Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior

Wilson Moseki Thupeng; Boikanyo Mokgweetsi; Thuto Mothupi

doi:doi:10.11648/j.ajtas.20190806.20

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Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior

Wilson Moseki Thupeng, Boikanyo Mokgweetsi, Thuto Mothupi

Published in American Journal of Theoretical and Applied Statistics (Volume 8, Issue 6)

Received: 14 October 2019 Accepted: 4 December 2019 Published: 25 December 2019

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Abstract

Extreme value modelling is widely applied in situations where accurate assessment of the behavior of a process at high levels is needed. The inherent scarcity of extreme value data, the natural objective of predicting future extreme values of a process associated with modelling of extremes and the regularity assumptions required by the likelihood and probability weighted moments methods of parameter estimation within the frequentist framework, make it imperative for a practitioner to consider Bayesian methodology when modelling extremes. Within the Bayesian paradigm, the widely used tool for assessing the fitness of a model is by using posterior predictive checks (PPCs). The method involves comparing the posterior predictive distribution of future observations to the historical data. Posterior predictive inference involves the prediction of unobserved variables in light of observed data.. This paper considers posterior predictive checks for assessing model fitness for the generalized Pareto model based on a Dirichlet process prior. The posterior predictive distribution for the Dirichlet process based model is derived. Threshold selection is done by minimizing the negative differential entropy of the Dirichlet distribution. Predictions are drawn from the Bayesian posterior distribution by Markov chain Monte Carlo simulation (Metropolis-Hastings Algorithm). Two graphical measures of discrepancy between the predicted observations and the observed values commonly applied in practical extreme value modelling are considered, the cumulative distribution function and quantile plots. To support these, the Nash-Sutcliffe coefficient of model efficiency, a numerical measure that evaluates the error in the predicted observations relative to the natural variation in the observed values is used. Finite sample performance of the proposed procedure is illustrated through simulated data. The results of the study suggest that posterior predictive checks are reasonable diagnostic tools for assessing the fit of the generalized Pareto distribution. In addition, the posterior predictive quantile plot seems to be more informative than the probability plot. Most interestingly, selecting the threshold by minimizing the negative differential entropy of a Dirichlet process has the added advantage of allowing the analyst to estimate the concentration parameter from the model, rather than specifying it as a measure of his/her belief in the proposed model as a prior guess for the unknown distribution that generated the observations. Lastly, the results of application to real life data show that the distribution of the annual maximal inflows into the Okavango River at Mohembo, Botswana, can be adequately described by the generalized Pareto distribution.

Published in	American Journal of Theoretical and Applied Statistics (Volume 8, Issue 6)
DOI	10.11648/j.ajtas.20190806.20
Page(s)	287-295
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Dirichlet Process Prior, Generalized Pareto Distribution, Markov Chain Monte Carlo, Peaks Over Threshold, Posterior Predictive Checks

References

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APA Style

Wilson Moseki Thupeng, Boikanyo Mokgweetsi, Thuto Mothupi. (2019). Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior. American Journal of Theoretical and Applied Statistics, 8(6), 287-295. https://doi.org/10.11648/j.ajtas.20190806.20

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ACS Style

Wilson Moseki Thupeng; Boikanyo Mokgweetsi; Thuto Mothupi. Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior. Am. J. Theor. Appl. Stat. 2019, 8(6), 287-295. doi: 10.11648/j.ajtas.20190806.20

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AMA Style

Wilson Moseki Thupeng, Boikanyo Mokgweetsi, Thuto Mothupi. Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior. Am J Theor Appl Stat. 2019;8(6):287-295. doi: 10.11648/j.ajtas.20190806.20

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@article{10.11648/j.ajtas.20190806.20,
  author = {Wilson Moseki Thupeng and Boikanyo Mokgweetsi and Thuto Mothupi},
  title = {Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {8},
  number = {6},
  pages = {287-295},
  doi = {10.11648/j.ajtas.20190806.20},
  url = {https://doi.org/10.11648/j.ajtas.20190806.20},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20190806.20},
  abstract = {Extreme value modelling is widely applied in situations where accurate assessment of the behavior of a process at high levels is needed. The inherent scarcity of extreme value data, the natural objective of predicting future extreme values of a process associated with modelling of extremes and the regularity assumptions required by the likelihood and probability weighted moments methods of parameter estimation within the frequentist framework, make it imperative for a practitioner to consider Bayesian methodology when modelling extremes. Within the Bayesian paradigm, the widely used tool for assessing the fitness of a model is by using posterior predictive checks (PPCs). The method involves comparing the posterior predictive distribution of future observations to the historical data. Posterior predictive inference involves the prediction of unobserved variables in light of observed data.. This paper considers posterior predictive checks for assessing model fitness for the generalized Pareto model based on a Dirichlet process prior. The posterior predictive distribution for the Dirichlet process based model is derived. Threshold selection is done by minimizing the negative differential entropy of the Dirichlet distribution. Predictions are drawn from the Bayesian posterior distribution by Markov chain Monte Carlo simulation (Metropolis-Hastings Algorithm). Two graphical measures of discrepancy between the predicted observations and the observed values commonly applied in practical extreme value modelling are considered, the cumulative distribution function and quantile plots. To support these, the Nash-Sutcliffe coefficient of model efficiency, a numerical measure that evaluates the error in the predicted observations relative to the natural variation in the observed values is used. Finite sample performance of the proposed procedure is illustrated through simulated data. The results of the study suggest that posterior predictive checks are reasonable diagnostic tools for assessing the fit of the generalized Pareto distribution. In addition, the posterior predictive quantile plot seems to be more informative than the probability plot. Most interestingly, selecting the threshold by minimizing the negative differential entropy of a Dirichlet process has the added advantage of allowing the analyst to estimate the concentration parameter from the model, rather than specifying it as a measure of his/her belief in the proposed model as a prior guess for the unknown distribution that generated the observations. Lastly, the results of application to real life data show that the distribution of the annual maximal inflows into the Okavango River at Mohembo, Botswana, can be adequately described by the generalized Pareto distribution.},
 year = {2019}
}

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TY  - JOUR
T1  - Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior
AU  - Wilson Moseki Thupeng
AU  - Boikanyo Mokgweetsi
AU  - Thuto Mothupi
Y1  - 2019/12/25
PY  - 2019
N1  - https://doi.org/10.11648/j.ajtas.20190806.20
DO  - 10.11648/j.ajtas.20190806.20
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 287
EP  - 295
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20190806.20
AB  - Extreme value modelling is widely applied in situations where accurate assessment of the behavior of a process at high levels is needed. The inherent scarcity of extreme value data, the natural objective of predicting future extreme values of a process associated with modelling of extremes and the regularity assumptions required by the likelihood and probability weighted moments methods of parameter estimation within the frequentist framework, make it imperative for a practitioner to consider Bayesian methodology when modelling extremes. Within the Bayesian paradigm, the widely used tool for assessing the fitness of a model is by using posterior predictive checks (PPCs). The method involves comparing the posterior predictive distribution of future observations to the historical data. Posterior predictive inference involves the prediction of unobserved variables in light of observed data.. This paper considers posterior predictive checks for assessing model fitness for the generalized Pareto model based on a Dirichlet process prior. The posterior predictive distribution for the Dirichlet process based model is derived. Threshold selection is done by minimizing the negative differential entropy of the Dirichlet distribution. Predictions are drawn from the Bayesian posterior distribution by Markov chain Monte Carlo simulation (Metropolis-Hastings Algorithm). Two graphical measures of discrepancy between the predicted observations and the observed values commonly applied in practical extreme value modelling are considered, the cumulative distribution function and quantile plots. To support these, the Nash-Sutcliffe coefficient of model efficiency, a numerical measure that evaluates the error in the predicted observations relative to the natural variation in the observed values is used. Finite sample performance of the proposed procedure is illustrated through simulated data. The results of the study suggest that posterior predictive checks are reasonable diagnostic tools for assessing the fit of the generalized Pareto distribution. In addition, the posterior predictive quantile plot seems to be more informative than the probability plot. Most interestingly, selecting the threshold by minimizing the negative differential entropy of a Dirichlet process has the added advantage of allowing the analyst to estimate the concentration parameter from the model, rather than specifying it as a measure of his/her belief in the proposed model as a prior guess for the unknown distribution that generated the observations. Lastly, the results of application to real life data show that the distribution of the annual maximal inflows into the Okavango River at Mohembo, Botswana, can be adequately described by the generalized Pareto distribution.
VL  - 8
IS  - 6
ER  -

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Author Information

Wilson Moseki Thupeng

Department of Statistics, University of Botswana, Gaborone, Botswana
Boikanyo Mokgweetsi

Department of Statistics, University of Botswana, Gaborone, Botswana
Thuto Mothupi

Department of Statistics, University of Botswana, Gaborone, Botswana

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APA Style

Wilson Moseki Thupeng, Boikanyo Mokgweetsi, Thuto Mothupi. (2019). Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior. American Journal of Theoretical and Applied Statistics, 8(6), 287-295. https://doi.org/10.11648/j.ajtas.20190806.20

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ACS Style

Wilson Moseki Thupeng; Boikanyo Mokgweetsi; Thuto Mothupi. Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior. Am. J. Theor. Appl. Stat. 2019, 8(6), 287-295. doi: 10.11648/j.ajtas.20190806.20

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AMA Style

Wilson Moseki Thupeng, Boikanyo Mokgweetsi, Thuto Mothupi. Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior. Am J Theor Appl Stat. 2019;8(6):287-295. doi: 10.11648/j.ajtas.20190806.20

Copy | Download

@article{10.11648/j.ajtas.20190806.20,
  author = {Wilson Moseki Thupeng and Boikanyo Mokgweetsi and Thuto Mothupi},
  title = {Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior},
  journal = {American Journal of Theoretical and Applied Statistics},
  volume = {8},
  number = {6},
  pages = {287-295},
  doi = {10.11648/j.ajtas.20190806.20},
  url = {https://doi.org/10.11648/j.ajtas.20190806.20},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajtas.20190806.20},
  abstract = {Extreme value modelling is widely applied in situations where accurate assessment of the behavior of a process at high levels is needed. The inherent scarcity of extreme value data, the natural objective of predicting future extreme values of a process associated with modelling of extremes and the regularity assumptions required by the likelihood and probability weighted moments methods of parameter estimation within the frequentist framework, make it imperative for a practitioner to consider Bayesian methodology when modelling extremes. Within the Bayesian paradigm, the widely used tool for assessing the fitness of a model is by using posterior predictive checks (PPCs). The method involves comparing the posterior predictive distribution of future observations to the historical data. Posterior predictive inference involves the prediction of unobserved variables in light of observed data.. This paper considers posterior predictive checks for assessing model fitness for the generalized Pareto model based on a Dirichlet process prior. The posterior predictive distribution for the Dirichlet process based model is derived. Threshold selection is done by minimizing the negative differential entropy of the Dirichlet distribution. Predictions are drawn from the Bayesian posterior distribution by Markov chain Monte Carlo simulation (Metropolis-Hastings Algorithm). Two graphical measures of discrepancy between the predicted observations and the observed values commonly applied in practical extreme value modelling are considered, the cumulative distribution function and quantile plots. To support these, the Nash-Sutcliffe coefficient of model efficiency, a numerical measure that evaluates the error in the predicted observations relative to the natural variation in the observed values is used. Finite sample performance of the proposed procedure is illustrated through simulated data. The results of the study suggest that posterior predictive checks are reasonable diagnostic tools for assessing the fit of the generalized Pareto distribution. In addition, the posterior predictive quantile plot seems to be more informative than the probability plot. Most interestingly, selecting the threshold by minimizing the negative differential entropy of a Dirichlet process has the added advantage of allowing the analyst to estimate the concentration parameter from the model, rather than specifying it as a measure of his/her belief in the proposed model as a prior guess for the unknown distribution that generated the observations. Lastly, the results of application to real life data show that the distribution of the annual maximal inflows into the Okavango River at Mohembo, Botswana, can be adequately described by the generalized Pareto distribution.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Posterior Predictive Checks for the Generalized Pareto Distribution Based on a Dirichlet Process Prior
AU  - Wilson Moseki Thupeng
AU  - Boikanyo Mokgweetsi
AU  - Thuto Mothupi
Y1  - 2019/12/25
PY  - 2019
N1  - https://doi.org/10.11648/j.ajtas.20190806.20
DO  - 10.11648/j.ajtas.20190806.20
T2  - American Journal of Theoretical and Applied Statistics
JF  - American Journal of Theoretical and Applied Statistics
JO  - American Journal of Theoretical and Applied Statistics
SP  - 287
EP  - 295
PB  - Science Publishing Group
SN  - 2326-9006
UR  - https://doi.org/10.11648/j.ajtas.20190806.20
AB  - Extreme value modelling is widely applied in situations where accurate assessment of the behavior of a process at high levels is needed. The inherent scarcity of extreme value data, the natural objective of predicting future extreme values of a process associated with modelling of extremes and the regularity assumptions required by the likelihood and probability weighted moments methods of parameter estimation within the frequentist framework, make it imperative for a practitioner to consider Bayesian methodology when modelling extremes. Within the Bayesian paradigm, the widely used tool for assessing the fitness of a model is by using posterior predictive checks (PPCs). The method involves comparing the posterior predictive distribution of future observations to the historical data. Posterior predictive inference involves the prediction of unobserved variables in light of observed data.. This paper considers posterior predictive checks for assessing model fitness for the generalized Pareto model based on a Dirichlet process prior. The posterior predictive distribution for the Dirichlet process based model is derived. Threshold selection is done by minimizing the negative differential entropy of the Dirichlet distribution. Predictions are drawn from the Bayesian posterior distribution by Markov chain Monte Carlo simulation (Metropolis-Hastings Algorithm). Two graphical measures of discrepancy between the predicted observations and the observed values commonly applied in practical extreme value modelling are considered, the cumulative distribution function and quantile plots. To support these, the Nash-Sutcliffe coefficient of model efficiency, a numerical measure that evaluates the error in the predicted observations relative to the natural variation in the observed values is used. Finite sample performance of the proposed procedure is illustrated through simulated data. The results of the study suggest that posterior predictive checks are reasonable diagnostic tools for assessing the fit of the generalized Pareto distribution. In addition, the posterior predictive quantile plot seems to be more informative than the probability plot. Most interestingly, selecting the threshold by minimizing the negative differential entropy of a Dirichlet process has the added advantage of allowing the analyst to estimate the concentration parameter from the model, rather than specifying it as a measure of his/her belief in the proposed model as a prior guess for the unknown distribution that generated the observations. Lastly, the results of application to real life data show that the distribution of the annual maximal inflows into the Okavango River at Mohembo, Botswana, can be adequately described by the generalized Pareto distribution.
VL  - 8
IS  - 6
ER  -

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