Gumbel Extreme Value Distribution Research Articles

Heavy-traffic extreme-value limits for queues

We consider the maximum waiting time along the first n customers in the G1/G/1 queue. We use strong approximations to prove, under regularity conditions, convergence of the normalized maximum wait to the Gumbel extreme-value distribution when the traffic intensity ϱ approaches 1 from below and n approaches infinity at a suitable rate. The normalization depends on the interarrival-time and service-time distributions only through their first two moments, corresponding to the iterated limit in which first ϱ approaches 1 and then n approaches infinity. We need n to approach infinity sufficiently fast so that n(1 − ϱ) 2 → ∞. We also need n to approach infinity sufficiently slowly: If the service time has a pth moment for ϱ > 2, then it suffices for (1 − ϱ)n 1 p to remain bounded; if the service time has a finite moment generating function, then it suffices to have (1 − ϱ)log n → 0. This limit can hold even when the normalized maximum waiting time fails to converge to the Gumbel distribution as n → ∞ for each fixed ϱ. Similar limits hold for the queue-length process.

Testing for a Change in the Parameter Values and Order of an Autoregressive Model

The problem of testing whether or not a change has occurred in the parameter values and order of an autoregressive model is considered. It is shown that if the white noise in the AR model is weakly stationary with finite fourth moments, then under the null hypothesis of no changepoint, the normalized Gaussian likelihood ratio test statistic converges in distribution to the Gumbel extreme value distribution. An asymptotically distribution-free procedure for testing a change of either the coefficients in the AR model, the white noise variance or the order is also proposed. The asymptotic null distribution of this test is obtained under the assumption that the third moment of the noise is zero. The proofs of these results rely on Horvath's extension of Darling-Erdos' result for the maximum of the norm of a $k$-dimensional Ornstein-Uhlenbeck process and an almost sure approximation to partial sums of dependent random variables.

Intermittent wind close to the ground within a grass canopy

Wind speed was measured at a height of 1 cm above the ground and at several other heights in and above a canopy of tall fescue grass (Festuca arundinacea) using single hot-wire and triple hot-film anemometers. The plant area density in the canopy was concentrated close to the ground, with ∼75% of the plant area standing belowz=15 cm, wherez is height above the ground. The frequency distributions of horizontal wind speeds,s, were sharply skewed towards positive values at all measurement heights, but were most highly skewed near the ground where the coefficient of skewness ranged from 1.6 to 2.9. Above mid-canopy height, the frequency distribution ofs was described reasonably well by a Gumbel extreme value distribution. Average wind speed,S, decreased exponentially with depth into the canopy with an exponential scale length of abouth/2.8, whereh is the height of the canopy. Atz=1 cm, the value ofS was about 11% of the surface-layeru *. The standard deviation of the fluctuations of the vertical and horizontal components of the wind speed also decreased exponentially with depth inside the canopy with a scale length of abouth/2.5. Inside the canopy, the Eulerian integral time scales for the vertical (Γ w ) and horizontal (Γ u ) components of wind speed were about 0.1 s and 1.0 s, respectively, and were approximately constant with height. Above the canopy, these time scales increased sharply and, atz=2.25h, Γ w and Γ u were approximately 1.0 and 3.0s, respectively. Turbulence length scales in the vertical and downwind directions, Γ u and Γ w ·U, respectively, were approximately 1 cm for heights between 1 to 10 cm above the ground inside the canopy, while atz=2.25h, they were about 55 cm and 277 cm. Relatively quiescent periods (lulls) in the air close to the ground were interrupted frequently by gusts. The frequency of occurrence of gusts appears to be correlated with the value of the local shear near the top of the canopy.

A Program of Frequency Analysis Using Gumbel's Method

AbstractA program (GUMBEL) has been written in QBASIC language to carry out the mathematical procedure for preparing frequency curves using Gumbel's extreme value distribution. The program can be used to analyze hydrologie and meteorologie events so as to establish a relationship between the magnitude of extreme events and their frequency through the use of probabilistic distributions.Six different plotting positions formulae are included in the program so as to select the most reliable formula of these graphical fitting methods in analyzing the observed hydrologie data. A sample input data file of annual rainfall data from Jordan has been tested, in order to show the applicability and usefulness of the program.

Reliability analysis of window glass failure pressure data

Results are presented for reliability analyses of 14 samples of new, full-size, simply supported monolithic glass plates and PVB laminated glass units that were tested to destruction. In all, the data base includes 318 specimens that were tested to failure under uniform lateral pressure and under controlled laboratory experimental conditions. Six probability distributions are employed in the reliability analyses. Results indicate that, a structural design perspective, the 2-parameter Weibull distribution consistently produces conservative (but not unduly low) estimates of sixty-second constant failure pressures ( P 60 ) at target reliabilities of 0.992 and 0.999. Normal and 3-parameter Weibull distributions appear to be usable for reliability analyses of architectural glass failure pressures, but they are not as consistent in producing reasonable lower bounds of glass failure pressures as is the 2-parameter Weibull distribution. Lognormal distributions (2- and 3-parameter) and the Gumbel extreme value distribution do not always produce meaningful results. At room temperature, the reliability analysis results do not support the presence of a 0.75 strength factor in building codes for specifying the lateral pressure resistance of PVB laminated glass units relative to that of monolitic glass plates. In fact, reliability analysis results herein indicate comparable nominal resistances to lateral pressure for monolithic glass plates and PVB laminated glass units at room temperature. At elevated temperatures, the “relative strength issue” becomes more complex; compared to test results at room temperature, a strength reduction clearly occurs in PVB laminated glass units tested at 170°F (77°C), but little strength reduction appears to occur at 120°F (49°C).

Revised ground snow loads for the 1990 National Building Code of Canada

The last systematic recalculation of ground snow loads in the Supplement to the National Building Code of Canada was made in 1977 and used data up to 1975. Data from three times as many stations are now available and there is also an additional 10 years of record. Using this expanded data base, ground snow loads have been recalculated for the 1990 Supplement.Several changes in methods have been utilized, the most significant of which is the use of an objective technique to estimate ground snow loads at Code (or other) locations. It explicitly incorporates an assumed dependence of the snow load on topographical elevation, and accounts for the magnitude of errors at snow depth observation sites. Other differences include (a) the use of the method of moments to fit the Gumbel extreme value distribution for the purpose of estimating the 30-year return period snow depth; (b) the use of geographically varying snow pack densities; and (c) using probabilistic rain components of the total snow load and estimating this component by use of a snow pack model.Results show an average national decrease of 6.6% in the 1990 loads compared with those in the 1985 Supplement. A regional exception is in the Northwest Territories where the use of a greater snow density has led to an average increase of about 16% in the loads. A reduction in the standard deviation about the mean load suggests a more spatially consistent set of values for the 1990 Supplement. Key words: snow, loads, building, code.

A note on the asymptotic normality of sums of extreme values

Let $X_1$, $X_2$,... be a sequence of independent random variables with common distribution function $F$ in the domain of attraction of a Gumbel extreme value distribution and for each integer $n\geq 1$, let $X_{1,n} \leq ... X_{n,n}$ denote the order statistics based on the first $n$ of these random variables. Along with related results it is shown that for any sequence of positive integers $k_n \rightarrow +\infty$ and $k_{n}/n \rightarrow 0$ as $n \rightarrow 0$ the sum of the upper $k_n$ extreme values $X_{n-k_{n},n}+...+X_{n,n}$, when properly centered and normalized, converges in distribution to a standard normal random variable $N(0, 1)$. These results constitute an extension of results by S. Cs\"{o}rg\H{o} and D.M. Mason (1985).

Some Graphical Techniques for Estimating Weibull Confidence Intervals

Many methods for estimating the parameter and percentile statistical confidence intervals for the Weibull and Gumbel (extreme value) distributions have been described in the literature. Most of these methods depend on extensive computer programs, require reference to tables which do not cover all sample sizes of interest and/or are not widely available. This paper describes a semi-empirical technique which permits rapid estimation of the 2-sided 90% statistical confidence intervals for the Weibull or Gumbel distribution parameters, as well as for the 1, 5, 10 percentiles. The estimates can be obtained for type II censoring and sample sizes to 25. The statistical confidence intervals calculated using this method are not exact, but are very good approximations and are useful to engineers who do not have ready access to programs or lengthy tables, or who require quick estimates. If more accurate statistical confidence intervals are required, then the more complicated methods described in the references should be used.

Stochastic seismic sliding of rigid mass supported through non‐symmetric friction

AbstractThe sliding behaviour of a rigid mass supported on a randomly vibrating foundation through a non‐symmetric Coulomb‐friction contact is studied both analytically and by numerical simulation. The analysis is based on a stationary solution of the associated Fokker‐Planck equation, and makes use of equivalent linearization and of a suitable decomposition of the non‐zero mean non‐stationary sliding process. It is shown that the analytical results yield several exact asymptotic expansions for both small and large values of time. An extensive Monte Carlo type numerical simulation study produces non‐stationary response statistics which are in very good accord with the analytical results. Furthermore, it is found that Gumbel's Extreme Value Distribution reproduces with remarkable accuracy the observed cumulative frequency of maximum slip displacement. The results of this paper may find application in seismic design of embankment dams, earth retaining walls and base ‘isolation’ systems.