On extremal dependence: some contributions

The usual coefficients of tail dependence are based on exceedances of high values. These extremal events are useful and widely used in literature but an adverse situation may also occur with the upcrossing of a high level. In this context we define upcrossings-tail dependence coefficients and analyze all types of dependence coming out. We will prove that these coefficients are related to multivariate tail dependence coefficients already known in literature. We shall see that the upcrossings-tail dependence coefficients have the interesting feature of congregating both ??temporal?? and ??spatial?? dependence. The coefficients of tail dependence can also be applied to stationary sequences and hence measure the tail dependence in time. Results concerning connections with the extremal index and the upcrossings index as well as with local dependence conditions will be stated. Several illustrative examples will be exploited and a small note on inference will be given by presenting estimators derived from the stated results and respective properties.     

Poisson Parameter Estimation with Arbitrary Factors of Proportionality and Arbitrary Points of Left Truncation

Two unbiased estimators of the Poisson parameter are given for a truncation situation in whirh Cohen has suggested a near-efficient procedure. The situation is one in which each observation has its own distinctive left. truncation point, c_i , and its own distinctive Poisson parameter λt _i — it is the common λ which is to be estimated. The proposed estimators are obtained as adaptations of a pltblished simple estimator for the case all c_i = c, all t_i = 1. Illustrative data from Cohen are used for intercomparing the various estimators. including the maximum likelihood estimator.     

Semi-parametric second-order reduced-bias high quantile estimation

In many areas of application, like, for instance, Climatology, Hydrology, Insurance, Finance, and Statistical Quality Control, a typical requirement is to estimate a high quantile of probability 1−p, a value high enough so that the chance of an exceedance of that value is equal to p, small. The semi-parametric estimation of high quantiles depends not only on the estimation of the tail index or extreme value index γ, the primary parameter of extreme events, but also on the adequate estimation of a scale first order parameter. Recently, apart from new classes of reduced-bias estimators for γ>0, new classes of the scale first order parameter have been introduced in the literature. Their use in quantile estimation enables us to introduce new classes of asymptotically unbiased high quantiles’ estimators, with the same asymptotic variance as the (biased) “classical” estimator. The asymptotic distributional properties of the proposed classes of estimators are derived and the estimators are compared with alternative ones, not only asymptotically, but also for finite samples through Monte Carlo techniques. An application to the log-exchange rates of the Euro against the Sterling Pound is also provided.     

Bias-corrected and robust estimation of the bivariate stable tail dependence function

The stable tail dependence function gives a full characterisation of the extremal dependence between two or more random variables. In this paper, we propose an estimator for this function which is robust against outliers in the sample. The estimator is derived from a bivariate second-order tail model together with a proper transformation of the bivariate observations, and its asymptotic properties are studied under some suitable regularity conditions. Our estimation procedure depends on two parameters: \(\alpha \), which controls the trade-off between efficiency and robustness of the estimator, and a second-order parameter \(\tau \), which can be replaced by a fixed value or by an estimate. In case where \(\tau \) has been replaced by the true value or by an external consistent estimator, our robust estimator is asymptotically unbiased, whereas in case where \(\tau \) is mis-specified, one loses this property, but still our estimator performs quite well with respect to bias. The finite sample performance of our robust and bias-corrected estimator of the stable tail dependence function is examined on a simulation study involving uncontaminated and contaminated samples. In particular, its behavior is illustrated for different values of the pair \((\alpha , \tau )\) and is compared with alternative estimators from the extreme value literature.     

Adaptive selection of polynomial degrees on a finite element mesh

The problem of finding a nearly optimal distribution of polynomial degrees on a fixed finite element mesh is discussed. An a posteriori error estimator based on the minimum complementary energy principle is proposed which utilizes the displacement vector field computed from the finite element solution. This estimator, designed for p- and hp-extensions, is conceptually different from estimators based on residuals or patch recovery which are designed for h-extension procedures. The quality of the error estimator is demonstrated by examples. The results show that the effectivity index is reasonably close to unity and the sequences of p-distributions obtained with the error indicators closely follow the optimal trajectory. © 1998 John Wiley & Sons, Ltd.     

Dynamic and thermodynamic characteristics of the moving-coil linear compressor for the pulse tube cryocooler: Part B – Experimental verifications

Experimental investigations are carried out to verify the theoretical analyses and modeling conducted in Part A. Some empirical corrections are made and two sets of experiments are arranged based on the same compressor coupled with two different coaxial cold fingers typically operating at 80 K and 60 K, respectively. The variations of φ, ΔP, I, θ; the input electric power, η_motor; the cooling capacity and ηCarnot with the operating frequency at the given cooling temperatures are tested and compared with the simulation results, and fairly good agreements are found in both cases. The effects of the cooling temperature on these characteristics are also tested and discussed. Experimental results verify the validity of the theoretical investigations in Part A. The results also indicate that the theoretical studies can apply to wide ranges of both the operating frequency and the cooling temperature.     

Counting level crossings by a stochastic process

The number of times, N, that a stochastic process has up-crossings of a fixed level within a fixed time interval, T, is investigated. Existing integral formulas for the moments of N for a stationary Gaussian process are shown also to apply to processes that are neither stationary nor Gaussian, and explicit formulas are given for approximating the probability distribution of N from the moment formulas. Particular attention is given to simplified results for the limiting situations of very small and very large values of T, and to the behavior of variance, skewness, and kurtosis of N. For small T, the number N approaches the well-known Poisson distribution, but the results for large T are significantly different. For many stationary processes it is shown that the variance of N tends to grow linearly with T when T is very large, but the large-T growth rate is sometimes much smaller than that of the small-T Poisson process. More detailed results and some numerical examples are presented for the special case of a stationary Gaussian process crossing its own mean value.     

DC Magnetic Properties of the Ho doped ZnO Compounds

In this work, we report on the DC magnetic properties of Ho doped ZnO, Zn_0.8?4x Ho _x O _y (0.05≤x≤0.10), polycrystalline samples prepared by the so-called solid state reaction method. Magnetic measurements, M–H and M–T, done by using a PPMS magnetometer, show a paramagnetic behavior including indirect antiferromagnetic coupling between some Ho³⁺ magnetic moments. In other words, the negative values of the Curie–Weiss temperatures, θ, for the polycrystalline samples with six different Ho concentrations indicate that there is an effective antiferromagnetic coupling among the magnetic moments of Ho³⁺ ions. However, the decrease in the magnitude of θ with increasing x indicates that this coupling decreases with increasing amount of Ho in the compounds. Furthermore, the effective magnetic moments, μ _eff, per Ho atom calculated from the 1/χ?T curves gradually increase with increasing concentration. Therefore, the trends of the magnitudes of θ, and μ _eff values clearly indicate that apart from the antiferromagnetic coupling there is also an increasing contribution of a ferromagnetic coupling with increasing x. Hence, one may also conclude that the compounds studied seem to have a kind of spin-glass behavior.     

Optimization of Composite Material System and Lay-up to Achieve Minimum Weight Pressure Vessel

The use of composite pressure vessels particularly in the aerospace industry is escalating rapidly because of their superiority in directional strength and colossal weight advantage. The present work elucidates the procedure to optimize the lay-up for composite pressure vessel using finite element analysis and calculate the relative weight saving compared with the reference metallic pressure vessel. The determination of proper fiber orientation and laminate thickness is very important to decrease manufacturing difficulties and increase structural efficiency. In the present work different lay-up sequences for laminates including, cross-ply [0 _m /90 _n ] _s , angle-ply [±θ] _ns , [90/±θ] _ns and [0/±θ] _ns , are analyzed. The lay-up sequence, orientation and laminate thickness (number of layers) are optimized for three candidate composite materials S-glass/epoxy, Kevlar/epoxy and Carbon/epoxy. Finite element analysis of composite pressure vessel is performed by using commercial finite element code ANSYS and utilizing the capabilities of ANSYS Parametric Design Language and Design Optimization module to automate the process of optimization. For verification, a code is developed in MATLAB based on classical lamination theory; incorporating Tsai–Wu failure criterion for first-ply failure (FPF). The results of the MATLAB code shows its effectiveness in theoretical prediction of first-ply failure strengths of laminated composite pressure vessels and close agreement with the FEA results. The optimization results shows that for all the composite material systems considered, the angle-ply [±θ] _ns is the optimum lay-up. For given fixed ply thickness the total thickness of laminate is obtained resulting in factor of safety slightly higher than two. Both Carbon/epoxy and Kevlar/Epoxy resulted in approximately same laminate thickness and considerable percentage of weight saving, but S-glass/epoxy resulted in weight increment.     

On properties of probability-based multivariate process capability indices

Multivariate process capability indices (MPCIs) have been proposed to measure multivariate process capability in real-world application over the past three decades. For the practitioner's point of view, the intention of this paper is to examine the performances and distributional properties of probability-based MPCIs. Considering issues of construction of capability indices in multivariate setup and computation with performance, we found that probability-based MPCIs are a proper generalization of univariate basic process capability indices (PCIs). In the beginning of this decade, computation of probability-based indices was a difficult and time-consuming task, but in the computer age statistics, computation of probability-based MPCIs is simple and quick. Recent work on the performance of MPCI NMC_pm and distributional properties of its estimator reasonably recommended this index, for use in practical situations. To study distributional properties of natural estimators of probability-based MPCIs and recommended index estimator, we conducted simulation study. Though natural estimators of probability-based indices are negatively biased, they are better with respect to mean, relative bias, mean square error. Probability-based MPCI MC_pm is better as compared with NMC_pm with respect to performance and as its estimator quality. Hence, in real-world practice, we recommend probability-based MPCIs as a multivariate analogue of basic PCIs.     

Internal constraint theories for the thermal expansion of viscous fluids

Viscous fluids in many processes are mechanically incompressible but experience significant thermally induced volume change. We focus here on models for viscous fluids that impose this behavior through a posited internal constraint, following the formalism which introduces constraint responses that produce no entropy. We investigate four different thermal expansion constraints, whereby the independent mechanical variable in the problem formulation (density ρ or pressure p) is assumed to be specified completely by the independent thermal variable (temperature θ or entropy η). The internal constraint approach yields simpler constitutive relations, and therefore easier material characterization; the resulting model equations are alternatives to the compressible Navier-Stokes equations for simulation and analysis of thermal expansion phenomena. However, whereas the internal constraint formalism preserves consistency with the thermomechanical balance laws, second law, and invariance by fiat, there is no a priori guarantee that stability of the rest state and nonnegativity of specific heat and bulk modulus are preserved. Here we first derive the four possible internal constraint models for thermal expansion. Next we show that each of these four models is equivalent to a specific constitutive limit of the compressible theory, namely two of no pressure dependence of density and their duals of no density dependence of pressure. From this connection, we deduce that two constraints, namely ρ=ρ(η) and p=p(θ), offer physically viable candidates for the modeling of thermal expansion, whereas two others (the customary density-temperature constraint ρ=ρ(θ), as well as p=p(η)) are unphysical in the absence of any other conditions on the process, predicting catastrophic instability of the rest state. This analysis reveals that the internal constraint formalism must be further conditioned to preserve fundamental irreversible (second law) and reversible (stability) inequalities.     

Extremes of integer-valued moving average sequences

This paper aims to analyze the extremal properties of integer-valued moving average sequences obtained as discrete analogues of conventional moving averages replacing scalar multiplication by binomial thinning. In particular, we consider the case in which the scalar coefficients are replaced by random coefficients since in real applications the thinning probabilities may depend on several factors changing in time. Furthermore, the extremal behavior of periodic integer-valued moving average sequences is also considered. In this case, we find that, when assessing their clustering tendency of high-threshold exceedances, the extremal index is the same as for the stationary case.     

The multivariate extremal index and the dependence structure of a multivariate extreme value distribution

LetH be the limiting distribution of a vector of maxima for ad-dimensional stationary sequnce with multivariate extremal index. We giv necessary and sufficient conditions forH to have independent or totally dependent margins by using relations between the multivariate extremal index and the univariate extremal indexes. A new functional family of multivariates extreme value distributions, containingH, is introduced. We apply the results to characterize the asymptotic independence of the maximum and the minimum and compute the multivariate extremal index of the Multivariate Maxima of Moving Maxima process.     

A class of asymptotically unbiased semi-parametric estimators of the tail index

In this paper we consider a class of consistent semi-parametric estimators of a positive tail index γ, parameterized in atuning orcontrol parameter α. Such a control parameter enables us to have access, for any available sample, to an estimator of γ with a null dominant component of asymptotic bias, and with a reasonably flatMean Squared Error pattern, as a functional ofk, the number of top order statistics considered. Moreover, we are able to achieve a high efficiency relatively to the classical Hill estimator, provided we may have access to a larger number of top order statistics than the number needed for optimal estimation through the Hill estimator. partially supported by FCT/POCTI/FEDER     

Strained coherent interface energy of the Guinier–Preston II phase in Al–Cu during stress aging

θ″-phase, with a formal stoichiometry of Al₃Cu, is a coherent, metastable precipitate (GP-II) phase commonly found in Al-based aerospace alloys. In this paper, we employed a first-principles based method to study the energetics of the Al/θ″ interface as response to external strains. The potential effects of temperature, Cu activity, and different strain modes on interface energy (γ _Al/θ″) were systematically investigated. Calculations show that (i) an unstrained γ _Al/θ″ always increases with temperature: as temperature increases from T = 298 to 498 K, γ _Al/θ″ increases by ~9.0 %; (ii) γ _Al/θ″ is more sensitive to compressive strains than to tensile strains of the same magnitude. In particular, for a parallel compressive strain increasing from 0 to 2 % at a typical aging temperature, γ _Al/θ″ decreases by ~6.6 %, while a vertical compressive strain of 2 % has a slightly stronger impact by decreasing γ _Al/θ″ by ~9.6 %. Different influences of applied strain/stress on the formation energies of different orientated interfaces can be further exaggerated by the Poisson effect, and eventually affect the preferential precipitation orientation of θ″ in the matrix.     

A principle of corresponding states for the viscosity of non-quantum mono- and diatomic condensed gases

The reduced viscosity coefficient η_R for the saturated liquid state of argon, neon, krypton, xenon, oxygen, nitrogen, and fluorine is examined as a function of T_R and as a function of 1/ρ_R ? 1/ρ_oR. The reduction parameters are the critical ones: η_R = η/η_c where η_c is the viscosity of the liquid calculated at the critical point supposing there is no anomalous behaviour; T_R = T/T_c, ρ_R = ρ/ρ_c and ρ_oR = ρ_o/ρ_c. The density ρ_o is the density of the liquid where η = ∞.For the inert gases there is a good η_R(T_R) correspondence. From this fact the viscosity of radon is calculated.Where η_R is plotted as a function of 1/ρ_R ? 1/ρ_oR, a principle of corresponding states is found for the investigated condensed gases. For densities greater than about 2ρ_c the reduced fluidity φ_R = 1.868 (1/ρ_R ? 1/ρ_oR). In this range the overall average deviation between the calculated and the experimental fluidity is 2.6%.     

A posteriori error estimates and adaptive procedures for the meshless Galerkin boundary node method for 3D potential problems

The Galerkin boundary node method (GBNM) is a boundary only meshless method that combines variational formulations of boundary integral equations with the moving least-squares approximations. This paper presents the mathematical derivation of a posteriori error estimates and adaptive refinement procedures for the GBNM for 3D potential problems. Two types of error estimators are developed in detail. One is a perturbation error estimator that is formulated based on the difference between numerical solutions obtained using two successive nodal arrangements. The other is a projection error estimator that is formulated based on the difference between the GBNM solution itself and its L²-orthogonal projection. The reliability and efficiency of both types of error estimators is established. That is, these error estimators are proven to have an upper and a lower bound by the constant multiples of the exact error in the energy norm. A localization technique is introduced to accommodate the non-local property of integral operators for the needed local and computable a posteriori error indicators. Convergence analysis results of corresponding adaptive meshless procedures are also given. Numerical examples with high singularities illustrate the theoretical results and show that the proposed adaptive procedures are simple, effective and efficient.     

Dependence of mixed-mode crack propagation on the variation of σr and τrθ

The variation of the direction θ_p of crack propagation in mixed mode with the angle β of the inclined crack is shown to fit very well to the variation of θ_r, of the bisector of the angle of directions θ_r1 and θ_r2 of max σ_r and sufficiently well to that of θ_rθ, bisector of the angle of directions θ_rθ1 and θ_rθ2 of min τ_rθ < 0 and max τ_rθ > 0. The variation of σ^cr/σ^cr_I, causing initiation of propagation, with β as predicted by the max σ_θ criterion, fits very well to that of a critical loading evaluated by considering the combined “opening” effect of σ and τ_rθ along the circumference of a small circle centered at the crack tip. The above results may be used as criteria of mixed-mode fracture based on σ_r and τ_rθ.     

Weiss-Hill estimator

In this paper the asymptotic distributional behaviour is derived for a new estimator for the extreme value index γ of distribution, which is a combination of two estimators proposed by Weiss and Hill (Weiss, 1971, and Hill, 1975). For |γ|>1/2, the estimator outperforms the Moment estimator (Dekkers and al., 1989). in the sense that it has a smaller asymptotic variance than the latter; moreover, for γ>1/2 (γ<0, resp.) the estimator behaves asymptoticaly like the Hillresp. Weiss—estimator; for |γ|<1/2 the estimator does not achieve the same rate of convergence as the Moment estimator. Simulation results concerning the comparison of the mentioned estimators are also presented. This research project was partially supported by FCT/PRAXIS XXI/FEDER and POCTI.     

Nonparametric density and survival function estimation in the multiplicative censoring model

Consider the multiplicative censoring model given by \(Y_i=X_iU_i\), \(i=1, \ldots ,n\) where \((X_i)\) are i.i.d. with unknown density f on \({\mathbb {R}}\), \((U_i)\) are i.i.d. with uniform distribution \({\mathcal {U}}([0,1])\) and \((U_i)\) and \((X_i)\) are independent sequences. Only the sample \((Y_i)\) is observed. We study nonparametric estimators of both the density f and the corresponding survival function \(\bar{F}\). First, kernel estimators are built. Pointwise risk bounds for the quadratic risk are given, and upper and lower bounds for the rates in this setting are provided. Then, in a global setting, a data-driven bandwidth selection procedure is proposed. The resulting estimator has been proved to be adaptive in the sense that its risk automatically realizes the bias-variance compromise. Second, when the \(X_i\)s are nonnegative, using kernels fitted for \({\mathbb {R}}^+\)-supported functions, we propose new estimators of the survival function which are also adaptive. By simulation experiments, we check the good performances of the estimators and compare the two strategies.