An improved method for the estimation of the Gumbel distribution parameters
Rubén Gómez González, M. Isabel Parra, Francisco Javier Acero, Jacinto Martín
AAn improved method for the estimation of the Gumbeldistribution parameters (cid:73)
Rub´en G´omez Gonz´alez a, ∗ , M. Isabel Parra b,c , Francisco Javier Acero a,d ,Jacinto Mart´ın b,e a Departamento de F´ısica, Universidad de Extremadura, Avenida de Elvas, 06006 Badajoz,Spain b Departamento de Matem´aticas, Universidad de Extremadura, Avenida de Elvas, 06006Badajoz, Spain c Instituto de Investigaci´on de Matem´aticas de la Universidad de Extremadura (IMUEX),Universidad de Extremadura, Avenida de Elvas, 06006 Badajoz, Spain d Instituto Universitario de Investigaci´on del Agua, Cambio Clim´atico y Sostenibilidad(IACYS), Universidad de Extremadura, Avenida de Elvas, 06006 Badajoz, Spain e Instituto Universitario de Computaci´on Cient´ıfica Avanzada de Extremadura (ICCAEX),Universidad de Extremadura, Avenida de Elvas, 06006 Badajoz, Spain
Abstract
Usual estimation methods for the parameters of extreme values distributionemploy only a few values, wasting a lot of information. More precisely, in thecase of the Gumbel distribution, only the block maxima values are used. In thiswork, we propose a method to seize all the available information in order toincrease the accuracy of the estimations. This intent can be achieved by takingadvantage of the existing relationship between the parameters of the baselinedistribution, which generates data from the full sample space, and the ones forthe limit Gumbel distribution. In this way, an informative prior distribution canbe obtained. Different statistical tests are used to compare the behaviour of ourmethod with the standard one, showing that the proposed method performs wellwhen dealing with very shortened available data. The empirical effectivenessof the approach is demonstrated through a simulation study and a case study.Reduction in the credible interval width and enhancement in parameter locationshow that the results with improved prior adapt to very shortened data better ∗ Corresponding author
Email addresses: [email protected] (Rub´en G´omez Gonz´alez), [email protected] (M. IsabelParra), [email protected] (Francisco Javier Acero), [email protected] (Jacinto Mart´ın)
Preprint submitted to Computational Statistics & Data Analysis February 22, 2019 a r X i v : . [ phy s i c s . d a t a - a n ] F e b han standard method does. Keywords:
Bayesian inference, Gumbel distribution, Metropoli-Hastingsalgorithm, Small dataset, Highly informative prior
1. Introduction
Extreme value theory (EVT) is a widely used statistical tool for modelingand forecasting an extreme value distribution of a given sample generated by abaseline distribution. EVT is employed in several scientific fields, for instancein climate change through studies of extreme events of temperature [1, 2, 3],precipitation [4, 5, 6, 7, 8, 9], and solar climatology [10, 11, 12], or engineeringwhere it is taken into account to design, for example, modern buildings [13].The extreme values depend upon the full sample space from which they havebeen drawn through its shape and size. Therefore, extremes variate accordingto the initial distribution and the sample size [14].The Gumbel distribution belongs to the Generalized Extreme Value distri-bution group (GEV) used to shape the maximum (or minimum) distributionof a sample space which can arise from various baseline distributions. By be-ing able to describe extreme events of the normal or exponential type [15], theGumbel distribution is greatly useful for modeling experimental and social data,the main two areas in which EVT is used. In order to estimate maximum datadistribution, both frequentist and Bayesian approaches have been developed[16, 17]. However, the knowledge of physical constraints, the historical evidenceof data behaviour, or previous assessments might be an extremely importantmatter for the adjustment of the data, particularly when it is not completelyrepresentative and further information is required. This fact leads to the use ofBayesian inference to address the extreme value estimation [18].Very few attempts have been made to incorporate Bayesian methodologyinto extreme value analysis. Nevertheless, practical use of Bayesian estimationis often associated with difficulties to choose prior information and prior dis-tribution for Gumbel parameters [19]. To fix this problem, several alternatives2ave been proposed, either by focusing exclusively on the selection of the priordensity [20, 21] or by improving the algorithm for the estimation of the parame-ters [22]. Even so, the lack of information still seems to be the weak point whenreferring to extreme value inference. In this paper, a known baseline distributionfor the complete dataset is assumed and a relationship between its parametersand the parameters of the Gumbel distribution is established through a bayesianiterative procedure. Thus, the aim of this work is to obtain a more accuratemodel for the estimation of the Gumbel parameters by considering a highly in-formative prior based on the connection between the distribution parameters.The use of the entire data, instead of the selected maximum data, provides anupgraded method which results to adapt quite well to very shortened availabledata.More precisely, a regression analysis is implemented to estimate a relationbetween the parameters of the initial distribution and the parameters of theextreme distribution for a given sample size. Once this connection is obtained,the next step is to obtain the posterior distribution of the parameters of theGumbel distribution. To perform this task, we need to use a MCMC method,concretely a Metropolis Hastings algorithm. Therefore, the proposed method isa regression Metropolis-Hastings algorithm (RMH). Several statistical analysisare performed to test the validity of our method (RMH) and to check its en-hancements in relation to the Standard Metropolis Hastings (SMH) algorithm.The paper is organized as follows. In Section 2 the asymptotic model usedto describe extremal behaviour is outlined. Specifically, Gumbel distributionand Bayesian inference for the estimation of its parameters are introduced. InSection 3, the regression analysis performed to improve MCMC procedure isproposed and implemented, and its coefficients are implemented into the genericmodel which is described in Section 4. The results of our analysis are presentedin Section 5. Here, we highlight the advantages that the knowledge of theregression coefficients has in this extreme value context over the using of non-informative priors. For the sake of completeness, Section 6 gives a real exampleof application of the model. 3 . The block maxima method in extreme value theory
In extreme value theory, there are two fundamental ways to model the limitdistribution: the block maxima (BM) method and the peaks-over-threshold(POT) method. We center our work on the former, which is a rather efficientmethod under usual practical conditions. BM method may be preferable thanPOT in several situations, specially when the only available info is block max-ima, seasonal periodicity is presented, or the block periods appear naturally.The BM approach consists of dividing the observation period into non-overlapping periods of equal size and look at the maximum observation in eachperiod. Let (cid:101) X , (cid:101) X , ..., (cid:101) X m be i.i.d. random variables with distribution function F . Given a fixed k ∈ N , we define the block maxima X i = max ( i − k 3. Bayesian estimation of the Gumbel distribution Many procedures have been suggested, optimized, or discarded to enhanceBayesian analysis of Gumbel distributed data. Some examples are the modelingof annual rainfall maximum intensities [23], the estimation of the probability ofexceedance of future flood discharge [24], or the forecasting of the extremes ofthe price distribution [25]. Some of these works are focused on the construc-tion of informative priors of the parameters for which data can provide littleinformation. Despite these previous efforts, it is well understood that someconstraints to quantify qualitative knowledge always appear when referring to4onstruct informative priors. For this reason, this paper focuses on BayesianMCMC techniques based on MH algorithms to estimate the parameters usingnon-informative prior [26].In order to make statistical inferences based on the Bayesian framework,after assuming a prior density for the parameters, π ( θ ), and combining thisdistribution with the information brought by the data which is quantified bythe likelihood function, L ( θ | x ), the posterior density function of the parameterscan be determined as follows π ( θ | x ) ∝ L ( θ | x ) π ( θ ) . (2)The remaining of the inference process is fulfilled based on the obtained posteriordistribution.Given the random sample x = ( x , ..., x n ) from a Gumbel ( µ, σ ) distribution,with cdf and density function given by F ( x | µ, σ ) = exp (cid:18) − exp (cid:18) − x − µσ (cid:19)(cid:19) (3)and f ( x | µ, σ ) = 1 σ exp (cid:18) − exp (cid:18) − x − µσ (cid:19) − x − µσ (cid:19) (4)respectively, where µ ∈ R , σ ∈ R ∗ + and x can take any value in R , the likelihoodfunction for θ = ( µ, σ ) is L ( µ, σ | x ) = 1 σ n exp (cid:32) − n (cid:88) i =1 exp (cid:18) − x i − µσ (cid:19) − n (cid:88) i =1 x i − µσ (cid:33) . (5)Based on Rostami and Adam [21], where eighteen pairs of priors were se-lected for Gumbel distribution and compared the posterior estimations by ap-plying Metropolis-Hastings algorithm, in this model we choose the combinationof Gumbel and Rayleigh as the most productive pair of priors, namely, π ( µ ) ∝ exp (cid:18) − exp (cid:18) − µ − µ σ (cid:19) − µ − µ σ (cid:19) ,π ( σ ) ∝ σ exp (cid:18) − σ λ (cid:19) . (6)5o, the posterior distribution is π ( µ, σ | x ) ∝ σ n − exp (cid:18) A − exp (cid:18) − µ − µ σ (cid:19) − µ − µ σ − σ λ (cid:19) , (7)and the marginal posterior distributions are given by π ( µ | x ) ∝ exp (cid:18) A − exp (cid:18) − µ − µ σ (cid:19) − µ − µ σ (cid:19) , (8)and π ( σ | x ) ∝ σ n − exp (cid:18) A − σ λ (cid:19) . (9)Here, A = − (cid:80) ni =1 exp (cid:18) − x i − µσ (cid:19) − (cid:80) ni =1 x i − µσ .The Metropolis-Hasting algorithm (MH) is a Markov chain Monte Carlo(MCMC) method for collecting a sequence of random samples from a probabilitydistribution where direct sampling is difficult [27]. In our context, the simplesteps of MH algorithm can be summarized as follows:1. Choose initial values: µ (0) , σ (0) 2. Given the chain is currently at µ ( j ) , σ ( j ) : • Draw a candidate µ can , σ can for the next sample, by picking from thenormal distribution for some suitable chosen variances v µ and v σ , µ can ∼ N ( µ ( j ) , v µ ) and σ can ∼ N ( σ ( j ) , v σ ) • Accept µ can with probability P µ = min (cid:26) , π ( µ can | σ ( j ) ) π ( µ ( j ) | σ ( j ) ) (cid:27) ;wherelog (cid:18) π ( µ can | σ ( j ) ) π ( µ ( j ) | σ ( j ) ) (cid:19) = nσ ( µ can − µ ( j ) ) + µ ( j ) − µ can σ +exp (cid:18) − µ ( j ) − µ σ (cid:19) − exp (cid:18) − µ can − µ σ (cid:19) + n (cid:88) i =1 (cid:18) exp (cid:18) − x i − µ ( j ) σ ( j ) (cid:19) − exp (cid:18) − x i − µ can σ ( j ) (cid:19)(cid:19) (10)6 Accept σ can with probability P σ = min (cid:26) , π ( σ can | µ ( j ) ) π ( σ ( j ) | µ ( j ) ) (cid:27) wherelog (cid:18) π ( σ can | µ ( j ) ) π ( σ ( j ) | µ ( j ) ) (cid:19) = (1 − n )(log σ can − log σ ( j ) ) + (cid:0) σ ( j ) (cid:1) − ( σ can ) λ + n (cid:88) i =1 (cid:18)(cid:16) x i − µ ( j ) (cid:17) (cid:18) σ ( j ) − σ can (cid:19)(cid:19) + n (cid:88) i =1 (cid:18) exp (cid:18) − x i − µ ( j ) σ ( j ) (cid:19) − exp (cid:18) − x i − µ ( j ) σ can (cid:19)(cid:19) (11)This is implemented by drawing u µ , u σ ∼ U (0 , 1) and taking µ ( j +1) = µ can if and only if u µ < P µ and σ ( j +1) = σ can if and only if u σ < P σ .3. Iterate the former procedure.When performing the MH algorithm some aspects must be considered. First,samples are highly correlated, so most of the MH draws have to be dropped bysetting a suitable number d (usually determined by analyzing the autocorrelationbetween samples), consequently, the n × d th draws are the only archived data.Second, the choice of the initial values could be far from the real one. As aresult, a burn-in period is necessary to pick values in the convergence plateau. 4. Regression analysis As aforementioned, the aim of this paper is to find a relationship betweenthe parameters of the Gumbel distribution and those ones of the baseline distri-bution. The easiest way to do so is by running the MH iterative procedure for abig enough sequence of parameters belonging to the distribution which generatethe maximum dataset. Convergence of the distribution function for the extremevalues is guaranteed for high sample sizes. Hence, the larger number of data is,the more accurate the regression coefficients will be. Furthermore, following the7nalysis done by Gumbel [14], the relation between the parameters will dependon the length of each sample or block ( k ) available to carry out the maximumselection and on the number of data ( n ), of course. One of the most distinctive features of the Gumbel distribution is its abil-ity to represent the limit distribution for the maximum values of a wide rangeof baseline distributions. normal and exponential distribution fall within thisrange and provide mono and multi-parametric performances examples since theydepend on the rate λ , and the mean, µ N , and standard deviation, σ N , param-eters, respectively. For the sake of convenience, a 90 sample size data for eachdistribution is selected in order to adapt the results to seasonal studies . Note that the sample size is a not alterable variable in all the studies and algorithms.Changes to this quantity will suppose a convergence to a Gumbel distribution with differentparameters and, as a consequence, regression coefficients will change. l m l m l s l s Figure 1: Relation between the parameter of the exponential distribution (with values for1 /λ from 0.1 to 10, by 0.1 regular increments) and µ (upper figures) and σ (lower figures)parameters of the Gumbel distribution from 10 replicates of n = 10 (left) and n = 1000 (right)blocks of size k = 90. Figures 1 and 2 illustrate the results from simulations for exponential andnormal distribution, respectively. These plots consist in 10 replicate simulationresults per parameter value with the aim of reducing the estimation uncertainty.As we can observe from the graphics, the existence of a clear relationship for theparameters is apparent. Moreover, their connections are not whichever but alinear one which will definitely simplify the priors implementation in successivealgorithms. In addition, while increasing the number of samples, data dispersionsubstantially decreases. In this sense, posterior studies will focus on the use ofinformation coming from data with high sample numbers. Regression resultsare summarized on Table 1. 9 s N m N µ s N m N µ −10103050 s N m N s −10103050 s N m N s Figure 2: Relation between the mean µ N (values from -20 to 50, by 1 regular increments)and standard deviation σ N (values from 0 to 10, by 0.5 regular increments) parameters of thenormal distribution and µ (upper figures) and σ (lower figures) parameters of the Gumbeldistribution from 10 replicates of n = 10 (left) or n = 1000 (right) samples of size k = 90. Outcomes given in Table 1 show a clear relationship between the parametersof the baseline distributions and the GEV ones. This information will be usedin posterior studies to obtain informative priors based on data. Moreover, thesepriors can be used to generate candidates on the MH algorithm with the aim toaccelerate convergence.Although in this work we only display the results obtained for the normaland exponential distributions, this strategy could be easily adapted to otherdistributions. 5. Regression Metropolis-Hasgtings method In the previous section, it has been established a relation between the pa-rameters of the two most widely used baseline distributions and the Gumbel10 able 1: Summarized samples from the posterior distribution of a linear regression to modelthe relationship between the Gumbel distribution parameters and exponential or normal base-line distribution parameters, obtained using the function MCMCregress by the R package MCM-Cpack [28]. Data frame is the table compound by n = 1000 samples of size k = 90. Exponential DistributionParameter of the Gumbel Distribution µ σ Summarized posterior sample Mean SD Mean SDIntercept -0.004409 0.012152 0.004305 0.00861361 /λ µ σ Summarized posterior sample Mean SD Mean SDIntercept 0.0001740 1.234e-03 -4.909e-04 9.153e-04 µ N σ N ones. Although the procedure can be applied to any baseline distribution wewant, in this section it will be kept normal and exponential distributions asexamples.The main objective of this section is to promote a generic model to importthe highly informative regression coefficients into the Bayesian inference proce-dure. To do so, a normal prior distribution for each parameter of the Gumbeldistribution is assumed in a manner that the mean and standard deviation areobtained from the regression coefficients π ( θ ) ∝ f ( α , (cid:101) x ) , (12)where f is a function of the regression coefficients vector α and the sample (cid:101) x .Then, f can be used to estimate the parameters of the baseline distribution(noted by γ ) via a non-informative prior. Both regression and baseline param-eters must be conveniently estimated. So, the full algorithm can be summarizeas follows:1. Choose initial values, ζ (0) = ( α (0) , γ (0) ), where α (0) are previously cal-11ulated regression coefficients (Table 1 showed examples for normal andexponential distributions) and γ (0) are baseline parameters.2. Given the chain is currently at ζ ( j ) = ( α ( j ) , γ ( j ) ): • Draw the candidates α can , γ can for the next sample by picking fromthe proper distribution. A normal distribution is used for α can withfixed mean and variance given by the regression coefficients, as wellas a suitable chosen posterior distribution, calculated on the basis ofa non-informative prior distribution and the entire dataset (cid:101) x , is usedfor γ can (this distribution is specified in next subsection in the caseof exponential and normal baseline distributions). • Due to the connection between γ and Gumbel parameters θ , α can and γ can must be acepted or rejected simultaneously. As usual, thisis implemented by drawing u θ ∼ U (0 , 1) and taking θ ( j +1) = θ can ifand only if u θ < P θ .Here, P θ = min (cid:40) , π ( ζ can | x ) q ( ζ ( j ) | ζ can ) π ( ζ ( j ) | x ) q ( ζ can | ζ ( j ) ) (cid:41) , (13)where θ can be µ or σ and ζ = ( α , γ ). θ is related with ζ by θ = α (1 , γ ) (cid:48) . (14)The conditional posterior distributions, π ( θ | x ), for the parameters µ and σ are obtained by ignoring all the terms that do not involvedparameter µ and σ , respectively. For the sake of simplicity, andnoting that the prior is highly informative, the candidates are drawnfrom the prior distribution. Thus, q ( ζ ( j ) | ζ can ) = π ( ζ ( j ) ) and so, P θ = min (cid:40) , L ( θ can | x ) L ( θ ( j ) | x ) (cid:41) . (15)Here, L ( θ | x ) is given by Eq. (5), since it depends only on the maxi-mum data. Despite this, the entire data (cid:101) x is still used for generatingdraws of γ .3. Iterate the former procedure. 12 .1. Exponential and normal distribution examples Proceeding with the examples given in Section 4, results of regression sum-marized in Table 1 are used. To draw the candidates in the MH step, it isnecessary to know the non-informative posterior distribution for the parameter λ of the exponential and, the µ and σ parameters of the normal that will beused as priors in the iterative procedure.With respect to the exponential case, it is assumed a non-informative Γ( α =1 , β = 1) prior for λ . Then, posterior distribution is (for a data size m ) π ( λ | (cid:101) x ) ∝ e − λ ( (cid:80) mi =1 x i +1 ) λ m ∝ Γ (cid:32) m + 1 , m (cid:88) i =1 x i + 1 (cid:33) . (16)Concerning the normal distribution, it is assumed the following prior distribu-tion for µ and σ µ | σ ∼ N ( µ = 0 , σ = 1) σ ∼ Γ( α = 1 , β = 1) . (17)Then , the posterior is π ( µ | σ, (cid:101) x ) ∝ N (cid:18) mm + 1 ¯ x, m + 1 (cid:19) π ( σ | (cid:101) x ) ∝ Γ (cid:32) m , m (cid:88) i =1 ( x i − ¯ x ) + m m + 1) ¯ x (cid:33) . (18)With prior distributions for the baseline parameters given by Eqs. (16) and(18), which will be used to draw the candidates in the MH steps, and likehoodfunction (5), generic method is ready to be implemented. It should be noted thatsome convergence and correlation tests must be done before making a definitiveanalysis. 6. Discussion The comparability of the two considered methods; the Standard-MH (SMH)algorithm and the Regression-MH (RMH) method introduced in Section 5, is13xplored via simulations, involving various sample sizes and values of the pa-rameter space.For all cases, the first 1000 iterations out of 10000 had to be discarded sincethe plot does not converge to the stable values before 1000th iteration for MHalgorithm. Moreover, to avoid autocorrelation between successive candidates, ithas been set down a thinning rate retaining each 60th element of the sequence[29].The main conclusion derived from this experiment is that an improvementhas been achieved. Figures 3 and 4 show the distribution of estimations ob-tained by the SMH and RMH methods for µ and σ , as particular examples ofgeneral behavior, which is more accused for large sample sizes. In the same way,the summary of the credible intervals for the estimated Gumbel parameters isshown in Table 2. Reduction in the credible interval width and, not less, enhace-ment in parameter location show the RMH results to adapt to very shorteneddata better than SMH does. Even outcomes show a distinct narrowing in thevalue range, this is most evident when looking at the scale parameter σ . Thescale parameter can provide more significant information when talking aboutstatistical inference. This means, to make better predictions of extremes eventsit is better to have a more accurate knowledge of the closeness of extreme valuesthan knowing the central tendency of them because it will reduce the probabilityinterval of finding these events. 14 able 2: 95% Credible interval summary for exponential and normal baseline distributionusing SMH and RMH methods, from 500 simulated samples with k = 90, n = 10, andbaseline parameters λ = 1 / , , µ N = 0 , 10, and σ N = 0 . , , Exponential DistributionParameter of the Gumbel Distribution µ σ Method SMH RMH SMH RMH λ = 1 / λ = 1 (3.952,5.256) (4.167,4.786) (0.711,1.899) (0.922,1.058) λ = 3 (1.333,1.791) (1.406,1.609) (0.245,0.672) (0.3149,0.359)Normal DistributionParameter of the Gumbel Distribution µ σ Method SMH RMH SMH RMH µ N = 0 , σ N = 0 . µ N = 0 , σ N = 1 (2.058,72.555) (2.129,2.391) (0.261,0.712) (0.356,0.389) µ N = 0 , σ N = 5 (10.096,12.663) (10.594,11.912) (1.297,3.232) (1.776,1.953) µ N = 10 , σ N = 0 . µ N = 10 , σ N = 1 (12.076,12.540) (12.145,12.404) (0.275,0.716) (0.358,0.392) µ N = 10 , σ N = 5 (20.111,22.438) (20.634,21.994) (1.313,3.265) (1.783,1.960) µ 12 13 14 15 . . . . . . SMH R MH s SMHRMH µ . . . . . . . s µ s Figure 3: Histograms for estimated Gumbel parameters obtained using SMH and RMH meth-ods, from 500 simulated samples with k = 90, n = 10 and baseline distribution exponential,with λ = 1 / SMH R MH µ µ . . . . s SMHRMH s s Figure 4: Histograms for estimated Gumbel parameters obtained using SMH and RMH meth-ods, from 500 simulated samples with k = 90, n = 10 and baseline distribution normal., withparameters µ N = 0 and σ N = 0 . 7. An experimental data example With the aim of giving a practical application of the RMH model, the sci-ence of climatology will be considered. This science and its practical applicationhave much to contribute to EVT. In particular, RMH algorithm will be usedto estimate the Gumbel parameters that fit the peak temperatures in summerseasons (June-August). These seasons will be treated separately because of thetime dependence of the normal distribution parameters that describe summer16emperatures. For this reason, actual data consists in 90 maximum daily sum-mer temperatures and the maximum value of each season. The dataset used inthis paper consists of a set of daily temperature records registered in the city ofC´aceres (Extremadura, Spain) in the 1908-1915 period.Results are displayed on Table 3. It can be noted that credible intervalsare extraordinary shorts, in view especially of the fact that only one maximumvalue per season have been used. However, this can be explained by the factthat not only one singular datum has been used, but all of the entire data.The capacities of the RMH model to adapt to very shortened data is the mainadvantage of this model with respect to those which target efforts on improvingprior information of the maximum distribution. Table 3: Credible interval summary for the Gumbel parameters arising from a normallydistributed annual temperature data. Year 1908 1909Parameter of the Gumbel Distribution ( ◦ C) µ σ µ σ 95% Credible Interval (42.026,42.052) (1.847,1.851) (44.304,44.336) (2.164,2.168)Year 1910 1911Parameter of the Gumbel Distribution ( ◦ C) µ σ µ σ 95% Credible Interval (42.429,42.456) (1.866,1.870) (44.531,44.562) (2.117,2.121)Year 1912 1913Parameter of the Gumbel Distribution ( ◦ C) µ σ µ σ 95% Credible Interval (41.243,41.271) (1.811,1.815) (43.414,43.441) (1.784,1.788)Year 1914 1915Parameter of the Gumbel Distribution ( ◦ C) µ σ µ σ 95% Credible Interval (41.737,41.761) (1.774,1.777) (42.847,42.872) (1.611,1.614) 8. Concluding remarks In this paper, a new method to incorporate Bayesian methodology into ex-treme value analysis is studied. A Bayesian MH algorithm based on highlyinformative priors obtained by a well-defined connection between Gumbel andbaseline parameters is presented. Simulations made show that the proposedmethod (RMH) can accurately estimate the Gumbel parameters and consider-ably shorten the credible interval width when compared with the standard MH17lgorithm. Experimental data fits show that the proposed model is also suitablewhen working with only one maximum value. It is found that RMH methodhas good performance and its usage is recommended in practice.Although we only show the results obtained for the normal and exponentialbaseline distributions and extreme distribution Gumbel, the strategy can beextended to other distributions. 9. Acknowledgments ReferencesReferences [1] M. Nogaj, P. Yiou, S. Parey, F. Malek, P. Naveau, Amplitude and fre-quency of temperature extremes over the north atlantic region, GeophysicalResearch Letters 33 (10). doi:10.1029/2005gl024251 .[2] C. A. S. Coelho, C. A. T. Ferro, D. B. Stephenson, D. J. Steinskog,Methods for exploring spatial and temporal variability of extreme eventsin climate data, Journal of Climate 21 (10) (2008) 2072–2092. doi:10.1175/2007jcli1781.1 .[3] F. J. Acero, M. I. Fern´andez-Fern´andez, V. M. S. Carrasco, S. Parey,T. T. H. Hoang, D. Dacunha-Castelle, J. A. Garc´ıa, Changes in heat wavecharacteristics over extremadura (sw spain), Theoretical and Applied Cli-matology (2017) 1–13 doi:10.1007/s00704-017-2210-x .184] J. Garc´ıa, M. C. Gallego, A. Serrano, J. Vaquero, Trends in block-seasonal extreme rainfall over the iberian peninsula in the second half ofthe twentieth century, Journal of Climate 20 (1) (2007) 113–130. doi:10.1175/jcli3995.1 .[5] M. Re, V. R. Barros, Extreme rainfalls in SE south america, ClimaticChange 96 (1-2) (2009) 119–136. doi:10.1007/s10584-009-9619-x .[6] F. J. Acero, J. A. Garc´ıa, M. C. Gallego, Peaks-over-threshold study oftrends in extreme rainfall over the iberian peninsula, Journal of Climate24 (4) (2011) 1089–1105. doi:10.1175/2010jcli3627.1 .[7] F. J. Acero, M. C. Gallego, J. A. Garc´ıa, Multi-day rainfall trends over theiberian peninsula, Theoretical and Applied Climatology 108 (3-4) (2011)411–423. doi:10.1007/s00704-011-0534-5 .[8] F. J. Acero, S. Parey, T. T. H. Hoang, D. Dacunha-Castelle, J. A. Garc´ıa,M. C. Gallego, Non-stationary future return levels for extreme rainfall overextremadura (southwestern iberian peninsula), Hydrological Sciences Jour-nal 62 (9) (2017) 1394–1411. doi:10.1080/02626667.2017.1328559 .[9] S. Wi, J. B. Vald´es, S. Steinschneider, T.-W. Kim, Non-stationaryfrequency analysis of extreme precipitation in south korea usingpeaks-over-threshold and annual maxima, Stochastic Environmental Re-search and Risk Assessment 30 (2) (2015) 583–606. doi:10.1007/s00477-015-1180-8 .[10] A. A. Ramos, Extreme value theory and the solar cycle, Astronomy &Astrophysics 472 (1) (2007) 293–298. doi:10.1051/0004-6361:20077574 .[11] F. J. Acero, V. M. S. Carrasco, M. C. Gallego, J. A. Garc´ıa, J. M. Vaquero,Extreme value theory and the new sunspot number series, The Astrophys-ical Journal 839 (2) (2017) 98. doi:10.3847/1538-4357/aa69bc .1912] F. J. Acero, M. C. Gallego, J. A. Garc´ıa, I. G. Usoskin, J. M. Vaquero, Ex-treme value theory applied to the millennial sunspot number series, The As-trophysical Journal 853 (1) (2018) 80. doi:10.3847/1538-4357/aaa406 .[13] E. Castillo, A. S. Hadi, N. Balakrishnan, J. M. Sarabia, Extreme Valueand Related Models with Applications in Engineering and Science, Wiley-Interscience, 2004.[14] E. J. Gumbel, Statistics of Extremes (Dover Books on Mathematics), DoverPublications, 2012.[15] E. Castillo Ron, Estad´ıstica de valores extremos. distribuciones asint´oticas,Estad´ıstica espa˜nola (116) (1987) 5–35.[16] R. L. Smith, J. C. Naylor, A comparison of maximum likelihood andbayesian estimators for the three- parameter weibull distribution, AppliedStatistics 36 (3) (1987) 358. doi:10.2307/2347795 .[17] S. Coles, L. R. Pericchi, S. Sisson, A fully probabilistic approach to extremerainfall modeling, Journal of Hydrology 273 (1-4) (2003) 35–50. doi:10.1016/s0022-1694(02)00353-0 .[18] J. M. Bernardo, A. F. M. Smith (Eds.), Bayesian Theory, John Wiley &Sons, Inc., 1994. doi:10.1002/9780470316870 .[19] S. Kotz, S. Nadarajah, Extreme Value Distributions: Theory and Applica-tions, ICP, 2000.[20] S. G. Coles, J. A. Tawn, A bayesian analysis of extreme rainfall data,Applied Statistics 45 (4) (1996) 463. doi:10.2307/2986068 .[21] M. Rostami, M. B. Adam, Analyses of prior selections for gumbel distri-bution, Matematika 29 (2013) 95–107. doi:10.11113/matematika.v29.n.582 . 2022] M.-H. Chen, Q.-M. Shao, J. G. Ibrahim, Monte Carlo Methods inBayesian Computation, Springer New York, 2000. doi:10.1007/978-1-4612-1276-8 .[23] I. Vidal, A bayesian analysis of the gumbel distribution: an application toextreme rainfall data, Stochastic Environmental Research and Risk Assess-ment 28 (3) (2013) 571–582. doi:10.1007/s00477-013-0773-3 .[24] L. M. Lye, Bayes estimate of the probability of exceedance of annual floods,Stochastic Hydrology and Hydraulics 4 (1) (1990) 55–64. doi:10.1007/bf01547732 .[25] M. Rostami, M. B. Adam, N. A. Ibrahim, M. H. Yahya, Slice samplingtechnique in bayesian extreme of gold price modelling, in: AIP ConferenceProceedings, Vol. 1557, AIP, 2013, pp. 473–477. doi:10.1063/1.4823959 .[26] S. R. W.R. Gilks, D. Spiegelhalter, Markov Chain Monte Carlo in Prac-tice (Chapman & Hall/CRC Interdisciplinary Statistics), Chapman andHall/CRC, 1996.[27] N. A. M. Amin, M. B. Adam, N. A. Ibrahim, Bayesian inference us-ing multiple-try metropolis hastings scheme for the efficiency of estimat-ing gumbel distribution parameters, Matematika 31 (1) (2015) 25–36. doi:10.11113/matematika.v31.n1.743 .[28] A. D. Martin, K. M. Quinn, J. H. Park, MCMCpack: Markov chain montecarlo in R, Journal of Statistical Software 42 (9) (2011) 22.URL [29] M. Plummer, N. Best, K. Cowles, K. Vines, Coda: Convergence diagnosisand output analysis for mcmc, R News 6 (1) (2006) 7–11.URL https://journal.r-project.org/archive/https://journal.r-project.org/archive/