Published

2019-07-01

Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach

Teoría de valores extremos aplicada a las r estadísticas de orden superior desde el punto de vista bayesiano

DOI:

https://doi.org/10.15446/rce.v42n2.70271

Keywords:

Markov chain monte carlo, Extreme value, Bayesian inference (en)
Monte Carlo para cadena de Markov, Valores extremos, Inferencia bayesiana (es)

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Authors

  • Renato Santos Silva Universidade Federal do Rio Grande do Norte (UFRN)
  • Fernando Ferraz Nascimento Universidade Federal do Piauí (UFPI)

Extreme Value Theory (EVT) is an important tool to predict efficient gains and losses. Its main areas of analyses are economic and environmental. Initially, for that form of event, it was developed the use of patterns of parametric distribution such as Normal and Gamma. However, economic and environmental data presents, in most cases, a heavy-tailed distribution, in contrast to those distributions. Thus, it was faced a great difficult to frame extreme events. Furthermore, it was almost impossible to use conventional models, making predictions about non-observed events, which exceed the maximum of observations. In some situations EVT is used to analyse only the maximum of some dataset, which provide few observations, and in those cases it is more effective to use the r largest-order statistics. This paper aims to propose Bayesian estimators' for parameters of the r largest-order statistics. During the research, it was used Monte Carlo simulation to analyze the data, and it was observed some properties of those estimators, such as mean, variance, bias and Root Mean Square Error (RMSE). The estimation of the parameters provided inference for its parameters and return levels. This paper also shows a procedure to the choice of the r-optimal to the r largest-order statistics, based on the Bayesian approach applying Markov chains Monte Carlo (MCMC). Simulation results reveal that the Bayesian approach has a similar performance to the Maximum Likelihood Estimation, and the applications were developed using the Bayesian approach and showed a gain in accurary compared with otherestimators.

La teoría de valores extremos (EVT) es una herramienta importante para predecir ganancias y pérdidas eficientes en ambientes económicos y ambientales. Además, la EVT se desarrolló inicialmente para uso con patrones de distribución paramétricos normales y gamma. Sin embargo, los datos económicos y ambientales presentan una distribución de cola pesada en la mayoría de los casos, lo que contrasta con los patrones anteriores. Así, la formulación de eventos extremos con EVT presenta grandes dificultades. Además, es casi imposible usar modelos convencionales para obtener predicciones sobre eventos no observados que excedieron el número máximo deobservaciones. En algunas situaciones, EVT es utilizado para analizar solamente los valores máximos de un conjunto de datos dado, que proporcionan poca información. En tales casos, es más eficiente usar las r estadísticas de orden superior. Este trabajo propone estimadores bayesianos para los parámetros de las r estadísticas de orden superior. Utilizamos simulaciónes de Monte Carlo para analizar los datos experimentales y observar ciertas propiedades del estimador como: media, intervalos de confianza y credibilidad, sesgo y error cuadrático medio (RMSE). Este tipo deestimación proporciona inferencias para estos parametros y niveles de retorno. Tambien, proponemos un procedimiento para seleccionar el r-óptimo de la distribución de las r estadísticas de orden superior basadas en el enfoque bayesiano y aplicando el método de Monte Carlo para cadenas de Markov (MCMC). Los resultados de la simulación muestran que el enfoque bayesiano produce un rendimiento similar al de la estimación de máxima verosemelianza. Finalmente, las aplicaciones desarrolladas utilizando el enfoque bayesiano mostraron una ganancia en la precisión en comparación con otros estimadores.

References

Bader, B. & Yan, J. (2016), ‘eva: Extreme value analysis with goodness-of-fit testing’. R package version 0.2.

Bader, B., Yan, J. & Zhang, X. (2017), ‘Automated selection of r for the r largest order statistics approach with adjustment for sequential testing’, Statistics and Computing 27(6), 1435–1451.

Balakrishnan, N., Kannan, N. & Nagaraja, H. N. (2007), Advances in ranking and selection, multiple comparisons, and reliability: methodology and applications, Springer Science & Business Media.

Benjamini, Y. (2010a), ‘Discovering the false discovery rate’, Journal of the Royal Statistical Society: series B (statistical methodology) 72(4), 405–416.

Benjamini, Y. (2010b), ‘Simultaneous and selective inference: Current successes and future challenges’, Biometrical Journal 52(6), 708–721.

Benjamini, Y. & Hochberg, Y. (1995), ‘Controlling the false discovery rate: a practical and powerful approach to multiple testing’, Journal of the royal statistical society. Series B (Methodological) pp. 289–300.

Benjamini, Y. & Yekutieli, D. (2001), ‘The control of the false discovery rate in multiple testing under dependency’, Annals of statistics pp. 1165–1188.

Coles, S. (2006), ‘Ismev: an introduction to statistical modeling of extreme values’. http://cran. r-project. org/web/packages/ismev/index.html.

Coles, S., Bawa, J., Trenner, L. & Dorazio, P. (2001), An introduction to statistical modeling of extreme values, Vol. 208, Springer.

Coles, S. G. & Tawn, J. A. (1996), ‘A bayesian analysis of extreme rainfall data’, Applied statistics pp. 463–478.

Do Nascimento, F. F. & Moura e Silva, W. V. (2016), ‘MCMC4Extremes: Posterior Distribution of Extreme Value Models in R’. R package version 1.1.

Fisher, R. A. & Tippett, L. H. C. (1928), ‘On the estimation of the frequency distributions of the largest and smallest sumber of a sample’, Proceedings of the Cambridge Philosophycal Society 24, 180–190.

Gamerman, D. & Lopes, H. F. (2006), Markov chain Monte Carlo: stochastic simulation for Bayesian inference, Chapman and Hall/CRC.

Gonçalves, K. C., Migon, H. S. & Bastos, L. S. (2019), ‘Dynamic quantile linear models: A bayesian approach’, Bayesian Analysis (online). https://arxiv.org/abs/1711.00162.

G’Sell, M. G., Wager, S., Chouldechova, A. & Tibshirani, R. (2016), ‘Sequential selection procedures and false discovery rate control’, Journal of the royal statistical society: series B (statistical methodology) 78(2), 423–444.

Hastings, W. K. (1970), ‘Monte carlo sampling methods using markov chains and their applications’, 57(l).

Huerta, G. & Sansó, B. (2007), ‘Time-varying models for extreme values’, Environmental and Ecological Statistics 14(3), 285–299.

Jenkinson, A. F. (1955), ‘The frequency distribution of the annual maximum (or minimum) values of meteorological elements’, Quarterly Journal of the Royal Meteorological Society 81(348), 158–171.

Kozumi, H. & Kobayashi, G. (2011), ‘Gibbs sampling methods for bayesian quantile regression’, Journal of statistical computation and simulation 81(11), 1565–1578.

Mises, R. v. (1936), ‘La distribution de la plus grande de n valeurs’, Revue Mathmatique de L’Union Interbalcanique 1, 141–160.

Nascimento, F. F. (2012), Modelos Probabilisticos para dados Extremos: Teoria e aplicacoes, Teresina: Piaui.

Nascimento, F. F., Gamerman, D. & Lopes, H. F. (2011), ‘Regression models for exceedance data via the full likelihood’, Environmental and ecological statistics 18(3), 495–512.

Nascimento, F. F., Gamerman, D. & Lopes, H. F. (2016), ‘Time-varying extreme pattern with dynamic models’, Test 25(1), 131–149.

Parmesan, C., Root, T. L. & Willig, M. R. (2000), ‘Impacts of extreme weather and climate on terrestrial biota’, Bulletin of the American Meteorological Society 81(3), 443–450.

Pirazzoli, P. (1982), ‘Maree estreme a venezia (periodo 1872–1981)’, Acqua Aria 10, 1023–1039.

Pirazzoli, P. (1983), ‘Flooding in venice: a worsening problem’. International Geographical Union Union, Bologna.

Sang, H. & Gelfand, A. E. (2009), ‘Hierarchical modeling for extreme values observed over space and time’, Environmental and ecological statistics 16(3), 407–426.

Shaffer, J. P. (1995), ‘Multiple hypothesis testing’, Annual review of psychology 46(1), 561–584.

Singh, V. P. (2013), Entropy theory and its application in environmental and water engineering, John Wiley & Sons.

Smith, R. L. (1984), Threshold methods for sample extremes, in ‘Statistical extremes and applications’, Springer, pp. 621–638.

Smith, R. L. (1986), ‘Extreme value theory based on the r largest annual events’, Journal of Hydrology 86(1-2), 27–43.

Soares, C. G. & Scotto, M. (2004), ‘Application of the r largest-order statistics for long-term predictions of significant wave height’, Coastal Engineering 51(5-6), 387–394.

Tawn, J. A. (1988), ‘An extreme-value theory model for dependent observations’, Journal of Hydrology 101(1-4), 227–250.

Yu, K. & Moyeed, R. A. (2001), ‘Bayesian quantile regression’, Statistics & Probability Letters 54(4), 437–447.

How to Cite

APA

Silva, R. S. & Nascimento, F. F. (2019). Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach. Revista Colombiana de Estadística, 42(2), 143–166. https://doi.org/10.15446/rce.v42n2.70271

ACM

[1]
Silva, R.S. and Nascimento, F.F. 2019. Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach. Revista Colombiana de Estadística. 42, 2 (Jul. 2019), 143–166. DOI:https://doi.org/10.15446/rce.v42n2.70271.

ACS

(1)
Silva, R. S.; Nascimento, F. F. Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach. Rev. colomb. estad. 2019, 42, 143-166.

ABNT

SILVA, R. S.; NASCIMENTO, F. F. Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach. Revista Colombiana de Estadística, [S. l.], v. 42, n. 2, p. 143–166, 2019. DOI: 10.15446/rce.v42n2.70271. Disponível em: https://revistas.unal.edu.co/index.php/estad/article/view/70271. Acesso em: 28 dec. 2025.

Chicago

Silva, Renato Santos, and Fernando Ferraz Nascimento. 2019. “Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach”. Revista Colombiana De Estadística 42 (2):143-66. https://doi.org/10.15446/rce.v42n2.70271.

Harvard

Silva, R. S. and Nascimento, F. F. (2019) “Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach”, Revista Colombiana de Estadística, 42(2), pp. 143–166. doi: 10.15446/rce.v42n2.70271.

IEEE

[1]
R. S. Silva and F. F. Nascimento, “Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach”, Rev. colomb. estad., vol. 42, no. 2, pp. 143–166, Jul. 2019.

MLA

Silva, R. S., and F. F. Nascimento. “Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach”. Revista Colombiana de Estadística, vol. 42, no. 2, July 2019, pp. 143-66, doi:10.15446/rce.v42n2.70271.

Turabian

Silva, Renato Santos, and Fernando Ferraz Nascimento. “Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach”. Revista Colombiana de Estadística 42, no. 2 (July 1, 2019): 143–166. Accessed December 28, 2025. https://revistas.unal.edu.co/index.php/estad/article/view/70271.

Vancouver

1.
Silva RS, Nascimento FF. Extreme Value Theory Applied to r Largest Order Statistics Under the Bayesian Approach. Rev. colomb. estad. [Internet]. 2019 Jul. 1 [cited 2025 Dec. 28];42(2):143-66. Available from: https://revistas.unal.edu.co/index.php/estad/article/view/70271

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CrossRef citations6

1. Salah H. Abid, Fadhl J. Kadhim. (2021). The maximum order statistic of doubly truncated Inverted Gamma distribution. Journal of Physics: Conference Series, 1999(1), p.012094. https://doi.org/10.1088/1742-6596/1999/1/012094.

2. Maashele Kholofelo Metwane, Daniel Maposa. (2023). Extreme Value Theory Modelling of the Behaviour of Johannesburg Stock Exchange Financial Market Data. International Journal of Financial Studies, 11(4), p.130. https://doi.org/10.3390/ijfs11040130.

3. Adewunmi O. Adeyemi, Ismail A. Adeleke, Eno E. E. Akarawak. (2023). Modeling Extreme Stochastic Variations using the Maximum Order Statistics of Convoluted Distributions. Journal of the Nigerian Society of Physical Sciences, , p.994. https://doi.org/10.46481/jnsps.2023.994.

4. Wyara Vanesa Moura e Silva, Fernando Ferraz do Nascimento, Marcelo Bourguignon. (2020). A change-point model for the r-largest order statistics with applications to environmental and financial data. Applied Mathematical Modelling, 82, p.666. https://doi.org/10.1016/j.apm.2020.01.064.

5. Fernando Ferraz Do Nascimento, Marcelo Bourguignon. (2020). Bayesian time‐varying quantile regression to extremes. Environmetrics, 31(2) https://doi.org/10.1002/env.2596.

6. Jax Li, Brook T. Russell, Whitney K. Huang, William C. Porter. (2024). Modeling nonstationary surface‐level ozone extremes through the lens of US air quality standards: A Bayesian hierarchical approach. Environmetrics, 35(8) https://doi.org/10.1002/env.2882.

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