Gumbel Type Research Articles

Maximum response statistics of MDoF linear structures excited by non-stationary random processes

The paper deals with the problem of predicting the maximum response statistics of Multi-Degree-of-Freedom (MDoF) linear structures subjected to non-stationary non-white noises. The extension of two different censored closures of Gumbel type, originally proposed by the authors for the response of Single-Degree-of-Freedom oscillators, it is presented. The improvement associated with the introduction in the closure of a consistent censorship factor, accounting for the response bandwidth, it is pointed out. Simple and effective step-by-step procedures are formulated and described in details. Numerical applications on a realistic 25-storey moment-resisting frame along with comparisons with classical approximations and Monte Carlo simulations are also included.

An Extreme Value Analysis of Pollutant Concentrations in Surface Soils Due to Atmospheric Deposition

Risk assessments often rely on deterministic models using long-term averages or “steady-state” values of input variables. Such models do not provide the information needed to estimate acute exposures. This study uses extreme value theory to examine the frequency and magnitude of daily pollutant concentrations in surface soils predicted at six U.S. locations. Concentrations are predicted using a deposition-leaching model and 50 years of historical precipitation data. A stochastic model also is used to generate 1000 years of precipitation data as modeling inputs for each location. The annual maximum concentrations at each site are fitted to a Gumbel type I distribution to estimate occurrence probability. For soluble pollutants, the predicted concentration varied substantially with precipitation, and the maximum daily concentrations exceeded annual averages by 4 to 8 times. Observed and synthetic precipitation data produced similar results at most study locations, though the synthetic data provided a slightly better fit to the Gumbel type I distribution. The precipitation model allows the generation of representative precipitation data that extend limited historical records. The extreme value analysis facilitates the evaluation of maximum pollutant concentrations, return periods, and other statistics that are important in evaluating acute exposures.

On the evaluation of probabilistic displacement spectra

In this paper, on the basis of a large set of ground motion records. the result of a statistical analysis on the elastic displacement spectra and the comparison between different theoretical probabilistic distribution will be presented for a selected number of vibration periods. The following four theoretical distributions have been taken into account: Normal. Lognormal. Gamma and Gumbel type I. Since the best fit to the data was provided from the Lognormal distribution, in this paper lognormal displacement spectra will be defined, for different levels of probability of non-exceedance. for the estimation of the displacement demands to structures located on different local soil condition. at different distance from the causative fault, and for different levels of magn~tude.

Sequence alignment: an approximation law for the Z-value with applications to databank scanning

The Z-value is an attempt to estimate the statistical significance of a Smith and Waterman dynamic programming alignment score ( H-score) through the use of a Monte-Carlo procedure. In this paper, we give an approximation for the Z-value law deduced from the Poisson clumping heuristic developed by Waterman and Vingron (Stat. Sci. 9 (1994) 367) in the case of independent and identically distributed sequences comparison. As for non-gapped alignment scores, our approximation is of Gumbel type but with parameters that are sequence independent. This result makes clear the related experimental results mentioned by Comet et al. (Comput. Chem. 23 (1999) 317). Using ‘quasi-real’ sequences (i.e. randomly shuffled sequences of the same length and amino acid composition as the real ones) we investigate the relevance of our approximation result. Since the Monte-Carlo approach we use generates a bias for the Gumbel decay parameter estimation, a correction procedure is proposed. Applications to real sequences are considered and we show how our results can be used to detect the potential biological relationships between real sequences.

Limit theorems for a class of tests of gradual changes

Abstract We study a class of asymptotic tests for detecting various types of gradual changes in a location model. The results of Jaruskova (1998. J. Statist. Plann. Inference 70, 263–276) and Huskova (1998. In: Szyszkowicz, B. (Ed.), Asymptotic Methods in Probability and Statistics. Elsevier, Amsterdam, pp. 577–584.) are extended to other families of change alternatives. Under the null hypothesis of “no change”, limiting extreme value distributions of Gumbel type are derived for some asymptotic log-likelihood ratio test statistics. Consistency under certain alternatives is also discussed.

Return periods of extreme water levels estimated for some vulnerable areas of Buenos Aires

In Argentina, particularly in the province of Buenos Aires, densely populated areas are affected by swell inundations. This paper estimates the return periods of extreme levels, including those fixed by the Dirección Nacional de Planeamiento y Protección Civil (DNPPC) as evacuation warnings. The analysis combines the probability density functions (pdf) of tides and surges, provided their independence. The tidal pdf is generated from an hourly prediction for 19 yr. The surge pdf is obtained applying the generalized extreme value (GEV) distribution to a set of extreme surges generated by removing tides from a series of 89 yr of annual maxima. This set is chosen in order to make best use of the historical information since hourly water levels are not available for a long enough period. It is shown that in this case, the probability distribution which best fits to the surge data is that of Gumbel Type I. For an evacuation warning level of 3.30 m the estimated return period is 5 yr. The last 89 yr maximum (4.44 m) has a recurrence of approximately 265 yr. Estimated return periods are compared with those computed by the classical annual maxima method for the same period. The extreme levels and return periods estimated herein strongly justify the undertaking of actions in order to mitigate the serious consequences caused by floods in these low-lying areas.

Random features of the fatigue limit

The classical fatigue limit is often an important characteristic in fatigue design regarding metallic material. The limit is usually obtained from a staircase test in combination with some assumption about the statistical distribution of the limit. This distribution can be of a normal, log-normal or of extreme value type and no particular physical argument gives favor to any specific distribution. This leads to a certain ambiguity in the evaluation of test results which forces the designer to introduce large safety factors. In order to find a physically based statistical distribution for use in staircase tests to determine the fatigue limit we present here a random model for the fatigue limit based on the following assumptions; (i) The square root area model according to Murakami and co-workers is valid, (ii) the randomness in the fatigue limit is induced by the randomness of the maximum defect size, (iii) the random maximum defect size has an extreme value distribution of Gumbel type. This leads to the fatigue limit distribution based on Gumbel (FLG), which is recommended to replace the normal distribution in the evaluation of staircase fatigue tests in case of hard materials. It turns out that the skewness of the resulting distribution depends on the coefficient of variation; with a normal-like non-skewed distribution at the coefficient of variation of five percent.

Risk of Extreme Events Under Nonstationary Conditions

The concept of the return period is widely used in the analysis of the risk of extreme events and in engineering design. For example, a levee can be designed to protect against the 100‐year flood, the flood which on average occurs once in 100 years. Use of the return period typically assumes that the probability of occurrence of an extreme event in the current or any future year is the same. However, there is evidence that potential climate change may affect the probabilities of some extreme events such as floods and droughts. In turn, this would affect the level of protection provided by the current infrastructure. For an engineering project, the risk of an extreme event in a future year could greatly exceed the average annual risk over the design life of the project. An equivalent definition of the return period under stationary conditions is the expected waiting time before failure. This paper examines how this definition can be adapted to nonstationary conditions. Designers of flood control projects should be aware that alternative definitions of the return period imply different risk under nonstationary conditions. The statistics of extremes and extreme value distributions are useful to examine extreme event risk. This paper uses a Gumbel Type I distribution to model the probability of failure under nonstationary conditions. The probability of an extreme event under nonstationary conditions depends on the rate of change of the parameters of the underlying distribution.

Frequency analysis of low flows at gaged points of the Ansung stream

The Power and SMEMAX transformation, Gumbel type III, Weibull, log-Pearson, type III, lognormal probability papers based on plotting position formula are considered to estimate 7-day and 30-day low flows for return periods at four gaged points of the Ansung stream in Korea. It is found that the Power transformation is the superior ability to the SMEMAX transformation. A trial is also performed to examine the applicability of the Power transformation by changing the parameters. The Kolmogorov-Smirnov, test (K-S test) and the Chi-square test (x 2 test) are adopted in the trial There may exist a range of acceptable parameters, which implies the high applicability of the Power transformation. The method of moments or the maximum likelihood procedure is adopted to calculate the parameters of the Gumbel type III, the Weibull and the log-Pearson type III distribution. The K-S test is performed to test the goodness of fit for each distribution. The lognormal distribution with the Weibull plotting position consistently performs well and the data separation technique based on the least median of squares (LMS) method improves the coefficients of skewness in the lognormal distribution. The LMS method, easily detects the outliers of the sample and contributes to improve the properties of the normal distribution. The Power transformation, the log-Pearson type III and the lognormal distribution provide good fit to low flows for the Ansung stream. The LMS method may enhance the reliability of low flow analysis. It is also found that the flows at the Ansung stream have been significantly influenced by controlled releases from the upper reservoirs.

Transformation of point rainfall to areal rainfall: Intensity-duration-frequency curves

Abstract Current approaches to constructing catchment intensity-duration-frequency (IDF) curves are dominated by the use of empirically-derived areal reduction factors (ARFs). In this paper we present an alternative methodology which is based on the spatial correlation structure of rainfall. It represents an attempt to link current scientific theories of space-time rainfall fields with design methods. The starting point is to derive the parent distribution of catchment average rainfall intensity from that of point rainfall intensity. The parameters of the two parent distributions are related through a variance reduction factor which is a function of the spatial correlation structure of rainfall and catchment area. Assuming that the parent distribution is of the “exponential type”, it is then transformed to an extreme value distribution of the Gumbel type. The crucial step is to match the parameters of the extreme rainfall distribution derived above, for the particular case of zero catchment area, with those of empirical point IDF curves which have also been fitted to the Gumbel distribution. With this match, the proposed theory then naturally generalises to yield catchment IDF curves for catchments of any size, and for rainfall of any spatial correlation structure. The new catchment IDF curves have the attractive property that, with a minimum number of assumptions, they can reproduce a range of observed properties of catchment rainfall. For example, not only the mean and the standard deviation of extreme rainfall, but also its coefficient of variation, decrease with increasing catchment area. We also find that computed ARFs using the new approach depend not only on catchment area and storm duration, but also on the return period. We estimate ARFs using the new methodology for two major observed storms in Austria, and find that these estimates compare favourably with our understanding of the rainfall generating mechanisms associated with these two particular storm types.

ON THE NUMBER OF RECORDS NEAR THE MAXIMUM

summaryRecent work has considered properties of the number of observations Xj, independently drawn from a discrete law, which equal the sample maximum X(n) The natural analogue for continuous laws is the number Kn(a) of observations in the interval (X(n)–a, X(n)], where a > 0. This paper derives general expressions for the law, first moment, and probability generating function of Kn(a), mentioning examples where evaluations can be given. It seeks limit laws for n→ and finds a central limit result when a is fixed and the population law has a finite right extremity. Whenever the population law is attracted to an extremal law, a limit theorem can be found by letting a depend on n in an appropriate manner; thus the limit law is geometric when the extremal law is the Gumbel type. With these results, the paper obtains limit laws for ‘top end’ spacings X(n) ‐ X(n‐j) with j fixed.

A Measure of Association for Bivariate Frailty Distributions

A general formula is derived for the variance of Kendall's coefficient of concordance for absolutely continuous bivariate distributions generated by frailties. The formula is specialized to the case of Gumbel's Type II distribution of extreme values, which arises when the frailty has a positive stable distribution. A new diagnostic is suggested for the goodness of fit of a bivariate frailty distribution, based on the value of Kendall's tau for a truncated sample. It is shown that as a function of the proportion of the sample truncated, this truncated tau characterizes the original frailty distribution. The methods are applied to some cable insulation failure data.

Stress-strength interference theory for a pin-loaded composite joint

Considering the increasing applications of composites as structural components, the reliability of these materials is an important issue. In this paper, stress-strength interference theory is used to determine the reliability of a component that is characterized having a Weibull strength distribution and is under the effects of an applied load that is not deterministic, but follows a probability distribution. The assumption that failure is caused by the maximum of a sequence of applied loads is the basis for describing the applied load distribution as a Gumbel Type 1 extreme value distribution. Reliability plots are given for a class of strength parameters that are typical of graphite/epoxy laminates. A numerical example for a pin-loaded composite laminate is shown using experimentally obtained data.

On the characteristics of extreme values for series systems

Abstract This paper investigates the characteristics of extreme events for series systems. A methodology is developed to determine the Gumbel Type of a series system, given that the Gumbel Types of the components are known. Determining the propagation of the Gumbel Types can be accomplished without knowing the exact probability density functions of the components and without calculating the analytic form for the distribution of the overall system. In addition, an analytical technique is developed to determine the parameters of the extreme value distribution—the characteristic largest value, and the inverse measure of dispersion—for the overall series system. Finally, these three pieces of information—the Gumbel Type, the characteristic largest value, and the inverse measure of dispersion—are combined to calculate a conditional expected value of extreme events of the overall series system.

Regional analysis of annual maxima precipitation using L-moments

Current precipitation frequency analysis used as a standard for urban runoff calculations are out of date. In this paper, the form of the distribution governing rainfall intensity of various durations in a regional context is investigated using the method of L-moments. The results indicate that the currently used Gumbel type I distribution is not generally valid for all the durations considered in this analysis. The form of the distribution is dependent on the duration of the event being considered.

Generalized Gumbel and likelihood ratio test statistics in the multivariate GEV model

Let X denote a sample from a multivariate Generalized Extreme Value – GEV( θ) – model. In this paper asymptotic properties of Gumbel type statistics – which properly balance both the upper and lower tail – are derived and asymptotic properties of such statistics for testing H 0: θ = 0 in the multivariate GEV( θ) model are obtained. A likelihood ratio test for H 0: θ = 0 is also developed. A suitable algorithm for obtaining the maximum likelihood estimates of the unknown parameters is implemented. Comparison of tests is made through simulation methods.

Mixture distributions - an alternative approach for estimating maximum magnitude earthquake occurrence

Summary. Gumbel’s theory of extreme value has been employed in the statistical forecasting of maximum-magnitude earthquake occurrence. The basic working hypothesis behind this method assumes that observations follow either Gumbel type I or type 111 asymptotic distributions. In certain cases, however, it is found that neither type of distribution fits the data well enough to produce accurate parameter-estimates, particularly in the larger earthquake range. This article proposes an alternative approach based on finite-mixture distributions whereby a more realistic prediction of upper earthquake magnitudes (at given return periods) is expected using a combined analysis of both Gumbel types I and I11 extremal distributions.

Probability Density Function of the Product and Quotient of Two Correlated Exponential Random Variables

AbstractThis article deals with the distributions of the product and the quotient of two correlated exponential random variables. We consider here three types of bivariate exponential distributions: Marshall-Olkin's bivariate exponential distribution, Gumbel's Type I bivariate exponential distribution, and Gumbel's Type II bivariate exponential distribution.

Bids for silvicultural services: estimating reasonable costs and identifying bidding errors

Econometric procedures were developed for (i) estimating the reasonable cost of a contract before advertising for bids and (ii) identifying abnormally low bids immediately after the bids tendered are opened. Two-order statistics were used in this initial study of low bidding in forestry. The ratio G (low bid minus mean of all bids divided by the standard deviation of all bids) follows Gumbel's type I extreme value distribution. The natural logarithm of the ratio M (the margin between the two lowest bids divided by the range in bids) has the same form. The "reasonable cost" of a contract not yet awarded can be estimated as a function of the distributions of G and M and the mean low bid and range of bids from prior contracts for the same service. Statistical tests for outliers from exponential distributions of order statistics can help identify abnormally low bids that may contain bidding errors. The moments of the G and M distributions reveal that half the silvicultural contractors are foregoing profits greater than 15% of the total contract cost, more than $6.79 per acre. The principal reasons are incomplete market information for preparing bids, heterogeneous cost structures among bidders, and cost advantages arising from work site location compared with the base of operations.

Statistical aspects of New England seismicity from 1975 to 1982 and implications for past and future earthquake activity

AbstractThe seismicity of New England from 1 October 1975 to 30 November 1982 has been studied to determine how well it represents past seismicity and what it might imply about future earthquakes. Just over 100 earthquakes with magnitudes between 2.0 and 4.7 were recorded during this modern time period, while the historic earthquake catalog of Chiburis (1981) lists 760 earthquakes in New England from 1568 to 1975, the largest having maximum Modified Mercalli intensity VII shaking. The spatial distribution of the recent seismicity reproduces that for the historical seismicity patterns quite well, with the larger earthquakes in the modern record having preferentially occurred along coastal and lowland New England or in central New Hampshire. Space-time plots show that the seismicity displays considerable spatial variation, and that there are only vague hints of migrations of epicenters. The number of earthquakes as well as the amount of seismic energy release appear essentially random in time, and the recurrence curve for the modern time period (log number of events versus magnitude) is a straight line for events between magnitudes 2 and 4.5 with a b value of 0.84 (±0.03). The sizes of the largest earthquakes in the historic record are those which would be expected by extrapolating the present seismicity rate backward in time using a Gumbel type I distribution. One major discrepancy between the statistics of the recent and historic catalogs is that the latter does not have as many moderate (magnitudes 5 to 5.8) earthquakes as would be expected from the recent data. This discrepancy could be due to inaccuracies in the historic catalog, to unusually high recent seismicity or to a real historical shortage of such moderate magnitude earthquakes. Large earthquakes may be more likely to happen in areas of past and present high seismicity, especially in central and southern New Hampshire, along coastal Maine, and along the south coast to New England.