Risk Analysis Method by the Extreme Data of Dependent Exogenous Variables


DOI: https://doi.org/10.14313/JAMRIS/3-2021/18
Keywords: exogenous variables, risk-oriented process approach, extreme value theory, tailed distribution


Many practical tasks of data multivariate statistical analysis from the standpoint of a risk-oriented process approach (in accordance with ISO 9001: 2015, 31000: 2018) requires the definition of the risk values for the dependent exogenous variables of some processes. This paper proposes the method, which consist of original stages sequence for calculating value-at-risk (VaR) or conditional-value-at-risk (CVaR) of dependent exogenous variables, presented of the extreme data frame of critical manufacture process parameters or other parameters, for example, extreme data of environmental monitoring and etc. Risk analysis method by the extreme data of dependent exogenous variables, presented of the data matrix, uses the result of solving the formalized problem of defines the tails parameters of the joint distributions of exogenous variables as components of a bivariate random variable. It can be argued that the tails parameters of the joint distributions of dependent exogenous variables make the validated corrections of the VaR and CVaR estimates for such variables. This method expands the practical application of extreme value theory for the value at risk analysis of any dependent variables as process parameters.

C. Lleras, “Path Analysis”. In: Encyclopedia of Social Measurement, Elsevier, 2005, 25–30, 10.1016/B0-12-369398-5/00483-7.

“ISO 9000, Introduction and Support Package: Guidance on the Concept and Use of the Process Approach for management systems”. ISO/TC 176/SC 2/N 544R3, https://www.iso.org/files/live/sites/isoorg/files/archive/pdf/en/04_concept_and_use_of_the_process_approach_for_management_systems.pdf. Accessed on: 2022-04-13.

“ISO 9001:2015 Quality management systems”. https://www.iso.org/standard/62085.html. Accessed on: 2022-04-13.

“The new ISO 31000 keeps risk management simple”. https://www.iso.org/cms/render/live/en/sites/isoorg/contents/news/2018/02/Ref2263.html. Accessed on: 2022-04-13.

A. Stephenson and E. Gilleland, “Software for the analysis of extreme events: The current state and future directions”, Extremes, vol. 8, 2005, 87–109, 10.1007/s10687-006-7962-0.

E. Gilleland, M. Ribatet and A. G. Stephenson, “A software review for extreme value analysis”, Extremes, vol. 16, no. 1, 2013, 103–119, 10.1007/s10687-012-0155-0.

M. I. Gomes and A. Guillou, “Extreme Value Theory and Statistics of Univariate Extremes: A Review”, International Statistical Review, vol. 83, no. 2, 2015, 263–292, 10.1111/insr.12058.

P. Abad, S. Benito and C. López, “A comprehensive review of Value at Risk methodologies”, The Spanish Review of Financial Economics, vol. 12, no. 1, 2014, 15–32, 10.1016/j.srfe.2013.06.001.

D. K. Dey and J. Yan, Extreme Value Modeling and Risk Analysis: Methods and Applications, Chapman and Hall/CRC, 2016, 10.1201/b19721.

N. G. Zrazhevska and A. G. Zrazhevsky, “Classification of methods for risk measures VaR and CVaR calculation and estimation”, System research and information technologies, no. 3, 2016, 126–141, 10.20535/SRIT.2308-8893.2016.3.11.

B. G. Peterson et al., “PerformanceAnalytics: Econometric Tools for Performance and Risk Analysis”, https://CRAN.R-project.org/package=PerformanceAnalytics. Accessed on: 2022-04-13.

J. Mina and J. Y. Xiao, “Return to RiskMetrics: The Evolution of a Standard”. https://www.msci.com/documents/10199/dbb975aa-5dc2-4441-aa2d-ae34ab5f0945. Accessed on: 2022-04-13.

B. Cordeiro and A. Kotoky, “MSCI to buy RiskMetrics for $1.55 billion | Reuters”. https://www.reuters.com/article/us-riskmetrics-msci-idUSTRE62041J20100301. Accessed on: 2022-04-13.

L. Rickenberg, “Tail Risk Targeting: Target VaR and CVaR Strategies”. https://papers.ssrn.com/abstract=3444999, DOI: 10.2139/ssrn.3444999. Accessed on: 2022-04-13.

D. Wuertz, T. Setz and Y. Chalabi, “fExtremes: Rmetrics - Modelling Extreme Events in Finance”. https://CRAN.R-project.org/package=fExtremes. Accessed on: 2022-04-13.

B. Pfaff et al., “evir: Extreme Values in R”. https:// CRAN.R-project.org/package=evir. Accessed on: 2022-04-13.

A. J. McNeil and R. Frey, “Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach”, Journal of Empirical Finance, vol. 7, no. 3-4, 2000, 271–300, 10.1016/S0927-5398(00)00012-8.

“ismev: An Introduction to Statistical Modeling of Extreme Values”. https://CRAN.R-project.org/package=ismev. Accessed on: 2022-04-13.

S. Coles, An Introduction to Statistical Modeling of Extreme Values, Springer London, 2001, 10.1007/978-1-4471-3675-0.

T. Reynkens et al., “ReIns: Functions from “Reinsurance: Actuarial and Statistical Aspects”.” https://CRAN.R-project.org/package=ReIns. Accessed on: 2022-04-13.

T. Reynkens, R. Verbelen, J. Beirlant and K. Antonio, “Modelling censored losses using splicing: A global fit strategy with mixed Erlang and extreme value distributions”, Insurance Mathematics and Economics, vol. 77, 2017, 65–77, 10.1016/j.insmatheco.2017.08.005.

H. Southworth et al., “texmex: Statistical Modelling of Extreme Values”. https://CRAN.R-project.org/package=texmex. Accessed on: 2022-04-13.

C. A. T. Ferro and J. Segers, “Inference for clusters of extreme values”, Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 65, no. 2, 2003, 545–556, 10.1111/1467-9868.00401.

M. Ferreira, “Analysis of estimation methods for the extremal index”, Electronic Journal of Applied Statistical Analysis, vol. 11, no. 1, 2018, 296–306. 10.1285/i20705948v11n1p296.

S. Coles, J. Heffernan and J. Tawn, “Dependence Measures for Extreme Value Analyses”, Extremes, vol. 2, no. 4, 1999, 339–365, 10.1023/A:1009963131610.

J. E. Heffernan and J. A. Tawn, “A conditional approach for multivariate extreme values (with discussion)”, Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 66, no. 3, 2004, 497–546, 10.1111/j.1467-9868.2004.02050.x.

T. Lugrin, “tsxtreme: Bayesian Modelling of Extremal Dependence in Time Series”. https://CRAN.R-project.org/package=tsxtreme. Accessed on: 2022-04-13.

E. Zivot and J. Wang, Modeling Financial Time Series with S-PLUS®, Springer New York, 2006, 10.1007/978-0-387-32348-0.

N. Bezak, M. Brilly and M. Šraj, “Comparison between the peaks-over-threshold method and the annual maximum method for flood frequency analysis”, Hydrological Sciences Journal, vol. 59, no. 5, 2014, 959–977, 10.1080/02626667.2013.831174.

S. N. Majumdar, A. Pal and G. Schehr, “Extreme value statistics of correlated random variables: A pedagogical review”, Physics Reports, vol. 840, 2020, 1–32, 10.1016/j.physrep.2019.10.005.

R. S. Tsay, “Testing serial correlations in high-dimensional time series via extreme value theory”, Journal of Econometrics, vol. 216, no. 1, 2020, 106–117, 10.1016/j.jeconom.2020.01.008.

E. C. Brechmann and U. Schepsmeier, “Modeling Dependence with C- and D-Vine Copulas: The R Package CDVine”, Journal of Statistical Software, vol. 52, no. 3, 2013, 10.18637/jss.v052.i03.

L. Deng, C. Ma and W. Yang, “Portfolio Optimization via Pair Copula-GARCH-EVT-CVaR Model”, Systems Engineering Procedia, vol. 2, 2011, 171–181, 10.1016/j.sepro.2011.10.020.

G. Meissner, Correlation Risk Modeling and Management: an applied guide including the Basel III correlation framework – with Interactive Correlation Models in Excel/VBA, John Wiley & Sons, 2014, 10.1002/9781118809204.

J. Carreau and G. Toulemonde, “Extra-parametrized extreme value copula: Extension to a spatial framework”, Spatial Statistics, vol. 40, 2020, 10.1016/j.spasta.2020.100410.

R. B. Nelsen, An Introduction to Copulas, Springer New York, 2006, 10.1007/0-387-28678-0.

N. Quinn, P. D. Bates, J. Neal, A. Smith, O. Wing, C. Sampson, J. Smith and J. Heffernan, “The Spatial Dependence of Flood Hazard and Risk in the United States”, Water Resources Research, vol. 55, no. 3, 2019, 1890–1911, 10.1029/2018WR024205.

C. A. T. Ferro, “Statistical Methods for Clusters of Extreme Values”, PhD Thesis, Lancaster University, September 2003, http://empslocal.ex.ac.uk/people/staff/ferro/Publications/Thesis/thesis.pdf. Accessed on: 2022-04-13.

L. de Haan and S. I. Resnick, “Limit theory for multivariate sample extremes”, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 40, no. 4, 1977, 317–337, 10.1007/ BF00533086.

Y. Bensalah, “Steps in Applying Extreme Value Theory to Finance: A Review”. https://www.banqueducanada.ca/wp-content/uploads/2010/01/wp00-20.pdf. Accessed on: 2022-04-13.

I. V. Tereshchenko, A. I. Tereshchenko and S. V. Shtangey, “Risks Estimation Method by Clustered Extreme Data of Process Covariates”, Radio Electronics, Computer Science, Control, no. 2, 2020, 51–64, 10.15588/1607-3274-2020-2-6.