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.

The Convergence Rate of Fréchet Distribution under the Second-Order Regular Variation Condition

In this article we consider the asymptotic behavior of extreme distribution with the extreme value index γ>0 . The rates of uniform convergence for Fréchet distribution are constructed under the second-order regular variation condition.

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.

ON VALUE DISTRIBUTIONS OF GENERAL DIFFERENTIAL MONOMIALS

Value distributions of the general differential monomials is discussed.The following theorem is obtained:Let f be a transcendental meromorphic function in the plane,F=f n 0 (f (i) ) n i …(f (k) ) n k -c,n i≥1,c≠0 be a constant then (n 0-2)T(r,f)≤(r,1F)+S(r,f) when n 0】2;T(r,f)≤7(i+1)i( i) (r,1f)+(r,1F))+S(r,f) when n 0=1;T(r,f)≤7(N(r,1f)+(r,1F))+S(r,f) when n 0=0.

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.

Research on the Spatial Distribution of Ecosystem Service Value in Guangxi and Its Ecological Protection Countermeasure

90 counties (cities) in Guangxi Province being taken as the tested region,the ecosystem service value of those counties (cities) was measured with the data of land-using in 2005 by means of the Table of Terrestrial Ecosystem Services Value in China.The result indicated that the value of the ecosystem services per unit was divided into 3 categories:the first category with high ecosystem services value,the second category with medium ecosystem services value and the third category with low ecosystem services value.The region of the first category was mainly distributed in the mountain area of northern,northwestern,eastern and northeast part of Guangxi;the second category,in the hilly area of southern part and the mountain area in the southwestern part of Guangxi and the third category,in the basin area of central Guangxi,the karst area of western and northwestern Guangxi Province.

Second Order Periodic Boundary Value Problems Involving the Distributional Henstock-Kurzweil Integral

We apply the distributional derivative to study the existence of solutions of the second order periodic boundary value problems involving the distributional Henstock-Kurzweil integral. The distributional Henstock-Kurzweil integral is a general intergral, which contains the Lebesgue and Henstock-Kurzweil integrals. And the distributional derivative includes ordinary derivatives and approximate derivatives. By using the method of upper and lower solutions and a fixed point theorem, we achieve some results which are the generalizations of some previous results in the literatures.

ON THE SINGULAR DIRECTIONS OF VALUE DISTRIBUTION OF HOLOMORPHIC CURVES IN P^n(C)

This article proves the existence of singular directions of value distribution theory for some transcendental holomorphic curves in the n-dimensional complex projective space P^n（C）.. An example is given to complement these results.

THE VALUE DISTRIBUTION OF RANDOM ANALYTIC DIRICHLET SERIES OF NEUTRAL GROWTH(I)

This paper deals with the value distribution of random Dirichlet series whose coefficients are a martingale difference sequence, and which is of neutral growth.

THE VALUE DISTRIBUTION OF RANDOM ANALYTIC DIRICHLET SERIES OF NEUTRAL GROWTH (Ⅱ)

This paper studies the value distribution of random analytic Dirichlet series f(s) = Zn()e-sn, where {Zn} is a sequence of independent random variables, n = 1 with moments zero, such that infE{Zn}/E1/2{Zn2≥ α > 0. Suppose [h*(σ)]2 = n converges for any α > 0, and diverges for = 0. It is shown that if = ρ E (0, ), then with probability one, where β is a constant depending only upon the constant α.

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.

Continuous-variable quantum key distribution based on continuous random basis choice

Gaussian-modulated coherent state quantum key distribution is gradually moving towards practical application. Generally, the involved scheme is based on the binary random basis choice. To improve the performance and security, we present a scheme based on a continuous random basis choice. The results show that our scheme obviously improves the performance, such as the secure communication distance. Our scheme avoids comparing the measurement basis and discarding the key bits, and it can be easily implemented with current technology. Moreover, the imperfection of the basis choice can be well removed by the known phase compensation algorithm.

THE DISTRIBUTION OF VALUES OF TAYLOR SERIES

Clarify some classical results and their proofs on the distribution of values of simple Taylor series and extend them to multiple Taylor series.

Importance of Generalized Logistic Distribution in Extreme Value Modeling

We consider a problem from stock market modeling, precisely, choice of adequate distribution of modeling extremal behavior of stock market data. Generalized extreme value (GEV) distribution and generalized Pareto (GP) distribution are the classical distributions for this problem. However, from 2004, [1] and many other researchers have been empirically showing that generalized logistic (GL) distribution is a better model than GEV and GP distributions in modeling extreme movement of stock market data. In this paper, we show that these results are not accidental. We prove the theoretical importance of GL distribution in extreme value modeling. For proving this, we introduce a general multivariate limit theorem and deduce some important multivariate theorems in probability as special cases. By using the theorem, we derive a limit theorem in extreme value theory, where GL distribution plays central role instead of GEV distribution. The proof of this result is parallel to the proof of classical extremal types theorem, in the sense that, it possess important characteristic in classical extreme value theory, for e.g. distributional property, stability, convergence and multivariate extension etc.

Further Extension on a Theorem of the Value Distribution of f′f^n

By using small function method, the following result is obtained. If f（z） is transcendental meromorphic and that ψ（z） is non-zero meromorphic and that T（r,ψ） = S（r, f）, then（n＋1）T（r,f）≤N^-（r,1/f＇f^n-ψ）＋2N^-（r,1/f）＋N^-（r,f）＋S（r,f）.

Diagnostic value of combined detection of myocardial markers, white blood cell count and platelet distribution width in patients with positive myocardial injury markers

Objective:To explore the diagnostic value of combined detection of myocardial markers, white blood cell (WBC) counts and platelet distribution width (PDW) in patients with positive myocardial injury markers.Methods: From January 2017 to January 2018, 100 patients with positive markers of myocardial injury in our hospital were selected as observation group, and 100 healthy people were selected as control group. Serum myocardial markers troponin I (cTnI), creatine kinase isoenzyme (CK-MB), myoglobin (MYO), WBC count, and PDW levels were measured at admission, and analyzed for individual indicators. And individual and combined detections of these indicators in early diagnosis of acute myocardial infarction (AMI) were analysed.Results: Serum cTnI, CK-MB, MYO, WBC count and PDW level were higher in the observation group than those in the control group, and the difference between the groups was statistically significant. Of the 100 patients with positive myocardial injury markers, 48 (48.00%) were diagnosed with AMI by final clinical diagnosis. Compared with the control group, the positive rate of serum index and the combined detection of five indicators in the observation group were significantly increased. The sensitivity and specificity of the five indicators combined detection and diagnosis of AMI were 95.83% and 94.23%, respectively, which were higher than the individual detection of each index, and the difference was statistically significant.Conclusions: The combined detection of serum cTnI, CK-MB, MYO, WBC count and PDW is helpful for early diagnosis of AMI and can improve the sensitivity and specificity of diagnosis.

Extreme value distributions of mixing two sequences with the same MDA

Suppose {Xi, i 1} and {Yi, i 1} are two independent sequences with distribution functions ()XFx and ()YFx, respectively. Zi,n is the combination of Xi and Yi with a probability np for each i with 1 in. The extreme value distribution GZ(x) of this particular triangular array of the i.i.d. random variables Z1,n, Z2,n, ,L Zn,n is discussed. We found a new form of the extreme value distributions i) 12()()AxxaaFF and ii) 12()()AxxaaYY (a1<a2), which are not max-stable. It occurs if FX and FY belong to the same MDA(? or MDA(?.

BACKGROUND VALUES AND TREND DISTRIBUTION OF Cu, Zn AND Ni IN SOILS IN CHINA

The background values of Cu, Zn and Ni are discussed based on the analytical data of 21 main soil types collected from various regions in China. According to statistics of many samples, the background value of Cu is 23.4±10.2 mg/ kg (482 samples), Zn 77.9±2.39 mg/ kg (474 samples), Ni 30.0±12.8 mg/kg (482 samples).