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.

Highlights

  • Modelling extreme values is widely applied in situations, like civil engineering design, where accurate assessment of the behavior of a process at high levels is required

  • Distribution Based on a Dirichlet Process Prior for the statistics practitioner to consider including other available prior information in the analysis, (ii) the main objective of an extreme value analysis is to estimate the probability of future events reaching extreme levels and predicting high quantiles and, one approach to prediction is through the predictive distribution, which is naturally Bayesian and, (iii) the Bayesian methodology does not require regularity assumptions required by the maximum likelihood and probability weighted moments methods of estimation

  • It is for these reasons that the present paper focusses on using posterior predictive checks for assessing model-fit for the generalized Pareto distribution, a family of continuous probability distributions that is popularly used to model the tails of distributions of extreme values

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Summary

Introduction

Modelling extreme values is widely applied in situations, like civil engineering design, where accurate assessment of the behavior of a process at high levels is required. Distribution Based on a Dirichlet Process Prior for the statistics practitioner to consider including other available prior information in the analysis, (ii) the main objective of an extreme value analysis is to estimate the probability of future events reaching extreme levels and predicting high quantiles and, one approach to prediction is through the predictive distribution, which is naturally Bayesian and, (iii) the Bayesian methodology does not require regularity assumptions required by the maximum likelihood and probability weighted moments methods of estimation It is for these reasons that the present paper focusses on using posterior predictive checks for assessing model-fit for the generalized Pareto distribution, a family of continuous probability distributions that is popularly used to model the tails of distributions of extreme values.

The Generalized Pareto Model
The Dirichlet Process
The Negative Differential Entropy of the Dirichlet Distribution
The Predictive Distribution
Model Assessment
Application
Conclusions
Full Text
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