Abstract
Usual estimation methods for the parameters of extreme value distributions only employ a small part of the observation values. When block maxima values are considered, many data are discarded, and therefore a lot of information is wasted. We develop a model to seize the whole data available in an extreme value framework. The key is to take advantage of the existing relation between the baseline parameters and the parameters of the block maxima distribution. We propose two methods to perform Bayesian estimation. Baseline distribution method (BDM) consists in computing estimations for the baseline parameters with all the data, and then making a transformation to compute estimations for the block maxima parameters. Improved baseline method (IBDM) is a refinement of the initial idea, with the aim of assigning more importance to the block maxima data than to the baseline values, performed by applying BDM to develop an improved prior distribution. We compare empirically these new methods with the Standard Bayesian analysis with non-informative prior, considering three baseline distributions that lead to a Gumbel extreme distribution, namely Gumbel, Exponential and Normal, by a broad simulation study.
Highlights
Extreme value theory (EVT) is a widely used statistical tool for modeling and forecasting the distributions which arise when we study events that are more extreme than any previously observed
Afterwards, making the transformation given by the relations we obtained in previous section, we can obtain new estimations for the parameters of block maxima distribution, which is the Gumbel in this case
To get a quick overview of how differences between baseline distribution data and block maxima data can affect the choice of the best method, we simulated a simple situation, when data come from a mixture of normal variables
Summary
Extreme value theory (EVT) is a widely used statistical tool for modeling and forecasting the distributions which arise when we study events that are more extreme than any previously observed. The knowledge of physical constraints, the historical evidence of data behavior or previous assessments might be an extremely important matter for the adjustment of the data, when they are not completely representative and further information is required This fact leads to the use of Bayesian inference to address the extreme value estimation [18]. Examples for its application are the modeling of annual rainfall maximum intensities [23], the estimation of the probability of exceedence of future flood discharge [24] and the forecasting of the extremes of the price distribution [25] Some of these works are focused on the construction of informative priors of the parameters for which data can provide little information. Several statistical analyses are performed to test the validity of our method and check its enhancements in relation to the standard Bayesian analysis without this information
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