Using Non-Homogeneous Poisson Models with Multiple Change- Points to Estimate the Number of Ozone Exceedances in Mexico City,” Environmetrics, Vol. 22, No. 1, 2011, pp. 1-12. doi:10.1002/env.1029

[32] E. R. Rodrigues, J. A. Achcar and J. Jara-Ettinger, “A Gibbs Sampling Algorithm to Estimate the Occurrence of Ozone Exceedances in Mexico City,” In: D. Popovic, Ed., Air Quality: Models and Applications, InTech Open Access Publishers, 2011, pp. 131-150.

[33] G. S. Muldholkar, D. K. Srivastava and H. Friemer, “The Exponentiated-Weibull Family: A Reanalysis of the Bus- Motor Failure Data,” Technometrics, Vol. 37, No. 4, 1995, pp. 436-445. doi:10.1080/00401706.1995.10484376

[34] J. E. Ramrez-Cid and J. A. Achcar, “Bayesian Inference for Nonhomogeneous Poisson Processes in Software Reliability Models Assuming Nonmonotonic Intensity Functions,” Computational Statistics and Data Analysis, Vol. 32, No. 2, 1999, pp. 147-159. doi:10.1016/S0167-9473(99)00028-6

[35] F. Famoye, C. Lee and O. Olumolade, “The Beta-Weibull Distribution,” Journal of Statistical Theory and Practice, Vol. 4, 2005, pp. 121-136.

[36] C. Lee, F. Famoye and O. Olumolade, “The Beta-Weibull Distributions: Some Properties and Applications to Censored Data,” Journal of Modern Applied Statistical Methods, Vol. 6, 2007, pp. 173-186.

[37] G. M. Cordeiro, et al., “Closed form Expressions for the Moments of the Beta-Weibull Distribution,” Annals of the Brazilian Academy of Sciences, Vol. 83, 2011, pp. 357- 373.

[38] J. D. Musa and K. Okumoto, “A Logarithmic Poisson Execution Time Model for Software Reliability Measurement,” Proceedings of Seventh International Conference on Software Engineering, Orlando, 1984, pp. 230- 238.

[39] A. L. Goel and K. Okumoto, “An Analysis of Recurrent Software Failures on a Real-Time Control System,” Proceedings of ACM Conference, Washington DC, 1978, pp. 496-500.

[40] W. Barreto-Souza, A. L. de Morais and G. M. Cordeiro, “The Weibull-Geometric Distribution,” Journal of Statistical Computation and Simulation, Vol. 81, No. 5, 2011, pp. 645-657. doi:10.1080/00949650903436554

[41] D. R. Cox and P. A. Lewis, “Statistical Analysis of Series Events,” Chapman and Hall, Methuen, 1966.

[42] J. F. Lawless, “Statistical Models and Methods for Lifetime Data,” John Wiley and Sons, New York, 1982.

[43] J. A. Achcar, E. Z. Martnez, A. Rufino-Neto , C. D. Paulino and P. Soares, “A Statistical Model Investigating the Prevalence of Tuberculosis in New York Using Counting Processes with Two Change-Points,” Epidemiology and Infection, Vol. 136, No. 12, 2008, pp. 1599-1605. doi:10.1017/S0950268808000526

[44] T. E. Yang and L. Kuo, “Bayesian Binary Segmentation Procedure for a Poisson Process with Multiple Change- Points,” Journal of Computational and Graphical Statistics, Vol. 10, No. 4, 2001, pp. 772-785. doi:10.1198/106186001317243449

[45] A. E. Gelfand and A. F. M. Smith, “Sampling-Based Approaches to Calculating Marginal Densities,” Journal of the American Statistical Association, Vol. 85, No. 410, 1990, pp. 398-409. doi:10.1080/01621459.1990.10476213

[46] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, “Equations of State Calculations by fast Computing Machine,” Journal of Chemical Physics, Vol. 21, No. 6, 1953, pp. 1087-1091. doi:10.1063/1.1699114

[47] W. K. Hastings, “Monte Carlo Sampling Methods Using Markov Chains and Their Applications,” Biometrika, Vol. 57, No. 1, 1970, pp. 97-109. doi:10.1093/biomet/57.1.97

[48] A. R. Didonato and A. H. Morris, “Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios,” Transactions on Mathematical Software, Vol. 18, No. 3, 1992, pp. 360-373. doi:10.1145/131766.131776

[49] A. Gelman and D. B. Rubin, “Inference from Iterative Simulation Using Multiple Sequences,” Statistical Sciences, Vol. 7, No. 4, 1992, pp. 457-511. doi:10.1214/ss/1177011136

[50] J. Geweke, “Evaluating the Accuracy of Sampling-Based Approaches to the Calculation of Posterior Moments,” In: J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, Eds., Bayesian Statistics, Vol. 4, Clarendon Press, Oxford, 1992, pp. 169-193.

[51] D. J. Spiegelhalter, N. G. Best, B. P. Carlin and A. van der Linde, “Bayesian Measures of Model Complexity and Fit,” Journal of the Royal Statistical Society Series B, Vol. 64, No. 4, 2002, pp. 583-639. doi:10.1111/1467-9868.00353

[52] A. E. Raftery, “Hypothesis Testing and Model Selection,” In: W. Gilks, S. Richardson and D. J. Speigelhalter, Eds., Markov Chain Monte Carlo in Practice, Chapman and Hall, London, 1996, pp. 163-187.

[53] J. A. Achcar, E. R. Rodrigues and M. H. Tarumoto, “Using Counting Processes to Estimate the Number of Ozone Exceedances: An Application to the Mexico City Measurements,” Proceedings of the 58th ISI World Statistics Congress, Dublin, 21-16 August 2011. http://isi2011.congressplanner.eu/25

[54] J. A. Achcar, E. Cepeda-Cuervo and E. R. Rodrigues, “Weibull and Generalised Exponential Overdispersion Models with an Application to Ozone Air Pollution,” Journal of Applied Statistics, Vol. 39, No. 9, 2012, pp. 1953-1963. doi:10.1080/02664763.2012.697132

[1] M. L. Bell, A. McDermontt, S. L. Zeger, J. M. Samet and F. Dominici, “Ozone and Short-Term Mortality in 95 US Urban Communities, 1987-2000,” Journal of the American Medical Society, Vol. 292, 2004, pp. 2372-2378.

[2] M. L. Bell, R. Peng and F. Dominici, “The Exposure- Response Curve for Ozone and Risk of Mortality and the Adequacy of Current Ozone Regulations,” Environmental Health Perspectives, Vol. 114, 2005, pp. 532-536. doi:10.1289/ehp.8816

[3] M. L. Bell, R. Goldberg, C. Rogrefe, P. L. Kinney, K. Knowlton, B. Lynn, J. Rosenthal, C. Rosenzwei and J. A. Patz, “Climate Change, Ambient Ozone, and Health in 50 US Cities,” Climate Change, Vol. 82, No. 1-2, 2007, pp. 61-76. doi:10.1007/s10584-006-9166-7

[4] W. J. Gauderman, E. Avol, F. Gililand, H. Vora, D. Tho- mas, K. Berhane, R. McConnel, N. Kuenzli, F. Lurmman, E. Rappaport, H. Margolis, D. Bates and J. Peter, “The Effects of Air Pollution on Lung Development from 10 to 18 Years of Age,” New England Journal of Medicine, Vol. 351, 2004, pp. 1057-1067. doi:10.1056/NEJMoa040610

[5] G. Likens (Lead Author), W. Davis, L. Zaikowski and S. C. Nodvin (Topic Editors), “Acid Rain, 2011,” In: J. Cutler, Ed., Encyclopedia of Earth, Environmental Information Coalition, National Council for Science and the Environment, Cleveland, 2011. http://www.eoearth.org/article/Acid_rain

[6] D. Loomis, V. H. Borja-Arbuto, S. I. Bangdiwala and C. M. Shy, “Ozone Exposure and Daily Mortality in Mexico City: A Time Series Analysis,” Health Effects Institute Research Report, Vol. 75, 1996, pp. 1-46.

[7] M. R. O’Neill, D. Loomis and V. H. Borja-Aburto, “Ozone, Area Social Conditions and Mortality in Mexico City,” Environmental Research, Vol. 94, No. 3, 2004, pp. 234-242. doi:10.1016/j.envres.2003.07.002

[8] WHO (World Health Organization), “Air Quality Guidelines—2005. Particulate Matter, Ozone, Nitrogen Dioxide and Sulfur Dioxide,” World Health Organization Regional Office for Europe, 2006.

[9] NOM, “Modificación a la Norma Oficial Mexicana NOM-020-SSA1-1993 (In Spanish),” Diario Oficial de la Federación, 2002.

[10] J. Horowitz, “Extreme Values from a Nonstationary Stochastic Process: An Application to Air Quality Analysis,” Technometrics, Vol. 22, No. 4, 1980, pp. 469-482. doi:10.1080/00401706.1980.10486195

[11] E. M. Roberts, “Review of Statistics Extreme Values with Applications to Air Quality Data. Part I. Review,” Journal of the Air Pollution Control Association, Vol. 29, No. 7, 1979, pp. 632-637. doi:10.1080/00022470.1979.10470856

[12] E. M. Roberts, “Review of Statistics Extreme Values with Applications to Air Quality Data. Part II. Applications,” Journal of the Air Pollution Control Association, Vol. 29, No. 6, 1979, pp. 733-740. doi:10.1080/00022470.1979.10470835

[13] R. L. Smith, “Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone,” Statistical Sciences, Vol. 4, No. 4, 1989, pp. 367-393. doi:10.1214/ss/1177012400

[14] J. B. Flaum, S. T. Rao and I. G. Zurbenko, “Moderating Influence of Meteorological Conditions on Ambient Ozone Concentrations,” Journal of the Air & Waste Management Association, Vol. 46, No. 1, 1996, pp. 33-46. doi:10.1080/10473289.1996.10467439

[15] N. Gouveia and T. Fletcher, “Time Series Analysis of Air Pollution and Mortality: Effects by Cause, Age and Socio-Economics Status,” Journal of Epidemiology and Community Health, Vol. 54, No. 10, 2000, pp. 750-755. doi:10.1136/jech.54.10.750

[16] U. Kumar, A. Prakash and V. K. Jain, “A Multivariate Time Series Approach to Study the Interdependence among O3, NOx and VOCs in Ambient Urban Atmosphere,” Environmental Modeling & Assessment, Vol. 14, No. 5, 2010, pp. 631-643. doi:10.1007/s10666-008-9167-1

[17] M. Lanfredi and M. Macchiato, “Searching for Low Dimensionality in Air Pollution Time Series,” Europhysics Letters, Vol. 40, No. 6, 1997, pp. 589-594. doi:10.1209/epl/i1997-00504-y

[18] J.-N. Pan and S.-T. Chen, “Monitoring Long-Memory Air Quality Data Using ARFIMA Model,” Environmetrics, Vol. 19, No. 2, 2008, pp. 209-219. doi:10.1002/env.882

[19] L. J. álvarez, A. A. Fernández-Bremauntz, E. R. Rodrigues and G. Tzintzun, “Maximum a Posteriori Estimation of the Daily Ozone Peaks in Mexico City,” Journal of Agricultural, Biological Statistics, Vol. 10, No. 3, 2005, pp. 276-290. doi:10.1198/108571105X59017

[20] J. Austin and H. Tran, “A Characterization of the Weekday-Weekend Behavior of Ambient Ozone Concentrations in California,” In: Air Pollution VII, WIT Press, Southampton, 1999, pp. 645-661.

[21] J. A. Achcar, H. C. Zozolotto and E. R. Rodrigues, “Bivariate Volatility Models Applied to Air Pollution Data,” Revista Brasileira de Biologia, Vol. 26, 2008, pp. 67-81.

[22] J. A. Achcar, H. C. Zozolotto and E. R. Rodrigues, “Bivariate Stochastic Volatility Models Applied to Mexico City Ozone Pollution Data,” In: G. C. Romano and A. G. Conti, Eds., Air Quality in the 21st Century, Nova Publishers, New York, 2010, pp. 241-267.

[23] J. A. Achcar, E. R. Rodrigues and G. Tzintzun, “Using Stochastic Volatility Models to Analyse Weekly Ozone Averages in Mexico City,” Environmental and Ecological Statistics, Vol. 18, No. 2, 2011, pp. 271-290. doi:10.1007/s10651-010-0132-1

[24] G. Huerta and B. Sansó, “Time-Varying Models for Extreme Values,” Environmental and Ecological Statistics, Vol. 14, No. 3, 2007, pp. 285-299. doi:10.1007/s10651-007-0014-3

[25] L. W. Davis, “The Effect of Driving Restrictions on Air Quality in Mexico City,” Journal of Political Economy, Vol. 116, No. 1, 2008, pp. 39-81. doi:10.1086/529398

[26] M. Zavala, S. C. Herndon, E. C. Wood, T. B. Onasch, W. B. Knighton, L. C. Marr, C. E. Kolb and L. T. Molina, “Evaluation of Mobile Emissions Contributions to Mexico City’s Emissions Inventory Using on Road and Cross-Road Emission Measurements and Ambient Data,” Atmospheric Chemistry and Physics, Vol. 9, No. 17, 2009, pp. 6305-6317. doi:10.5194/acp-9-6305-2009

[27] J. S. Javits, “Statistical Interdependencies in the Ozone National Ambient Air Quality Standard,” Journal of the Air Pollution Control Association, Vol. 30, No. 1, 1980, pp. 58-59. doi:10.1080/00022470.1980.10465918

[28] A. E. Raftery, “Extreme Value Analysis of Environmental Time Series: An Application to Trend Detection in Ground-Level Ozone,” Statistical Sciences, Vol. 4, No. 4, 1989, pp. 378-381. doi:10.1214/ss/1177012401

[29] J. A. Achcar, A. A. Fernández-Bremauntz, E. R. Rodrigues and G. Tzintzun, “Estimating the Number of Ozone Peaks in Mexico City Using a Non-Homogeneous Poisson Model,” Environmetrics, Vol. 19, No. 5, 2008, pp. 469-485. doi:10.1002/env.890

[30] J. A. Achcar, E. R. Rodrigues, C. D. Paulino and P. Soares, “Non-Homogeneous Poisson Processes with a Change-Point: An Application to Ozone Exceedances in Mexico City,” Environmental and Ecological Statistics, Vol. 17, No. 4, 2010, pp. 521-541. doi:10.1007/s10651-009-0114-3

[31] J. A. Achcar, E. R. Rodrigues and G. Tzintzun, “Using Non-Homogeneous Poisson Models with Multiple Change- Points to Estimate the Number of Ozone Exceedances in Mexico City,” Environmetrics, Vol. 22, No. 1, 2011, pp. 1-12. doi:10.1002/env.1029

[32] E. R. Rodrigues, J. A. Achcar and J. Jara-Ettinger, “A Gibbs Sampling Algorithm to Estimate the Occurrence of Ozone Exceedances in Mexico City,” In: D. Popovic, Ed., Air Quality: Models and Applications, InTech Open Access Publishers, 2011, pp. 131-150.

[33] G. S. Muldholkar, D. K. Srivastava and H. Friemer, “The Exponentiated-Weibull Family: A Reanalysis of the Bus- Motor Failure Data,” Technometrics, Vol. 37, No. 4, 1995, pp. 436-445. doi:10.1080/00401706.1995.10484376

[34] J. E. Ramrez-Cid and J. A. Achcar, “Bayesian Inference for Nonhomogeneous Poisson Processes in Software Reliability Models Assuming Nonmonotonic Intensity Functions,” Computational Statistics and Data Analysis, Vol. 32, No. 2, 1999, pp. 147-159. doi:10.1016/S0167-9473(99)00028-6

[35] F. Famoye, C. Lee and O. Olumolade, “The Beta-Weibull Distribution,” Journal of Statistical Theory and Practice, Vol. 4, 2005, pp. 121-136.

[36] C. Lee, F. Famoye and O. Olumolade, “The Beta-Weibull Distributions: Some Properties and Applications to Censored Data,” Journal of Modern Applied Statistical Methods, Vol. 6, 2007, pp. 173-186.

[37] G. M. Cordeiro, et al., “Closed form Expressions for the Moments of the Beta-Weibull Distribution,” Annals of the Brazilian Academy of Sciences, Vol. 83, 2011, pp. 357- 373.

[38] J. D. Musa and K. Okumoto, “A Logarithmic Poisson Execution Time Model for Software Reliability Measurement,” Proceedings of Seventh International Conference on Software Engineering, Orlando, 1984, pp. 230- 238.

[39] A. L. Goel and K. Okumoto, “An Analysis of Recurrent Software Failures on a Real-Time Control System,” Proceedings of ACM Conference, Washington DC, 1978, pp. 496-500.

[40] W. Barreto-Souza, A. L. de Morais and G. M. Cordeiro, “The Weibull-Geometric Distribution,” Journal of Statistical Computation and Simulation, Vol. 81, No. 5, 2011, pp. 645-657. doi:10.1080/00949650903436554

[41] D. R. Cox and P. A. Lewis, “Statistical Analysis of Series Events,” Chapman and Hall, Methuen, 1966.

[42] J. F. Lawless, “Statistical Models and Methods for Lifetime Data,” John Wiley and Sons, New York, 1982.

[43] J. A. Achcar, E. Z. Martnez, A. Rufino-Neto , C. D. Paulino and P. Soares, “A Statistical Model Investigating the Prevalence of Tuberculosis in New York Using Counting Processes with Two Change-Points,” Epidemiology and Infection, Vol. 136, No. 12, 2008, pp. 1599-1605. doi:10.1017/S0950268808000526

[44] T. E. Yang and L. Kuo, “Bayesian Binary Segmentation Procedure for a Poisson Process with Multiple Change- Points,” Journal of Computational and Graphical Statistics, Vol. 10, No. 4, 2001, pp. 772-785. doi:10.1198/106186001317243449

[45] A. E. Gelfand and A. F. M. Smith, “Sampling-Based Approaches to Calculating Marginal Densities,” Journal of the American Statistical Association, Vol. 85, No. 410, 1990, pp. 398-409. doi:10.1080/01621459.1990.10476213

[46] N. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller and E. Teller, “Equations of State Calculations by fast Computing Machine,” Journal of Chemical Physics, Vol. 21, No. 6, 1953, pp. 1087-1091. doi:10.1063/1.1699114

[47] W. K. Hastings, “Monte Carlo Sampling Methods Using Markov Chains and Their Applications,” Biometrika, Vol. 57, No. 1, 1970, pp. 97-109. doi:10.1093/biomet/57.1.97

[48] A. R. Didonato and A. H. Morris, “Algorithm 708: Significant Digit Computation of the Incomplete Beta Function Ratios,” Transactions on Mathematical Software, Vol. 18, No. 3, 1992, pp. 360-373. doi:10.1145/131766.131776

[49] A. Gelman and D. B. Rubin, “Inference from Iterative Simulation Using Multiple Sequences,” Statistical Sciences, Vol. 7, No. 4, 1992, pp. 457-511. doi:10.1214/ss/1177011136

[50] J. Geweke, “Evaluating the Accuracy of Sampling-Based Approaches to the Calculation of Posterior Moments,” In: J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, Eds., Bayesian Statistics, Vol. 4, Clarendon Press, Oxford, 1992, pp. 169-193.

[51] D. J. Spiegelhalter, N. G. Best, B. P. Carlin and A. van der Linde, “Bayesian Measures of Model Complexity and Fit,” Journal of the Royal Statistical Society Series B, Vol. 64, No. 4, 2002, pp. 583-639. doi:10.1111/1467-9868.00353

[52] A. E. Raftery, “Hypothesis Testing and Model Selection,” In: W. Gilks, S. Richardson and D. J. Speigelhalter, Eds., Markov Chain Monte Carlo in Practice, Chapman and Hall, London, 1996, pp. 163-187.

[53] J. A. Achcar, E. R. Rodrigues and M. H. Tarumoto, “Using Counting Processes to Estimate the Number of Ozone Exceedances: An Application to the Mexico City Measurements,” Proceedings of the 58th ISI World Statistics Congress, Dublin, 21-16 August 2011. http://isi2011.congressplanner.eu/25

[54] J. A. Achcar, E. Cepeda-Cuervo and E. R. Rodrigues, “Weibull and Generalised Exponential Overdispersion Models with an Application to Ozone Air Pollution,” Journal of Applied Statistics, Vol. 39, No. 9, 2012, pp. 1953-1963. doi:10.1080/02664763.2012.697132