Saddlepoint approximations to sensitivities of tail probabilities of random sums and comparisons with Monte Carlo estimators

This article proposes computing sensitivities of upper tail probabilities of random sums by the saddlepoint approximation. The considered sensitivity is the derivative of the upper tail probability with respect to the parameter of the summation index distribution. Random sums with Poisson or Geometric distributed summation indices and Gamma or Weibull distributed summands are considered. The score method with importance sampling is considered as an alternative approximation. Numerical studies show that the saddlepoint approximation and the method of score with importance sampling are very accurate. But the saddlepoint approximation is substantially faster than the score method with importance sampling. Thus, the suggested saddlepoint approximation can be conveniently used in various scientific problems.

Importance sampling approximations to various probabilities of ruin of spectrally negative Lévy risk processes

This article provides importance sampling algorithms for computing the probabilities of various types ruin of spectrally negative Lévy risk processes, which are ruin over the infinite time horizon, ruin within a finite time horizon and ruin past a finite time horizon. For the special case of the compound Poisson process perturbed by diffusion, algorithms for computing probabilities of ruins by creeping (i.e. induced by the diffusion term) and by jumping (i.e. by a claim amount) are provided. It is shown that these algorithms have either bounded relative error or logarithmic efficiency, as t,x→∞t,x→∞, where t>0t>0 is the time horizon and x>0x>0 is the starting point of the risk process, with y=t/xy=t/x held constant and assumed either below or above a certain constant.

Saddlepoint Approximations to the Probability of Ruin in Finite Time for the Compound Poisson Risk Process Perturbed by Diffusion

A large deviations type approximation to the probability of ruin within a finite time for the compound Poisson risk process perturbed by diffusion is derived. This approximation is based on the saddlepoint method and generalizes the approximation for the non-perturbed risk process by Barndorff-Nielsen and Schmidli (Scand Actuar J 1995(2):169–186, 1995). An importance sampling approximation to this probability of ruin is also provided. Numerical illustrations assess the accuracy of the saddlepoint approximation using importance sampling as a benchmark. The relative deviations between saddlepoint approximation and importance sampling are very small, even for extremely small probabilities of ruin. The saddlepoint approximation is however substantially faster to compute.

Saddlepoint approximations

The saddlepoint method provides accurate approximations for the distributions of many test statistics, estimators and for important probabilities arising in various stochastic models. The saddlepoint approximation is a large deviations technique which is substantially more accurate than limiting normal or Edgeworth approximations, especially in presence of very small sample sizes or very small probabilities. The outstanding accuracy of the saddlepoint approximation can be explained by the fact that it has bounded relative error.

CT based volume measurement and estimation in cases of pericardial effusion

The measurement of fluid volumes in cases of pericardial effusion is a necessary procedure during autopsy. With the increased use of virtual autopsy methods in forensics, the need for a quick volume measurement method on computed tomography (CT) data arises, especially since methods such as CT angiography can potentially alter the fluid content in the pericardium. We retrospectively selected 15 cases with hemopericardium, which underwent post-mortem imaging and autopsy. Based on CT data, the pericardial blood volume was estimated using segmentation techniques and downsampling of CT datasets. Additionally, a variety of measures (distances, areas and 3D approximations of the effusion) were examined to find a quick and easy way of estimating the effusion volume. Segmentation of CT images as shown in the present study is a feasible method to measure the pericardial fluid amount accurately. Downsampling of a dataset significantly increases the speed of segmentation without losing too much accuracy. Some of the other methods examined might be used to quickly estimate the severity of the effusion volumes.

Maximum flight velocity of blood drops in analysing blood traces

When analysing blood spatters, traces often occur which regarding the collision angle, cannot be allocated to any supposed centre of origin. Drops following highly curved (ballistic) trajectories usually form these types of traces. The reconstruction of such trajectories requires knowledge of the mass, the diameter (of which approximations are known) and the velocity of the blood drops. This article provides an upper range of the velocity in relation to the diameter of the blood drops based on physical laws. This is very helpful in analysing ballistic trajectories.

Misleading Population Estimates: Biases and Consistency of Visual Surveys and Matrix Modelling in the Endangered Bearded Vulture

Conservation strategies for long-lived vertebrates require accurate estimates of parameters relative to the populations' size, numbers of non-breeding individuals (the “cryptic” fraction of the population) and the age structure. Frequently, visual survey techniques are used to make these estimates but the accuracy of these approaches is questionable, mainly because of the existence of numerous potential biases. Here we compare data on population trends and age structure in a bearded vulture (Gypaetus barbatus) population from visual surveys performed at supplementary feeding stations with data derived from population matrix-modelling approximations. Our results suggest that visual surveys overestimate the number of immature (<2 years old) birds, whereas subadults (3–5 y.o.) and adults (>6 y.o.) were underestimated in comparison with the predictions of a population model using a stable-age distribution. In addition, we found that visual surveys did not provide conclusive information on true variations in the size of the focal population. Our results suggest that although long-term studies (i.e. population matrix modelling based on capture-recapture procedures) are a more time-consuming method, they provide more reliable and robust estimates of population parameters needed in designing and applying conservation strategies. The findings shown here are likely transferable to the management and conservation of other long-lived vertebrate populations that share similar life-history traits and ecological requirements.

Radiation doses along selected flight profiles during two extreme solar cosmic ray events

The radiation dose rates at flight altitudes may be hazardously increased during solar cosmic ray events. Within the scope of this paper we investigate the total accumulated radiation doses, i.e. the contribution of galactic and solar cosmic rays, during the two extreme solar cosmic ray events on 29 September 1989 and on 20 January 2005 along selected flight profiles. In addition, the paper discusses the consequences of possible solar cosmic ray flux approximations on the results of the radiation dose computations.

Stress free configuration of the human eye

Numerical simulations of eye globes often rely on topographies that have been measured in vivo using devices such as the Pentacam or OCT. The topographies, which represent the form of the already stressed eye under the existing intraocular pressure, introduce approximations in the analysis. The accuracy of the simulations could be improved if either the stress state of the eye under the effect of intraocular pressure is determined, or the stress-free form of the eye estimated prior to conducting the analysis. This study reviews earlier attempts to address this problem and assesses the performance of an iterative technique proposed by Pandolfi and Holzapfel [1], which is both simple to implement and promises high accuracy in estimating the eye's stress-free form. A parametric study has been conducted and demonstrated reliance of the error level on the level of flexibility of the eye model, especially in the cornea region. However, in all cases considered 3-4 analysis iterations were sufficient to produce a stress-free form with average errors in node location <10(-6)mm and a maximal error <10(-4)mm. This error level, which is similar to what has been achieved with other methods and orders of magnitude lower than the accuracy of current clinical topography systems, justifies the use of the technique as a pre-processing step in ocular numerical simulations.