Asymptotic Extreme Distribution Research Articles

Hydrological study and analysis for proposed Al-Arkhama Dam, Iraq

The hydrological study and analysis for the proposed Al-Arkhama Dam which is located 90 km south west of Kirkuk City is presented in this work. The main aim of this research is to estimate the probable maximum flood (PMF) to perform the main objective of constructing the proposed dam through designing the necessary hydraulic structures. For the analysis of the maximum annual rainfall, the asymptotic extreme distribution type I was adopted as theoretical model. The computed PMF discharge for return periods of 5, 10, 50, 100, and 1000 years for the proposed Al-Arkhama Dam catchment area were 48, 66, 112, 134 and 212 m3/s, respectively. The PMF hydrographs of the 25, 50, 100, and 1000 years return periods were computed. The results show that uncertainty of using the SCS-CN method for estimating PMF increases if the values of its parameters are not optimized. In addition, to determine the peak runoff in the study area, this method is better than the rational method, which is permissive in estimation of peak runoff. However, in the proposed location of the Al-Arkhama Dam, considerable amount of water can be harvested. Consequently, this quantity can contribute in solving the problem of water shortage within the region. Moreover, this dam could be very useful in mitigation of the drought and flood risks in the study area and Tigris River. However, assessment of the surface runoff and sediment yield spatiotemporal distribution and the method of mitigation of these yields should be investigated.

Return Level Estimation of Extreme Rainfall over the Iberian Peninsula: Comparison of Methods

Different ways to estimate future return levels (RLs) for extreme rainfall, based on extreme value theory (EVT), are described and applied to the Iberian Peninsula (IP). The study was done for an ensemble of high quality rainfall time series observed in the IP during the period 1961–2010. Two approaches, peaks-over-threshold (POT) and block maxima (BM) with the generalized extreme value (GEV) distribution, were compared in order to identify which is the more appropriate for the estimation of RLs. For the first approach, which identifies trends in the parameters of the asymptotic distributions of extremes, both all-days and rainy-days-only datasets were considered because a major fraction of values of daily rainfall over the IP is zero. For the second approach, rainy-days-only data were considered showing how the mean, variance and number of rainy days evolve. The 20-year RLs expected for 2020 were estimated using these methods for three seasons: autumn, spring and winter. The GEV is less reliable than the POT because fixed blocks lead to the selection of non-extreme values. Future RLs obtained with the POT are greater than those estimated with the GEV, mainly because some gauges show significant positive trends for the number of rainy days. Autumn, rather than winter, is currently the season with the heaviest rainfall for some regions.

New examples of partial samples from the uniform AR(1) process and asymptotic distributions of extremes

The joint limiting distribution of maximum of the specific sub-sample and maximum of the complete sample from the first-order auto-regressive process with uniform marginal distributions is obtained in this article. There are considered several examples of partial samples, consisted of non-randomly selected terms of the full sample. It is well known that the uniform AR(1) process is strictly stationary random sequence; it doesn?t satisfy condition of weak dependency, that prohibits clustering of extremes. As a consequence of this property, some interesting conclusions about joint asymptotic distributions are reached.

A Probabilistic Estimate of the Most Perceptible Earthquake Magnitudes in the NW Himalaya and Adjoining Regions

NW Himalaya and its neighboring region (25°–40°N and 65°–85°E) is one of the most seismically hazardous regions in the Indian subcontinent, a region that has historically experienced large to great damaging earthquakes. In the present study, the most perceptible earthquake magnitudes, M p, are estimated for intensity I = VII, horizontal peak ground acceleration a = 300 cm/s2 and horizontal peak ground velocity v = 10 cm/s in 28 seismogenic zones using the two earthquake recurrence models of Kijko and Sellevoll (Bulletin of the Seismological Society of America 82(1):120–134 1992 ) and Gumbel’s third asymptotic distribution of extremes (GIII). Both methods deal with maximum magnitudes. The earthquake perceptibility is calculated by combining earthquake recurrence models with ground motion attenuation relations at a particular level of intensity, acceleration and velocity. The estimated results reveal that the values of M p for velocity v = 10 cm/s show higher estimates than corresponding values for intensity I = VII and acceleration a = 300 cm/s2. It is also observed that differences in perceptible magnitudes calculated by the Kijko–Sellevoll method and GIII statistics show significantly high values, up to 0.7, 0.6 and 1.7 for intensity, acceleration and velocity, respectively, revealing the importance of earthquake recurrence model selection. The estimated most perceptible earthquake magnitudes, M p, in the present study vary from M W 5.1 to 7.7 in the entire zone of the study area. Results of perceptible magnitudes are also represented in the form of spatial maps in 28 seismogenic zones for the aforementioned threshold levels of intensity, acceleration and velocity, estimated from two recurrence models. The spatial maps show that the Quetta of Pakistan, the Hindukush–Pamir Himalaya, the Caucasus mountain belt and the Himalayan frontal thrust belt (Kashmir-Kangra-Uttarkashi-Chamoli regions) exhibit higher values of the most perceptible earthquake magnitudes (M W > 6.0). These spatially-distributed values indicate good agreement with high seismic hazard zones in previously published hazard maps based on seismicity, maximum expected earthquake magnitudes during specific time intervals and maximum expected acceleration. These maps have useful implications in preparing earthquake selection criteria for the anti-seismic design of non-critical structures in the study region.

Seismicity and seismic hazard assessment for greater Tehran region using Gumbel first asymptotic distribution

Considering the history of severe earthquakes and the presence of active faults in the greater Tehran region, the possibility of a destructive earthquake occurring is high and seismic hazard analysis is crucial. Gumbel distributions are commonly-used statistical distributions in earthquake engineering and seismology. Their main advantage is their basis on the largest earthquake magnitudes selected from an equal-time predefined set. In this study, the first asymptotic distribution of extremes is used to estimate seismicity parameters and peak ground acceleration (PGA). By assuming a Poisson distribution for the earthquakes, after estimation of seismicity parameters, the mean return period and the probable maximum magnitude within a given time interval are obtained. A maximum probable magnitude of 7.0 has a mean return period of 100 years in this region. For a return period of 475 years, the PGA in the greater Tehran region is estimated to be 0.39g to 0.42g, depending on local site conditions. This value is greater than that of the Iranian Code for Seismic Design of Buildings, indicating that a revision of the code is necessary.

Seismicity assessment for the different regions in and around Turkey based on instrumental data: Gumbel first asymptotic distribution and Gutenberg-Richter cumulative frequency law

Abstract. The evaluation of the seismicity in 24 seismic regions, in which Turkey and adjacent areas divided, is carried out. For this purpose two methods are adopted. The first is the "whole process" which follows the Gutenberg and Richter distribution frequency-magnitude law, while the second one is the "part process" which is well known as the theory of extreme values. This theory was developed by Gumbel in order to solve many geophysical problems. The first asymptotic distribution of extremes was used in the present study. The advantage of the method is that it does not required analysis of the whole data set. It uses, instead, the sequence of earthquakes with the largest magnitudes in a set of predetermined equal-time intervals. The parameters a and b were estimated from both methods. For the goodness of fit, to the Gutenberg-Richter frequency-magnitude law, the maximum likelihood approach is applied. The b-values calculated from Gutenberg and Richter frequency-magnitude law, reveal a better fit to the tectonic environment of the 24 seismic regions of Turkey and its surroundings examining in this study. On the other hand b-values evaluated from Gumbel's first distribution, do not adjust to the particular tectonics of the 24 seismic regions. The modal value a1/b adopted from Gutenberg-Richter for the 24 seismic regions were calculated, as well. An effort made to correlate the tectonics of the area with the spatial distribution of the various computed seismic parameter, while maps were produced for this purpose. These maps provide a detail image of seismicity and local tectonics for the whole investigated area. The results showed that the Aegean arc and the North Anatolian fault zone ranked among to the first positions between the 24 seismic regions researched.

A comparison of scores of two protein structures with foldings

Let $\{X_i;\, i\geq 1\},\,\{Y_i;\,i\geq 1\},\,\{U, U_i;\, i\geq 1\}$ and $\{V, V_i;\, i\geq 1\}$ be four i.i.d. sequences of random variables. Suppose U and V are uniformly distributed on $[0,1]^3.$ For each realization of $\{U_j;\, 1\leq j\leq n\},\ \{X_{i,p};\break \, 1\leq p \leq n\}$ is constructed as a certain permutation of $\{X_p;\, 1\leq p\leq n\}$ for any $1\leq i \leq n.$ Also, $\{Y_{j,p};\, 1\leq p \leq n\}, 1\leq j\leq n,$ are constructed the same way, based on $\{Y_j\}$ and $\{V_j\}.$ For a score function F, we show that \begin{eqnarray*} W_n:= \max_{1\leq i, j,m \leq n}\sum_{p=1}^m F(X_{i,p},Y_{j, p}) \end{eqnarray*} has an asymptotic extreme distribution with the same parameters as in the one-dimensional case. This model is constructed for a comparison of scores of protein structures with foldings.

The first and third asymptotic distributions of extremes as applied to the seismic source regions of India and adjacent areas

The seismic source regions are identified on the basis of spatial and temporal distributions of shocks (1900–1989), recurrence relations and the tectonic architecture of the Indian subcontinent and adjoining areas. The probable occurrence of the maximum magnitude earthquake is estimated using the theory of extreme values of Gumbel. The parameters of the first and third asymptotic distributions of extremes and their uncertainty values are computed for the seven identified seismic source regions of India and adjacent areas. The third-type distribution curve is preferable to the first type in all the regions, as revealed by the χ2 test. The results of the third asymptotic distribution indicate the upper bound to earthquake magnitude w is equal to 8.94 ± 0.21 for Assam, 8.56 ± 0.29 for Bihar-Nepal, 8.43 ± 0.10 for Kangra, 8.97 ± 0.27 for Hindukush, 7.61 ± 0.24 for Pakistan-Cutch, 7.34 ± 0.12 for Koyna and 8.98 ± 0.27 for Andaman Sea seismic source regions. The predicted most probable largest earthquake magnitude is computed for return periods of 10, 20, 50, 75 and 100 yr in each source region.

Asymptotic distributions of extremes of extremal Markov sequences

Let {ξn, n ≧1} be a sequence of independent real random variables, F denote the common distribution function of identically distributed random variables ξn, n ≧1 and let ξ1 have an arbitrary distribution. Define Xn+1 = k max(Xn, ξ n+1), Yn+ 1 = max(Yn, ξ n+1) – c, Un+1 = l min(Un, ξ n+1), Vn+1 = min(Vn, ξ n+1) + c, n ≧ 1, 0 < k < 1, l > 1, 0 < c < ∞, and X1 = Υ1= U1 = V1 = ξ1. We establish conditions under which the limit law of max(X1, · ··, Xn) coincides with that of max(ξ2, · ··, ξ n+1) when both are appropriately normed. A similar exercise is carried out for the extreme statistics max(Y1, · ··, Yn), min(U1,· ··, Un) and min(V1, · ··, Vn).

Asymptotic distributions of extremes of extremal Markov sequences

Let {ξ n , n ≧1} be a sequence of independent real random variables, F denote the common distribution function of identically distributed random variables ξ n , n ≧1 and let ξ 1 have an arbitrary distribution. Define Xn+ 1 = k max(Xn, ξ n +1), Yn + 1 = max(Yn, ξ n +1) – c, Un +1 = l min(Un, ξ n +1), Vn+ 1 = min(Vn, ξ n +1) + c, n ≧ 1, 0 < k < 1, l > 1, 0 < c < ∞, and X 1 = Υ 1 = U 1 = V 1 = ξ 1. We establish conditions under which the limit law of max(X 1, · ··, Xn ) coincides with that of max(ξ 2, · ··, ξ n+ 1) when both are appropriately normed. A similar exercise is carried out for the extreme statistics max(Y 1, · ··, Yn ), min(U 1,· ··, Un ) and min(V 1, · ··, Vn ).

Seismic risk assessment for structures with the Monte Carlo simulation

A systematic study is presented of random factors involved in risk assessment for structures subjected to strong seismic actions. Various aspects of the randomization of relevant parameters in the numerical simulation procedures are discussed, among them the randomization of maximum accelerations and spectral characteristics of ground motions, and resistance properties of structure. Asymptotic distributions of extremes are applied to obtain more reliable estimates of the seismic risk in the area of rare events in the presence of multiple restrictions on the dynamic response of a structure.

Theoretical Consideration of Asymptotic Distributions of Extremes for the Case of Log-Normal Distribution.

In this study, we investigate asymptotic distributions of extremes for the case when the initial distribution is log-normal. We show theoretically by means of logarithmic transformation that the second asymptotic distribution is derived for the largest value. This result seems to be in contradiction to Gumbel's statement that the first asymptotic distribution is derived for the largest value, since it is not possible for two different types of asymptotic distributions of the largest value to correspond to the same initial distribution. This apparent contradiction is explained by the finding that the resulting first and second asymptotic distributions coincide with each other due to some relation between their distribution parameters. This result is also demonstrated graphically using probability paper. We also examine the asymptotic distribution of the smallest value.

The Asymptotic Theory of Extreme Order Statistics

Abstract. Let X j denote the life length of the j th component of a machine. In reliability theory, one is interested in the life length Z n of the machine where n signifies its number of components. Evidently, Z n = min (X j : 1 ≤ j ≤ n). Another important problem, which is extensively discussed in the literature, is the service time W n of a machine with n components. If Y j is the time period required for servicing the j th component, then W n = max (Y j : 1 ≤ j ≤ n). In the early investigations, it was usually assumed that the X's or Y's are stochastically independent and identically distributed random variables. If n is large, then asymptotic theory is used for describing Z n or W n . Classical theory thus gives that the (asymptotic) distribution of these extremes (Z n or W n ) is of Weibull type. While the independence assumptions are practically never satisfied, data usually fits well the assumed Weibull distribution. This contradictory situation leads to the following mathematical problems: (i) What type of dependence property of the X's (or the Y's) will result in a Weibull distribution as the asymptotic law of Z n (or W n )? (ii) given the dependence structure of the X's (or Y's), what type of new asymptotic laws can be obtained for Z n (or W n )? The aim of the present paper is to analyze the recent development of the (mathematical) theory of the asymptotic distribution of extremes in the light of the questions (i) and (ii). Several dependence concepts will be introduced, each of which leads to a solution of (i). In regard to (ii), the following result holds: the class of limit laws of extremes for exchangeable variables is identical to the class of limit laws of extremes for arbitrary random variables. One can therefore limit attention to exchangeable variables. The basic references to this paper are the author's recent papers in Duke Math. J. 40 (1973), 581–586, J. Appl. Probability 10 (1973, 122–129 and 11 (1974), 219–222 and Zeitschrift fur Wahrscheinlichkeitstheorie 32 (1975), 197–207. For multivariate extensions see H. A. David and the author, J. Appl. Probability 11 (1974), 762–770 and the author's paper in J. Amer. Statist. Assoc. 70 (1975), 674–680. Finally, we shall point out the difficulty of distinguishing between several distributions based on data. Hence, only a combination of theoretical results and experimentations can be used as conclusive evidence on the laws governing the behavior of extremes.

Comparison of Reliability of Ocean Vehicles Using Asymptotic Wave Load Distribution

Asymptotic extreme distribution of wave loads acting on a marine vehicle is derived and used in a reliability framework to estimate the safety of the structure. Both exact and approximate methods of reliability analysis are used and the resulting probabilities of failure are compared. The effect of correlation between the wave load and the still water load on the risk of failure is also examined. The results of the comparison showed that the approximate reliability analysis underestimates the failure probability compared to the exact method, and that the correlation effect between the wave and still water loads is not significant.

Some properties of multivariate extreme value distributions and multivariate tail equivalence

Denote byH ak-dimensional extreme value distribution with marginal distributionHi(x)=Λ(x)=exp(−e−x),x∈R1. Then it is proved thatH(x)=Λ(x1)...Λ(xk) for anyx=(x1, ...,xk) ∈Rk, if and only if the equation holds forx=(0,...,0). Next some multivariate extensions of the results by Resnick (1971,J. Appl. Probab.,8, 136–156) on tail equivalence and asymptotic distributions of extremes are established.

Bayesian Estimation of Parameters of Life Distributions and Reliability from Type II Censored Samples

The Bayesian approach to reliability estimation from Type II censored samples is discussed here with emphasis on obtaining natural conjugate prior distributions. The underlying sampling distribution from which the censored samples are drawn follows a generalized life model (GLM) which includes a model proposed by Epstein and Sobel, Weibull, exponential, and Rayleigh distributions as special cases. Results are given for the Type II asymptotic distribution of largest values, Pareto, and Limited distribution. The natural conjugate prior, Bayes estimate for the generalized scale parameter, posterior risk, Bayes risk and Bayes estimate of the reliability function were derived for the distributions studied. In every case the natural conjugate prior is a 2-parameter family which provides a wide range of possible prior knowledge. Conjugate diffuse priors were derived. A diffuse prior, also called a quasi-pdf, is not a pdf because its integral is not unity. It represents roughly an informationless prior state of knowledge. The proper choice of the parameter for the diffuse prior leads to maximum likelihood, classical uniform minimum-variance unbiased estimator, and an admissible biased estimator with minimum mean square error as the generalized Bayes estimate. A feature of the GLM is the increasing function g(·) with possible applications in accelerated testing. KG(·) is a s-complete s-sufficient statistic for ?, and KG(·)/m is a maximum likelihood estimate for ?. Similar results were obtained for the Pareto, Type II asymptotic distribution of extremes, Pareto (associated with Pearl-Reed growth distribution) and others.

Statistical Methodology for Large Claims

The question of large claims in insurance is, evidently, a very important one, chiefly if we consider it in relation with reinsurance. To a statistician it seems that it can be approached, essentially, in two different ways.The first one can be the study of overpassing of a large bound, considered to be a critical one. If N(t) is the Poisson process of events (claims) of intensity v, each claim having amounts Yi, independent and identically distributed with distribution function F(x), the compound Poisson processwhere a denotes the critical level, can describe the behaviour of some problems connected with the overpassing of the critical level. For instance, if h(Y, a) = H(Y − a), where H(x) denotes the Heavside jump function (H(x) = o if x < o, H(x) = 1 if x ≥ o), M(t) is then the number of claims overpassing a; if h(Y, a) = Y H(Y − a), M(t) denotes the total amount of claims exceeding the critical level; if h(Y, a) = (Y − a) H(Y − a), M(t) denotes the total claims reinsured for some reinsurance policy, etc.Taking the year as unit of time, the random variables M(1), M(2) − M(1), … are evidently independent and identically distributed; its distribution function is easy to obtain through the computation of the characteristic function of M(1). For details see Parzen (1964) and the papers on The ASTIN Bulletin on compound processes; for the use of distribution functions F(x), it seems that the ones developed recently by Pickands III (1975) can be useful, as they are, in some way, pre-asymptotic forms associated with tails, leading easily to the asymptotic distributions of extremes.

A probabilistic approach to the problem of electron localization in disordered systems and sharpness of the mobility edge

By applying the theory of the asymptotic distribution of extremes and a certain stability criterion to the question of the domain of convergence in the probability sense, of the renormalized perturbation expansion (RPE) for the site self-energy in a cellularly disordered system, an expression has been obtained in closed form for the probability of nonconvergence of the RPE on the real-energy axis. Hence, the intrinsic mobility mu (E) as a function of the carrier energy E is deduced to be given by mu (E)= mu 0exp(-exp( mod E mod -Ec) Delta ), where Ec is a nominal 'mobility edge' and Delta is the width of the random site-energy distribution. Thus mobility falls off sharply but continuously for mod E mod >Ec, in contradistinction with the notion of an abrupt 'mobility edge' proposed by Cohen et al. and Mott. Also, the calculated electrical conductivity shows a temperature dependence in qualitative agreement with experiments on disordered semiconductors.

Extreme-value statistics of human red blood cells

Paper is devoted to answer the question: “How large is the largest red blood cell in a blood sample?” The red cell diameter was measured by photographing the cell in an interferometric microscope and processing the hologram by a numerical procedure of superresolution, which yields an improved resolution of 0.02 μm. Data on the largest cells in samples of specific size were collected and analyzed according to the theory of extreme-value statistics. It was found that the statistical distribution of the largest cells follows the asymptote derived for initial distribution of the exponential type. When the parameters of the asymptotic extreme distribution are determined from the observed data, the expected largest red cell in a sample of arbitrary size n can be easily determined as a function of n. For example, for the group of young adult male students examined the predicted largest diameter in a population of 10 8 red blood cells lies between 15.66 and 17.06 μm.

The kinetics and thermodynamics of death in multicellular organisms

The survivorship curves for Drosophila melanogaster, determined from experiments in which the flies are individually housed, are better fitted by a two-constant power law of the form, −1/ N(d N/d t) = At n , than by the Gompertz function: −1/ N(d N/d t) = R 0e αt . Such curves, taken for different ambient temperatures, show that A is a strong function of temperature, while n is not. Over the limited temperature range of the experiments (25–33°C), and Arrhenius plot is obtained for A, yielding an activation enthalpy for death ( ΔH † ) of 190 kcal/mole. This value is much higher than that reported by Strehler of 18.4 kcal/mole, but is consistent with the high activation enthalpies reported for unicellular organisms and mammalian cells in tissue culture. The power law plots, in either differential or integral form, appear to be generally applicable to mammalian age-related death statistics as shown by data on mice, rats and humans. The values of n are approximately 4–5 for all species tested. For human mortality rates, the three major individual causes of death are also well represented by power law plots. The ordinary distribution laws of the theory of probability do not give the correct forms for the mortality rates. However, the asymptotic distribution of extremes in order statistics do yield both the power law and the Gompertz function distributions. We speculate that the existence of high values of activation enthalpies for death predict that small decreases in core body temperatures of homeothermic mammals should produce marked increases in life expectancy.