An Alternative to the Weibull Function for Some Cases

1991 ◽  
Vol 113 (2) ◽  
pp. 195-199 ◽  
Author(s):  
D. J. Neville ◽  
J. B. Kennedy
Keyword(s):  
Fundamental Problem ◽  
Weak Link ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Distribution Functions ◽  
Weibull Function ◽  
Link Function ◽  
Statistical Parameters ◽  
Weakest Link ◽  
Large Sets

Doubt about the applicability of the Weibull function has been expressed by various workers, some of whom have suggested modifications to the Weibull function. Such modifications usually involve more parameters than the original Weibull function being thus much more flexible and thereby, in some cases, providing a good fit if the numerous (up to six) parameters can be estimated. These functions are not valid as asymptotic extreme-value distribution functions and thus represent a departure from the so-called weak-link principle. A fundamental problem with the Weibull approach, the lack of statistical independence of volume elements, will be briefly discussed. For cases where failure is caused by sharp defects a new extreme-value (weakest-link) function has been developed on the basis of the mechanics of the near-tip regions of such defects. The new function has only two statistical parameters which can be measured easily from plots, graphically or by least-squares fitting. Several large sets of data, fracture toughness, and fracture stress from several different materials will be shown, to which the new function provides a much better fit than the Weibull function.

DIscrepancy of Gear Tooth Allowable Bending Stress Using the Extreme Value Distribution From Different Test Methods

1991 ◽  
Author(s):  
Chienann A. Hou ◽  
Shijun Ma
Keyword(s):  
Gear Tooth ◽  
Extreme Value Distribution ◽  
Test Methods ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
Test Method ◽  
Bending Stress ◽  
Gear Design ◽  
Experimental Approaches

Abstract The allowable bending stress Se of a gear tooth is one of the basic factors in gear design. It can be determined by either the pulsating test or the gear-running test. However, some differences exist between the allowable bending stress Se obtained from these different test methods. In this paper, the probability distribution functions corresponding to each test method are analyzed and the expressions for the minimum extreme value distribution are presented. By using numerical integration, Se values from the population of the same tested gear tooth are obtained. Based on this investigation it is shown that the differences in Se obtained from the different test methods are significant. A proposed correction factor associated with the different experimental approaches is also presented.

scholarly journals Optimization of Parameters in the Generalized Extreme-Value Distribution Type 1 for Three Populations Using Harmonic Search

Atmosphere ◽  
2019 ◽  
Vol 10 (5) ◽  
pp. 257 ◽  
Author(s):  
Juan Pablo Molina-Aguilar ◽  
Alfonso Gutierrez-Lopez ◽  
Jose Angel Raynal-Villaseñor ◽  
Luis Gabriel Garcia-Valenzuela
Keyword(s):  
Extreme Values ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
Generalized Extreme Value Distribution ◽  
Type I ◽  
Generalized Extreme Value ◽  
Heuristic Technique ◽  
Distribution Type

Due to its geographical position, Mexico is exposed annually to cold fronts and tropical cyclones, registering extremely high values that are atypical in the series of maximum annual flows. Univariate mixed probability distribution functions have been developed based on the theory of extreme values, which require techniques to determine their parameters. Therefore, this paper explores a function that considers three populations to analyze maximum annual flows. According to the structure of the Generalized Extreme-Value Distribution (GEV), the simultaneous definition of nine parameters is required: three of location, three of scale, and three of probability of occurrence. Thus, the use of a meta-heuristic technique was proposed (harmonic search). The precision of the adjustment was increased through the optimization of the parameters, and with it came a reduction in the uncertainty of the forecast, particularly for cyclonic events. It is concluded that the use of an extreme value distribution (Type I) structured with three populations and accompanied by the technique of harmonic search improves the performance in respect to classic techniques for the determination of its parameters.

Tail equivalence and its applications

1971 ◽  
Vol 8 (01) ◽  
pp. 136-156 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
Normalizing Constants ◽  
Independent Identically Distributed

If for two c.d.f.'s F(·) and G(·), 1 – F(x)/1 – G(x) → A, 0 <A <∞ , as x → ∞, then for normalizing constants an > 0, bn, n > 1, Fn (anx + bn ) → φ(x), φ(x) non-degenerate, iff Gn (anx + bn )→ φ A−1(x). Conversely, if Fn (anx+bn )→ φ(x), Gn (anx + bn ) → φ'(x), φ(x) and φ'(x) non-degenerate, then there exist constants C >0 and D such that φ'(x) =φ(Cx + D) and limx→∞ 1 — F(x)/1 — G(x) exists and is expressed in terms of C and D, depending on which type of extreme value distribution φ(x) is. These results are used to study domain of attraction questions for products of distribution functions and to reduce the limit law problem for maxima of a sequence of random variables defined on a Markov chain (M.C.) to the independent, identically distributed (i.i.d.) case.

DETERMINATION OF EXTREME VALUES OF WAVES, WINDS CURRENTS, AND TIDES FOR THE DESIGN OF OFFSHORE INSTALLATIONS

1974 ◽  
Vol 14 (1) ◽  
pp. 166
Author(s):  
P. M. Aagaard
Keyword(s):  
Wave Height ◽  
Extreme Values ◽  
Meteorological Data ◽  
Extreme Value Distribution ◽  
Relevant Information ◽  
Extreme Value ◽  
Distribution Functions ◽  
Design Parameters ◽  
Final Design ◽  
Wave Heights

Frequently the only relevant information available to a designer about a propective offshore platform site is its location, the water depth, and whatever can be gleaned from oceanographic atlases. In spite of this lack of data the platform designer is faced with the problem of selecting design parameters such that the proposed platform will not fail during its exposed life. He therefore needs to know what are the greatest wave height, current speed, etc., the platform will experience, and must specify studies that can provide the needed information on extreme values. This paper discusses methods used in such studies and their associated uncertainties.The method for acquiring extreme value data should be chosen on the basis of available oceanographic and meteorological data for the site, reliability requirements, time available before final design, and cost. Wave height is usually the most critical design parameter. Data over a long time span (e.g. greater than ten years) are needed to achieve reliable extreme values. Measured wave data covering such time spans are almost never available for a site of interest, and schedules seldom permit lengthy data-collection periods. Frequently the most reliable extreme wave heights can be obtained by calculating wave heights (i.e. hindcasting) from windfields derived from historical weather charts and fitting certain extreme-value distribution functions to the hindcast results. This preferred approach should include calibration of the wave height calculation method with local measured data. Alternative approaches, usually involving greater uncertainties in predicted extremes, are also appropriate for particular cases. Methods for determining extreme winds, currents, and tides are similar to those used for extreme waves, but some differences result from the nature of the phenomena and the type of data typically available.

Statistical Modeling for One Hour Rainfall Data in Kuala Lumpur and Selangor

2014 ◽  
Vol 1030-1032 ◽  
pp. 665-668
Author(s):  
Amanda Lee Sean Peik ◽  
Choong Wee Kang ◽  
Andy Chan
Keyword(s):  
Extreme Value Distribution ◽  
Extreme Rainfall ◽  
Extreme Value ◽  
Distribution Functions ◽  
Rainfall Data ◽  
Kuala Lumpur ◽  
Maximum Rainfall ◽  
Two Phases ◽  
Annual Maximum Rainfall ◽  
The One

The purpose of this study is to assess patterns of extreme rainfall and this study focused on the changes between two phases for extreme rainfall, for the period of 1971 to 2011 and from 1995 to 2011 in Kuala Lumpur and Selangor. The generalised extreme value distribution appears to outperform other distribution functions such as two-parameter Gumbel and lognormal and the three-parameter generalized extreme value (GEV), lognormal (LN3) and log Pearson (LP3) in modeling the one-hour annual maximum rainfall series from 14 stations. The estimated return period of 20, 50, 100-year for each stations based on the best fitting model for the periods of entire record data and from 1995-2011 have been computed. More than 70% of estimated quantiles using rainfall data from 1995-2011 are higher compared to estimation using the entire recorded data.

Modeling the wind speed in North Morocco

2021 ◽  
pp. 0309524X2110639
Author(s):  
Zuhair Bahraoui
Keyword(s):  
Distribution Function ◽  
Wind Speed ◽  
Ground Cover ◽  
Extreme Value Distribution ◽  
Value Theory ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
Generalized Extreme Value Distribution ◽  
Generalized Extreme Value

The change of the wind speed is strictly related to several natural factors such as local topographical and the ground cover variations, then any adjustment has to take into account the statistical variation for each specific region under study. Unlike the Weibull distribution, which is most used in wind speed modeling, we investigate two alternative distribution functions for wind speed by using the extreme value theory. The generalized Champernowne distribution function and the mixture Log-normal-Pareto distribution function are considered. We demonstrate that the proper generalized extreme value distribution gives a good fit for wind speed in the North Moroccan. In order to validate the models, a comparison of the produced aggregate wind energy in the aeolian wind turbine was being established. The empirical study shows that the generalized extreme value distribution reflects better the intensity of the wind power energy.

Tail equivalence and its applications

1971 ◽  
Vol 8 (1) ◽  
pp. 136-156 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
Normalizing Constants ◽  
Independent Identically Distributed

If for two c.d.f.'s F(·) and G(·), 1 – F(x)/1 – G(x) → A, 0 <A <∞, as x → ∞, then for normalizing constants an > 0, bn, n > 1, Fn(anx + bn) → φ(x), φ(x) non-degenerate, iff Gn(anx + bn)→ φ A−1(x). Conversely, if Fn(anx+bn)→ φ(x), Gn(anx + bn) → φ'(x), φ(x) and φ'(x) non-degenerate, then there exist constants C >0 and D such that φ'(x) =φ(Cx + D) and limx→∞ 1 — F(x)/1 — G(x) exists and is expressed in terms of C and D, depending on which type of extreme value distribution φ(x) is. These results are used to study domain of attraction questions for products of distribution functions and to reduce the limit law problem for maxima of a sequence of random variables defined on a Markov chain (M.C.) to the independent, identically distributed (i.i.d.) case.

Sea Ice Distribution in the Barents Sea

2016 ◽  
Author(s):  
Sindre M. Fritzner ◽  
Trond Sagerup
Keyword(s):  
Sea Ice ◽  
Normal Distribution ◽  
Barents Sea ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Distribution Functions ◽  
Value Distribution ◽  
The Barents Sea ◽  
Ice Edge ◽  
Sea Ice Edge

This paper provides a statistical description of the sea ice occurrence in the Barents Sea, using yearly maximum sea ice data for the last 36 years from the European Centre for Medium-Range Weather Forecasts (ECMWF). A set of four distribution functions have been estimated with the maximum likelihood method. The distribution functions used were Extreme Value distribution, Gumbel distribution, Normal distribution and kernel density estimation. The normal distribution was found to fit the data best and provide the most likely result. Our results verify dependency of the North Atlantic current on the sea ice edge. Warm water northwards prevents the ice from extending south; this makes the extreme value distribution unlikely since this will prevent long tailed distributions. The results for sea ice occurrence are compared to the boundaries given in the proposed revision to NORSOK N-003. These boundaries were found to be too simplistic and not necessarily conservative. Here we have proposed new and more accurate boundaries for the sea ice occurrence. We have found trends indicating northwards movement of the sea ice edge in the Norwegian Sea and eastern parts of the Barents Sea. These trends are mostly due to less ice in the last ten years and not trends for the whole period. In the south-western parts of the Barents Sea where oil and gas operations are imminent no trends have been discovered. The lack of trend is related to the islands in the western Barents Sea.

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