Bivariate Analysis of Pollutants Monthly Maxima in Mexico City Using Extreme Value Distributions and Copula

In the present work, we are interested in studying the joint distributions of pairs of the monthly maxima of the pollutants used by the environmental authorities in Mexico City to classify the air quality in the metropolitan area. In order to obtain the joint distributions a copula will be considered. Since we are analyzing the monthly maxima, the extreme value distributions of Weibull and Fréchet are taken into account. Using these two distributions as marginal distributions in the copula a Bayesian inference was made in order to estimate the parameters of both distributions and also the association parameters appearing in the copula model. The pollutants taken into account are ozone, nitrogen dioxide, sulphur dioxide, carbon monoxide, and particulate matter with diameters smaller than 10 and 2.5 microns obtained from the Mexico City monitoring network. The estimation was performed by taking samples of the parameters generated through a Markov chain Monte Carlo algorithm implemented using the software OpenBugs. Once the algorithm is implemented it is applied to the pairs of pollutants where one of the coordinates of the pair is ozone and the other varies on the set of the remaining pollutants. Depending on the pollutant and the region where they were collected, different results were obtained. Hence, in some cases we have that the best model is that where we have a Fréchet distribution as the marginal distribution for the measurements of both pollutants and in others the most suitable model is the one assuming a Fréchet for ozone and a Weibull for the other pollutant. Results show that, in the present case, the estimated association parameter is a good representation to the correlation parameters between the pair of pollutants analyzed. Additionally, it is a straightforward task to obtain these correlation parameters from the corresponding association parameters.

Extreme value distribution and reliability of nonlinear stochastic structures

A new approach to evaluate the extreme value distribution （EVD） of the response and reliability of general multi-DOF nonlinear stochastic structures is proposed. The approach is based on the recently developed probability density evolution method, which enables the instantaneous probability density functions of the stochastic responses to be captured. In the proposed method, a virtual stochastic process is first constructed to satisfy the condition that the extreme value of the response equals the value of the constructed process at a certain instant of time. The probability density evolution method is then applied to evaluate the instantaneous probability density function of the response, yielding the EVD. The reliability is therefore available through a simple integration over the safe domain. A numerical algorithm is developed using the Number Theoretical Method to select the discretized representative points. Further, a hyper-ball is imposed to sieve the points from the preceding point set in the hypercube. In the numerical examples, the EVD of random variables is evaluated and compared with the analytical solution. A frame structure is analyzed to capture the EVD of the response and the dynamic reliability. The investigations indicate that the proposed approach provides reasonable accuracy and efficiency.

Estimation of Poisson-Generalized Pareto Compound Extreme Value Distribution by Probability-Weighted Moments and Empirical Analysis

This paper puts forward a Poisson-generalized Pareto (Poisson-GP) distribution. This new form of compound extreme value distribution expands the existing application of compound extreme value distribution, and can be applied to predicting financial risk, large insurance settlement and high-grade earthquake, etc. Compared with the maximum likelihood estimation (MLE) and compound moment estimation (CME), probability-weighted moment estimation (PWME) is used to estimate the parameters of the distribution function. The specific formulas are presented. Through Monte Carlo simulation with sample sizes 10, 20, 50, 100, 1 000, it is concluded that PWME is an efficient method and it behaves steadily. The mean square errors (MSE) of estimators by PWME are much smaller than those of estimators by CME, and there is no significant difference between PWME and MLE. Finally, an example of foreign exchange rate is given. For Dollar/Pound exchange rates from 1990-01-02 to 2006-12-29, this paper formulates the distribution function of the largest loss among the investment losses exceeding a certain threshold by Poisson-GP compound extreme value distribution, and obtains predictive values at different confidence levels.

Extreme value distributions of mixing two sequences with different MDA＇s

Suppose {Xi, i≥1} and {Yi, i≥1} are two independent sequences with distribution functions FX(x) and FY(x), respectively. Zi is the combination of Xi and Yi with a probability pn for each i with 1≤i≤n. The extreme value distribution ,n GZ(x) of this particular triangular array of the i.i.d. random variables Z1, , Z2, ,…, Zn n n ,nis discussed. We found a new form of the extreme value distribution ΛA(ρx)Λ(x)(0<ρ <1), which is not max-stable. It occurs if FX(x) and FY(x) belong to the same MDA(Λ). GZ(x) does not exist as mixture forms of the different types of extreme value distributions.

More general results on mixed extreme value distributions

The sequences {Zi,n, 1≤i≤n}, n≥1 are multi-nomial distribution among i.i.d, random variables {X1,i, i≥1}, {X2,i, i≥1 } {Xm,i, i≥1 }. The extreme value distribution Gz（x） of this particular triangular array of i.i,d, random variables Z1,n, Z2 n,...,Zn,n is discussed. A new type of not max-stable extreme value distributions which are Fréchet mixture, Gumbel mixture and Weibull mixture has been found if Fj,…… Fm belong to the same MDA. Whether mixtures of different types of extreme value distributions exist or not and the more general case are discussed in this paper. We found that Gz（x） does not exist as mixture forms of the different types of extreme value distributions after we investigated all cases.

A class of not max-stable extreme value distributions

The sequences {Zi , 1≤i≤n}, n≥1 have multi-nomial distribution among i.i.d. random variables {X1, , i≥1}, {X2, , ,n i i i≥1}, …, {Xm , i≥1}. The extreme value distribution GZ(x) of this particular triangular array of i.i.d. random variables Z1, , Z2, , …, ,i n n r ?1 Zn is discussed in this paper. We found a new type of not max-stable extreme value distributions, i) GZ (x) = ,n ∏Φα Ai(x)×Φαr (x); i i=1 r ?1 r?1 ii) GZ (x) = ∏Ψα Ai(x)×Ψαr (x); iii) GZ (x) = ∏Λ Ai(λix)×Λ(x), r≥2, 0<α1≤α2≤…≤αr and λi∈(0,1] for i, 1≤i≤r?1 which occur if i i=1 i=1 Fj, …, Fm belong to the same MDA.

Applications of Bootstrap in Analyzing General Extreme Value Distributions

The bootstrap method is one of the new ways of studying statistical math which this article uses but is a major tool for studying and evaluating the values of parameters in probability distribution.Our research is concerned overview of the theory of infinite distribution functions.The tool to deal with the problems raised in the paper is the mathematical methods of random analysis(theory of random process and multivariate statistics).In this article,we introduce the new function to find out the bias and standard error with jackknife method for Generalized Extreme Value distributions.

General Regular Variation of n-th Order and the 2nd Order Edgeworth Expansion of the Extreme Value Distribution (Ⅰ)

In Part Ⅰ the concept of the general regular variation of n-th order is proposed and its construction is discussed. The uniqueness of the standard expression and the higher order regularity of the auxiliary functions are proved.

MULTIVARIATE EXTREME VALUE DISTRIBUTION AND ITS FISHER INFORMATION MATRIX

The paper is concerned with the basic properties of multivariate extreme value distribution (in the Logistic model). We obtain the characteristic function and recurrence formula of the density function. The explicit algebraic formula for Fisher information matrix is indicated. A simple and accurate procedure for generating random vector from multivariate extreme value distribution is presented.

Stationary distributions for two-dimensional sticky Brownian motions:Exact tail asymptotics and extreme value distributions

Sticky Brownian motions can be viewed as time-changed semimartingale reflecting Brownian motions,which find applications in many areas including queueing theory and mathematical finance.In this paper,we focus on stationary distributions for sticky Brownian motions.Main results obtained here include tail asymptotic properties in the marginal distributions and joint distributions.The kernel method,copula concept and extreme value theory are the main tools used in our analysis.

General Regular Variation of the n-th Order and 2nd Order Edgeworth Expansions of the Extreme Value Distribution （Ⅱ）

In this part II the fundamental inequality of the third order general regular variation is proved and the second order Edgeworth expansion of the distribution of the extreme values is discussed.

Analyzing the Annual Maximum Magnitude of Earthquakes in Japan by Extreme Value Theory

One of the most important and interesting issues associated with the earthquakes is the long-term trend of the extreme events. Extreme value theory provides methods for analysis of the most extreme parts of data. We estimated the annual maximum magnitude of earthquakes in Japan by extreme value theory using earthquake data between 1900 and 2019. Generalized extreme value (GEV) distribution was applied to fit the extreme indices. The distribution was used to estimate the probability of extreme values in specified time periods. The various diagnostic plots for assessing the accuracy of the GEV model fitted to the magnitude of maximum earthquakes data in Japan gave the validity of the GEV model. The extreme value index, ξ was evaluated as −0.163, with a 95% confidence interval of [−0.260, −0.0174] by the use of profile likelihood. Hence, the annual maximum magnitude of earthquakes has a finite upper limit. We obtained the maximum return level for the return periods of 10, 20, 50, 100 and 500 years along with their respective 95% confidence interval. Further, to get a more accurate confidence interval, we estimated the profile log-likelihood. The return level estimate was obtained as 7.83, 8.60 and 8.99, with a 95% confidence interval of [7.67, 8.06], [8.32, 9.21] and [8.61, 10.0] for the 10-, 100- and 500-year return periods, respectively. Hence, the 2011 off the Pacific coast of Tohoku Earthquake, which was the largest in the observation history of Japan, had a magnitude of 9.0, and it was a phenomenon that occurs once every 500 year.

Empirical Analysis of Value-at-Risk Estimation Methods Using Extreme Value Theory

This paper investigates methods of value-at-risk (VaR) estimation using extreme value theory (EVT). It compares two different estimation methods, 'two-step subsample bootstrap' based on moment estimation and maximum likelihood estimation (MLE), according to their theoretical bases and computation procedures. Then, the estimation results are analyzed together with those of normal method and empirical method. The empirical research of foreign exchange data shows that the EVT methods have good characters in estimating VaR under extreme conditions and 'two-step subsample bootstrap' method is preferable to MLE.

Red Tide Frequency Analysis Using the Extreme Value Theory

In recent years, the red tide erupted frequently, and caused a great economic loss. At present, most literatures emphasize the academic research on the growth mechanism of red tide alga. In order to find out the characters of red tide in detail and improve the precision of forecast, this paper gives some new approaches to dealing with the red tide. By the extreme values, we deal with the red tide frequency analysis and get the estimation of T-times red tide level U (T), which is the level once the consistence of red tide alga exceeds on the average in a period of T times.

A New Method for Determining Threshold in Using PGCEVD to Calculate Return Values of Typhoon Wave Height

In using the PGCEVD （Poisson-Gumbel Compound Extreme Value Distribution） model to calculate return values of typhoon wave height, the quantitative selection of the threshold has blocked its application. By analyzing the principle of the threshold selection of PGCEVD model and in combination of the change point statistical methods, this paper proposes a new method for quantitative calculation of the threshold in PGCEVD model. Eleven samples from five engineering points in several coastal waters of Guangdong and Hainan, China, are calculated and analyzed by using PGCEVD model and the traditional Pearson type III distribution （P-III） model, respectively. By comparing the results of the two models, it is shown that the new method of selecting the optimal threshold is feasible. PGCEVD model has more stable results than that of P-III model and can be used for the return wave height in every direction.

Non-conventional modeling of extreme significant wave height through random sets

The analysis and design of offshore structures necessitates the consideration of wave loads. Realistic modeling of wave loads is particularly important to ensure reliable performance of these structures. Among the available methods for the modeling of the extreme significant wave height on a statistical basis, the peak over threshold method has attracted most attention. This method employs Poisson process to character- ize time-varying properties in the parameters of an extreme value distribution. In this paper, the peak over threshold method is reviewed and extended to account for subjectivity in the modeling. The freedom in selecting the threshold and the time span to separate extremes from the original time series data is incorpo- rated as imprecision in the model. This leads to an extension from random variables to random sets in the probabilistic model for the extreme significant wave height. The extended model is also applied to different periods of the sampled data to evaluate the significance of the climatic conditions on the uncertainties of the parameters.

Distributional expansion of maximum from logarithmic general error distribution

Logarithmic general error distribution is an extension of lognormal distribution. In this paper, with optimal norming constants the higher-order expansion of distribution of partial maximum of logarithmic general error distribution is derived.

Probabilistic Distribution of Ice Forces on a Solitary Pile in the Liaodong Bay

The level ice thickness and compressive strength at the four measuring stations in the Liaodong Bay are inferred according to the hydrologic and meteorologic data there, then the yearly extreme ice forces on a solitary pile are calculated by the use of appropriate formula of ice forces and its probabilistic distribution is determined. Generally, the yearly extreme ice force follows Weibull distribution best as compared with Normal, Lognormal, and Extreme Value I distribution. On the other hand, the short-term distribution of ice forces on a solitary pile is obtained from the model experiment data analysis: It does not refuse Extreme Value I distribution.

A fuzzy quantification approach of uncertainties in an extreme wave height modeling

A non-traditional fuzzy quantification method is presented in the modeling of an extreme significant wave height. First, a set of parametric models are selected to fit time series data for the significant wave height and the extrapolation for extremes are obtained based on high quantile estimations. The quality of these results is compared and discussed. Then, the proposed fuzzy model, which combines Poisson process and gener-alized Pareto distribution （GPD） model, is applied to characterizing the wave extremes in the time series data. The estimations for a long-term return value are considered as time-varying as a threshold is regarded as non-stationary. The estimated intervals coupled with the fuzzy theory are then introduced to construct the probability bounds for the return values. This nontraditional model is analyzed in comparison with the traditional model in the degree of conservatism for the long-term estimate. The impact on the fuzzy bounds of extreme estimations from the non stationary effect in the proposed model is also investigated.

Assesment of Frequency-Magnitude of Extreme Rainfall Events-Case Study of the MeKong River Delta

Extreme rainfall events are primary natural hazards, which cause a severe threat to people and their properties in populated cities, which are normally located in coastal areas in Vietnam. Analysing these events by using a data series observed over years will support us to draw a picture of how the climate change impact on local environments. The purpose of this report is to understand the characteristics of the extreme rainfall events in MEKONG river delta （south VietNam）. Daily rainfall data in the period of 30 years for a meteorological station in each area were collected from the Vietnam National Hydro-meteorological Service. The extreme rainfall events were defined as those exceeding the 95th percentile for each station. The analytical results show that the rainfall values （95th percentile） are 37.4 mm/day at Nam Can station, 27 mm/day at My Thanh station, 22.4 mm/day at Hoa Binh station, 23.8 mm/day at Binh Dai station and 22.7 mm/day at Ben Trai station. The highest rainfall data ever recorded are 246.4 mm/day （Nam Can）, 174.5 mm/day （My Thanh）, 179 mm/day （Hoa Bin_h）, 187.3 mm/day （Binh Dai） and 136.3 mm/day （Ben Trai） during 1983-2012. The result of the Mann-Kendall tests show that there was a significant creasing of the rainfall at Nam Can, My Thanh station in two periods （1983-2012, 1998-2012） while no clear trend of the rainfall was recoreded at Hoa Birth, Binh Dai, Ben Trai station. In order to estimate the return period of the extreme rainfall events, the method General Extreme Value Distribution was used to calculate frequent distribution. The magnitudes of daily maximum rainfall were from 2 to 100 years. The results of return period show that maximum rainfalls are 46.6 mm at Nam Can station （highest） and 31.4 mm at Hoa Birth station （lowest） during 50 years. Similarly, maximum rainfalls are expected to be about 55.1 mm at Nam Can station and 37.2 mm at Hoa Birth station for 100 years.