Limit theorem for maximum of the storage process with fractional Brownian motion as input

AbstractThe maximum MT of the storage process Y(t)=sups⩾t(X(s)-X(t)-c(s-t)) in the interval [0,T] is dealt with, in particular, for growing interval length T. Here X(s) is a fractional Brownian motion with Hurst parameter, 0<H<1. For fixed T the asymptotic behaviour of MT was analysed by Piterbarg (Extremes 4(2) (2001) 147) by determining an approximation for the probability P{MT>u} for u→∞. Using this expression the convergence P{MT<uT(x)}→G(x) as T→∞ is derived where uT(x)→∞ is a suitable normalization and G(x)=exp(-exp(-x)) the Gumbel distribution. Also the relation to the maximum of the process on a dense grid is analysed

A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases

A satellite-based snow cover climatology (1985–2011) for the European Alps derived from AVHRR data

Seasonal snow cover is of great environmental and socio-economic importance for the European Alps. Therefore a high priority has been assigned to quantifying its temporal and spatial variability. Complementary to land-based monitoring networks, optical satellite observations can be used to derive spatially comprehensive information on snow cover extent. For understanding long-term changes in alpine snow cover extent, the data acquired by the Advanced Very High Resolution Radiometer (AVHRR) sensors mounted onboard the National Oceanic and Atmospheric Association (NOAA) and Meteorological Operational satellite (MetOp) platforms offer a unique source of information. <br><br> In this paper, we present the first space-borne 1 km snow extent climatology for the Alpine region derived from AVHRR data over the period 1985–2011. The objective of this study is twofold: first, to generate a new set of cloud-free satellite snow products using a specific cloud gap-filling technique and second, to examine the spatiotemporal distribution of snow cover in the European Alps over the last 27 yr from the satellite perspective. For this purpose, snow parameters such as snow onset day, snow cover duration (SCD), melt-out date and the snow cover area percentage (SCA) were employed to analyze spatiotemporal variability of snow cover over the course of three decades. On the regional scale, significant trends were found toward a shorter SCD at lower elevations in the south-east and south-west. However, our results do not show any significant trends in the monthly mean SCA over the last 27 yr. This is in agreement with other research findings and may indicate a deceleration of the decreasing snow trend in the Alpine region. Furthermore, such data may provide spatially and temporally homogeneous snow information for comprehensive use in related research fields (i.e., hydrologic and economic applications) or can serve as a reference for climate models

On the exceedance point process for a stationary sequence

It is known that the exceedance points of a hiqh level by a stationary sequence are asymptotically Poisson as the level increases, under appropriate lone range and local dependence conditions. When the local dependence conditions are relaxed, clustering of exceedances may occur, based on Poisson positions for the clusters. In this paper a detailed analysis of the exceedance point process is given, and shows that, under wide conditions, any limiting point process for exceedances is necessarily compound Poisson. Sufficient conditions are also qiven for the existence of such a limit. The limiting distributions of extreme order statistics are derived as corollaries

Relationship between internal accuracy and load-bearing capacity of minimally invasive lithium disilicate occlusal veneers

Purpose: To test whether internal accuracy affects the load-bearing capacity of 0.5-mm-thick occlusal veneers made out of milled or heat-pressed lithium disilicate (LS2). Materials and methods: Extracted human molars (N = 80) were divided into four groups (n = 20 each) depending on the bonding substrate (enamel [E] or dentin [D]) and the fabrication method (milling [CAD] or heat pressing [PRE]) for the occlusal LS2 veneers: (1) E-CAD, (2) D-CAD, (3) E-PRE, or (4) D-PRE. After restoration fabrication, the abutment teeth and the corresponding restorations were scanned and superimposed in order to measure the marginal and internal accuracy. After adhesive cementation, the specimens were thermomechanically aged and thereafter loaded until fracture. The load-bearing capacities (Fmax) were measured. Fmax and the marginal and internal accuracy between the groups were compared using Kruskal-Wallis test (P < .05) and pairwise group comparisons. In addition, the relationship between Fmax and the internal accuracy was analyzed using Spearman rank correlation. Results: Median Fmax values (and first and third quartiles) per group were as follows: 1,495 N (Q1: 932; Q3: 2'318) for E-CAD; 1,575 N (Q1: 1,314; Q3: 1,668) for E-PRE; 1,856 N (Q1: 1,555; Q3: 2,013) for D-CAD; and 1,877 N (Q1: 1,566; Q3: 2,131) for D-PRE. No statistical difference was found between the groups (P = .0981). Overall, the internal accuracy in the areas of the cusp (P < .0007) and fossa (P < .0001) showed significant differences. While no significant differences were detected in the marginal area (P = .3518), a significant correlation with a negative linear relationship was found between the 3D internal accuracy and the Fmax values (P = .0007). Conclusion: An increase in the internal accuracy raised the load-bearing capacity of minimally invasive LS2 occlusal veneers. In general, the restorations bonded to dentin in the occlusal regions showed a better accuracy compared to those bonded to enamel

A nonparametric urn-based approach to interacting failing systems with an application to credit risk modeling

In this paper we propose a new nonparametric approach to interacting failing systems (FS), that is systems whose probability of failure is not negligible in a fixed time horizon, a typical example being firms and financial bonds. The main purpose when studying a FS is to calculate the probability of default and the distribution of the number of failures that may occur during the observation period. A model used to study a failing system is defined default model. In particular, we present a general recursive model constructed by the means of inter- acting urns. After introducing the theoretical model and its properties we show a first application to credit risk modeling, showing how to assess the idiosyncratic probability of default of an obligor and the joint probability of failure of a set of obligors in a portfolio of risks, that are divided into reliability classes

Temporal factors in violence related injuries—An 11year trend analysis of violence-related injuries from a Swiss Emergency Department

Summary: Background: Injury from interpersonal violence is a major social and medical problem in the industrialized world. Little is known about the trends in prevalence and injury pattern or about the demographic characteristics of the patients involved. Methods: In this retrospective analysis, we screened the database of the Emergency Department of a large university hospital for all patients who were admitted for injuries due to interpersonal violence over an 11year period. For all patients identified, we gathered data on age, country of origin, quality of injury, and hospitalization or outpatient management. A trend analysis was performed using Kendall's tau-b correlation coefficients for regression analysis. Results: The overall number of patients admitted to our Emergency Department remained stable over the study period. Non-Swiss nationals were overrepresented in comparison to the demographics of the region where the study was conducted. There was a trend toward a more severe pattern of injury, such as an increase in the number of severe head injuries. Conclusions: Although the overall number of patients remained stable over the study period, there was an alarming trend toward a more severe pattern of injury, expressed by an increase in severe head trauma

On the modelling of the excesses of galaxy clusters over high-mass thresholds

In this work we present for the first time an application of the Pareto approach to the modelling of the excesses of galaxy clusters over high-mass thresholds. The distribution of those excesses can be described by the generalized Pareto distribution (GPD), which is closely related to the generalized extreme value (GEV) distribution. After introducing the formalism, we study the impact of different thresholds and redshift ranges on the distributions, as well as the influence of the survey area on the mean excess above a given mass threshold. We also show that both the GPD and the GEV approach lead to identical results for rare, thus high-mass and high-redshift, clusters. As an example, we apply the Pareto approach to ACT-CL J0102-4915 and SPT-CL J2106-5844 and derive the respective cumulative distribution functions of the exceedance over different mass thresholds. We also study the possibility to use the GPD as a cosmological probe. Since in the maximum likelihood estimation of the distribution parameters all the information from clusters above the mass threshold is used, the GPD might offer an interesting alternative to GEV-based methods that use only the maxima in patches. When comparing the accuracy with which the parameters can be estimated, it turns out that the patch-based modelling of maxima is superior to the Pareto approach. In an ideal case, the GEV approach is capable to estimate the location parameter with a percent level precision for less than 100 patches. This result makes the GEV based approach potentially also interesting for cluster surveys with a smaller area.Comment: 10 pages, 8 figures, MNRAS accepted, minor modifications to match the accepted versio

Extremes of Gaussian random fields with regularly varying dependence structure

Let be a centered Gaussian random field with variance function sigma (2)(ai...) that attains its maximum at the unique point , and let . For a compact subset of a"e, the current literature explains the asymptotic tail behaviour of under some regularity conditions including that 1 - sigma(t) has a polynomial decrease to 0 as t -&gt; t (0). In this contribution we consider more general case that 1 - sigma(t) is regularly varying at t (0). We extend our analysis to Gaussian random fields defined on some compact set , deriving the exact tail asymptotics of for the class of Gaussian random fields with variance and correlation functions being regularly varying at t (0). A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics

Statistical modeling of ground motion relations for seismic hazard analysis

We introduce a new approach for ground motion relations (GMR) in the probabilistic seismic hazard analysis (PSHA), being influenced by the extreme value theory of mathematical statistics. Therein, we understand a GMR as a random function. We derive mathematically the principle of area-equivalence; wherein two alternative GMRs have an equivalent influence on the hazard if these GMRs have equivalent area functions. This includes local biases. An interpretation of the difference between these GMRs (an actual and a modeled one) as a random component leads to a general overestimation of residual variance and hazard. Beside this, we discuss important aspects of classical approaches and discover discrepancies with the state of the art of stochastics and statistics (model selection and significance, test of distribution assumptions, extreme value statistics). We criticize especially the assumption of logarithmic normally distributed residuals of maxima like the peak ground acceleration (PGA). The natural distribution of its individual random component (equivalent to exp(epsilon_0) of Joyner and Boore 1993) is the generalized extreme value. We show by numerical researches that the actual distribution can be hidden and a wrong distribution assumption can influence the PSHA negatively as the negligence of area equivalence does. Finally, we suggest an estimation concept for GMRs of PSHA with a regression-free variance estimation of the individual random component. We demonstrate the advantages of event-specific GMRs by analyzing data sets from the PEER strong motion database and estimate event-specific GMRs. Therein, the majority of the best models base on an anisotropic point source approach. The residual variance of logarithmized PGA is significantly smaller than in previous models. We validate the estimations for the event with the largest sample by empirical area functions. etc