$U{\hbox{-}}\textrm{max}$ statistics were introduced by Lao and Mayer in 2008. Such statistics are natural in stochastic geometry. Examples are the maximal perimeters and areas of polygons and polyhedra formed by random points on a circle, ellipse, etc. The main method to study limit theorems for $U{\hbox{-}}\textrm{max}$ statistics is via a Poisson approximation. In this paper we consider a general class of kernels defined on a circle, and we prove a universal limit theorem with the Weibull distribution as a limit. Its parameters depend on the degree of the kernel, the structure of its points of maximum, and the Hessians of the kernel at these points. Almost all limit theorems known so far may be obtained as simple special cases of our general theorem. We also consider several new examples. Moreover, we consider not only the uniform distribution of points but also almost arbitrary distribution on a circle satisfying mild additional conditions.

Flight delays may be decreased in a predictable way if the Weibull wind speed parameters of a runway, which are an important aspect of safety during the take-off and landing phases of aircraft, can be determined. One aim of this work is to determine the wind profile of Hasan Polatkan Airport (HPA) as a case study. Numerical methods for Weibull parameter determination perform better when the average wind speed estimation is the main objective. In this paper, a novel objective function that minimises the root-mean-square error by employing the cumulative distribution function is proposed based on the genetic algorithm and particle swarm optimisation. The results are compared with well-known numerical methods, such as maximum-likelihood estimation, the empirical method, the graphical method and the equivalent energy method, as well as the available objective function. Various statistical tests in the literature are applied, such as R2, Root-Mean-Square Error (RMSE) and $\chi$2. In addition, the Mean Absolute Error (MAE) and total elapsed time calculated using the algorithms are compared. According to the results of the statistical tests, the proposed methods outperform others, achieving scores as high as 0.9789 and 0.9996 for the R2 test, as low as 0.0058 and 0.0057 for the RMSE test, 0.0036 and 0.0045 for the MAE test and 3.53 × 10−5 and 3.50 × 10−5 for the $\chi$2 test. In addition, the determination of the wind speed characteristics at HPA show that low wind speed characteristics and regimes throughout the year offer safer take-off and landing schedules for target aircraft. The principle aim of this paper is to help establish the correct orientation of new runways at HPA and maximise the capacity of the airport by minimising flight delays, which represent a significant impediment to air traffic flow.

Dielectric breakdown in a thin oxide is presented in terms of an interacting particle system on a two-dimensional lattice. All edges in the system are initially assumed to be closed. An edge between two adjacent vertices will open according to an exponentially distributed random variable. Breakdown occurs at the time an open path connects the top layer of the lattice to the bottom layer. Using the extreme value theory, we show that the time until breakdown is asymptotically Weibull distributed.

In this paper we prove that a parallel system consisting of Weibull components with different scale parameters ages faster than a parallel system comprising Weibull components with equal scale parameters in the convex transform order when the lifetimes of components of both systems have different shape parameters satisfying some restriction. Moreover, while comparing these two systems, we show that the dispersive and the usual stochastic orders, and the right-spread order and the increasing convex order are equivalent. Further, some of the known results in the literature concerning comparisons of k-out-of-n systems in the exponential model are extended to the Weibull model. We also provide solutions to two open problems mentioned by Balakrishnan and Zhao (2013) and Zhao et al. (2016).

A high-resolution column of 57 loess samples was collected from the Dry Creek archaeological site in the Nenana River Valley in central Alaska. Numerical grain-size partitioning using a mixed Weibull function was performed on grain-size distributions to obtain a reconstructed record of wind intensity over the last ~15,000 yr. Two grain-size components were identified, one with a mode in the coarse silt range (C1) and the other ranging from medium to very coarse sand (C2). C1 dominates most samples and records regional northerly winds carrying sediment from the Nenana River. These winds were strong during cold intervals, namely, the Carlo Creek glacial readvance (14.2–14 ka), a late Holocene Neoglacial period (4.2–2.7 ka), and recent glacier expansion; weak during the Allerød (14–13.3 ka) and Younger Dryas (12.9–11.7 ka); and variable during the Holocene thermal maximum (11.4–9.4 ka). Deposition of C2 was episodic and represents locally derived sand deposited by southerly katabatic winds from the Alaska Range. These katabatic winds occurred mainly prior to 12 ka and after 4 ka. This study shows that numerical grain-size partitioning is a powerful tool for reconstructing paleoclimate and that it can be successfully applied to Alaskan loess.

Here, we present two high-resolution records of macroscopic charcoal from high-elevation lake sites in the Sierra Nevada, California, and evaluate the synchroneity of fire response for east- and west-side subalpine forests during the past 9200 yr. Charcoal influx was low between 11,200 and 8000 cal yr BP when vegetation consisted of sparse Pinus-dominated forest and montane chaparral shrubs. High charcoal influx after 8000 cal yr BP marks the arrival of Tsuga mertensiana and Abies magnifica, and a higher-than-present treeline that persisted into the mid-Holocene. Coeval decreases in fire episode frequency coincide with neoglacial advances and lower treeline in the Sierra Nevada after 3800 cal yr BP. Independent fire response occurs between 9200 and 5000 cal yr BP, and significant synchrony at 100- to 1000-yr timescales emerges between 5000 cal yr BP and the present, especially during the last 2500 yr. Indistinguishable fire-return interval distributions and synchronous fires show that climatic control of fire became increasingly important during the late Holocene. Fires after 1200 cal yr BP are often synchronous and corroborate with inferred droughts. Holocene fire activity in the high Sierra Nevada is driven by changes in climate linked to insolation and appears to be sensitive to the dynamics of the El Niño–Southern Oscillation.

In stable populations with constant demographic rates, size distributions reflect size-dependent patterns of growth and mortality. However, population growth can also affect size distributions, which may not be aligned with current growth and mortality. Using 25 y of demographic data from the 50-ha Barro Colorado Island plot, we examined how interspecific variation in diameter distributions of over 150 tropical trees relates to growth–diameter and mortality–diameter curves and to population growth rates. Diameter distributions were more skewed in species with faster increases/slower decreases in absolute growth and mortality with diameter and higher population growth rates. The strongest predictor of the diameter distribution shape was the exponent governing the scaling of growth with diameter (partial R2 = 0.20–0.34), which differed among growth forms, indicating a role of life history variation. However, interspecific variation in diameter distributions was also significantly related to population growth rates (partial R2 = 0.03–0.23), reinforcing that many populations are not at equilibrium. Consequently, although fitted size distribution parameters were positively related to theoretical predictions based on current size-dependent growth and mortality, there was considerable deviation. These analyses show that temporally variable demographic rates, probably related to cyclic climate variation, are important influences on forest structure.

One of the pricing strategies for Bonus–Malus (BM) systems relies on the decomposition of the claims’ randomness into one part accounting for claims’ frequency and the other part for claims’ severity. By mixing an exponential with a Lévy distribution, we focus on modelling the claim severity component as a Weibull distribution. For a Negative Binomial number of claims, we employ the Bayesian approach to derive the BM premiums for Weibull severities. We then conclude by comparing our explicit formulas and numerical results with those for Pareto severities that were introduced by Frangos & Vrontos.

The main target of this paper is to determine the seasonal features of the wind speed inthe northwestern of Senegal and to assess the cost of annual energy output using windturbines provided from several manufactures. The wind data used for this study werecollected in four sites located in the northwestern coast of Senegal and cover the periodof one year. The seasonal mean wind speed and the annual distribution curves were obtainedby using the Weibull distribution function. A technical assessment of electricitygeneration from five big wind turbines which power is between 640 kW and 2000 kW and fromsix small wind turbines which power is between 250 W and 10 kW were carried out. Theresults obtained show that the lower cost (0.0265 €/kWh in Sokhar) was observed for thewind turbine Ecotècnia 62. The highest cost was 0.5103 €/kWh (in Gandon) observed for thetechnology of Alize. All wind turbines was better suited to the site of Sokhar with thehighest capacity factor and the lowest cost of output energy.

Consider a generic data unit of random size L that needs to be transmitted over a channel of unit capacity. The channel availability dynamic is modeled as an independent and identically distributed sequence {A, Ai}i≥1 that is independent of L. During each period of time that the channel becomes available, say Ai, we attempt to transmit the data unit. If L <Ai, the transmission is considered successful; otherwise, we wait for the next available period Ai+1 and attempt to retransmit the data from the beginning. We investigate the asymptotic properties of the number of retransmissions N and the total transmission time T until the data is successfully transmitted. In the context of studying the completion times in systems with failures where jobs restart from the beginning, it was first recognized by Fiorini, Sheahan and Lipsky (2005) and Sheahan, Lipsky, Fiorini and Asmussen (2006) that this model results in power-law and, in general, heavy-tailed delays. The main objective of this paper is to uncover the detailed structure of this class of heavy-tailed distributions induced by retransmissions. More precisely, we study how the functional relationship ℙ[L>x]-1 ≈ Φ (ℙ[A>x]-1) impacts the distributions of N and T; the approximation ‘≈’ will be appropriately defined in the paper based on the context. Depending on the growth rate of Φ(·), we discover several criticality points that separate classes of different functional behaviors of the distribution of N. For example, we show that if log(Φ(n)) is slowly varying then log(1/ℙ[N>n]) is essentially slowly varying as well. Interestingly, if log(Φ(n)) grows slower than e√(logn) then we have the asymptotic equivalence log(ℙ[N>n]) ≈ - log(Φ(n)). However, if log(Φ(n)) grows faster than e√(logn), this asymptotic equivalence does not hold and admits a different functional form. Similarly, different types of distributional behavior are shown for moderately heavy tails (Weibull distributions) where log(ℙ[N>n]) ≈ -(log Φ(n))1/(β+1), assuming that log Φ(n) ≈ nβ, as well as the nearly exponential ones of the form log(ℙ[N>n]) ≈ -n/(log n)1/γ, γ>0, when Φ(·) grows faster than two exponential scales log log (Φ(n)) ≈ nγ.

Let X1, X2,… and Y1, Y2,… be two sequences of absolutely continuous, independent and identically distributed (i.i.d.) random variables with equal means E(Xi)=E(Yi), i=1,2,… In this work we provide upper bounds for the total variation and Kolmogorov distances between the distributions of the partial sums ∑i=1nXi and ∑i=1nYi. In the case where the distributions of the Xis and the Yis are compared with respect to the convex order, the proposed upper bounds are further refined. Finally, in order to illustrate the applicability of the results presented, we consider specific examples concerning gamma and normal approximations.

La fiabilité des matériels mécaniques est largement conditionnée par les caractéristiques propres à chaque système, à ses conditions d'utilisation, son environnement... De plus, les mécanismes de dégradations comme la fatigue et le stress créent des phénomènes de vieillissement. Les bases de données de fiabilité, en fournissant des taux de défaillance génériques et constants, ne permettent pas de prendre en considération ces particularités. L'utilisation de telles bases entraîne par conséquent de grandes incertitudes quant aux résultats des évaluations fiabilistes. Nous proposons dans cet article une modélisation des taux de défaillance, fonction du temps, qui prend en compte les facteurs d'influence. Une application permet de valider le modèle.

It is critical to understand the responses of tropical tree species to ongoing anthropogenic disturbances. Given the longevity of large trees and the scarcity of appropriately long-term demographic data, standing size distributions are a potential tool for predicting species' responses to disturbances and resultant changes in population structure. Here we test the utility of several different measures of size distribution for predicting subsequent population changes at the intraspecific level using demographic records from two subsampled 50-ha tree plots in Malaysia (Pasoh and Lambir). Most measures of size distribution failed to successfully predict population change better than random; however, the ‘coefficient of skewness’ (a measure of the relative proportion of small vs. large stems in a population) was able to correctly predict the direction of population change for approximately three-quarters of species at both sites. At Pasoh, the magnitude of this relationship decreased with adult stature and rate of turnover, but was unrelated to sapling growth rates at either site. Finally, using data for species common at both forests, we found that size distributions were generally uninformative of subsequent differences in population change between sites (only median dbh correctly predicted the direction of change for more species than random). Based on these results we conclude that some measures of intraspecific differences in size distribution are potentially informative of population trends within forests but have limited utility across broader spatial scales.

Two cumulative damage models are considered, the inverse gamma process and a composed gamma process. They can be seen as ‘continuous’ analogues of Poisson and compound Poisson processes, respectively. For these models the first passage time distribution functions are derived. Inhomogeneous versions of these processes lead to models closely related to the Weibull failure model. All models show interesting size effects.

The estimation of P(Sn>u) by simulation, where Sn is the sum of independent, identically distributed random varibles Y1,…,Yn, is of importance in many applications. We propose two simulation estimators based upon the identity P(Sn>u)=nP(Sn>u, Mn=Yn), where Mn=max(Y1,…,Yn). One estimator uses importance sampling (for Yn only), and the other uses conditional Monte Carlo conditioning upon Y1,…,Yn−1. Properties of the relative error of the estimators are derived and a numerical study given in terms of the M/G/1 queue in which n is replaced by an independent geometric random variable N. The conclusion is that the new estimators compare extremely favorably with previous ones. In particular, the conditional Monte Carlo estimator is the first heavy-tailed example of an estimator with bounded relative error. Further improvements are obtained in the random-N case, by incorporating control variates and stratification techniques into the new estimation procedures.

We consider the lifetimes of systems that can be modeled as particles that move within a bounded region in ℝn. Particles move within the set according to a random walk, and particles that leave the set are lost. We divide the set into equal cells and define the lifetime of the set as the time required for the number of particles in one of the cells to fall below a predetermined threshold. We show that the lifetime of the system, given a sufficiently large number of particles, is Weibull distributed.

Macroparasites are almost always aggregated across their host populations, hence the Negative Binomial Distribution (NBD) with its exponent parameter k is widely used for modelling, quantifying or analysing parasite distributions. However, many studies have pointed out some drawbacks in the use of the NBD, with respect to the sensitivity of k to the mean number of parasites per host or the under-representation of the heavily infected hosts in the estimate of k. In this study, we compare the fit of the NBD with 4 other widely used distributions on observed parasitic gastrointestinal nematode distributions in their sheep host populations (11 datasets). Distributions were fitted to observed data using maximum likelihood estimator and the best fits were selected using the Akaike's Information Criterion (AIC). A simulation study was also conducted in order to assess the possible bias in parameter estimations especially in the case of small sample sizes. We found that the NBD is seldom the best fit for gastrointestinal nematode distributions. The Weibull distribution was clearly more appropriate over a very wide range of degrees of aggregation, mainly because it was more flexible in fitting the heavily infected hosts. Moreover, the Weibull distribution estimates are less sensitive to sample size. Thus, when possible, we suggest to carefully check on observed data if the NBD is appropriate before conducting any further analysis on parasite distributions.

We compare R1(t), the reliability function of a redundant m-of-n system operating within the laboratory, with RD(t), the reliability function of the same system operating subject to environmental effects. Within the laboratory, all component lifetimes are independent and identically distributed according to G(α + 1, λ), a gamma distribution with index α + 1 and scale λ. Outside the laboratory, we adopt the model of Lindley and Singpurwalla (J. Appl. Prob. 23 (1986), 418-431) and assume that, conditional on a positive random variable η which models the effect of the common environment, all component lifetimes are independent and identically distributed according to G(α + 1, λη). When α is a non-negative integer we prove that for RD(t) to underestimate (resp. overestimate) R1(t) for all t sufficiently close to zero, it is necessary and sufficient that E(η(n-m + 1)(α+1)) > 1 (resp. E(η(n-m + 1)(α+1)) < 1). In the case in which n = 2, m= 1 and α = 0 we obtain a special case of a result of Currit and Singpurwalla (J. Appl. Prob. 26 (1988), 763-771). As an application, we obtain a necessary and sufficient condition under which RD(t) initially understimates (or overestimates) R1(t) when η follows a gamma or an inverse Gaussian distribution.

A consecutive-k-out-of-n: F system consists of n components ordered on a line. Each component, and the system as a whole, has two states: it is either functional or failed. The system will fail if and only if at least k consecutive components fail. The components are not necessarily equal and we assume that components' failures are stochastically independent. Using a result of Barbour and Eagleson (1984) we find a bound for the distance of the distribution of system's lifetime from the Weibull distribution. Subsequently, using this bound limit theorems are derived under quite general conditions.

A consecutive k-within-m-out-of-n system consists of n identical and stochastically independent components arranged on a line. The system will fail if and only if within m consecutive components, there are at least k failures. Let Tn be the system's lifetime. Then, under quite general conditions we prove that there is a positive constant a such that the random variable n1/kaTn converges to a Weibull distribution as n →∞.