Search engines have revolutionised the access to information to the general public. Today search engines are the most important promotional method on the Internet. Sponsored search dominates the revenue model behind this growth. The rise in popularity and the auction pricing mechanism of sponsored advertising have increased the average cost-per-click. Marketing managers need tools to enable them to increase return on investment in this medium. The application of Anderson’s (2004) long tail distribution holds great promise to solve this dilemma. The current study used causal research in a two by two factorial design. Here data from an online property portal in a developing market was collected in order to examine the effect of a long tail (LT) distribution in keyword selection on return on investment (ROI) with sponsored search. Sponsored search allows for individualised targeting of the users behaviour. The application of the long tail (LT) enables further matching the advert text to the users search query. The results provide strong support for the significant impact on cost-per-click and by implication the return on investment that keyword selection and targeted advert text have when used in conjunction with the principles of the long tail. The interaction of the independent variables of long tail and sponsored search is significant, contributing to a 430% increase in click-through (CTR) rates and 61% reduction in cost-per-click, translating into a 61% increase in return on investment. / Dissertation (MBA)--University of Pretoria, 2010. / Gordon Institute of Business Science (GIBS) / unrestricted

Αρκετές φορές σε προβλήματα αξιοπιστίας, το ενδιαφέρον μας εστιάζεται στην ουρά κατανομής του χρόνου ζωής. Συνήθως οι ουρές κατανομών ακόμη και των πλέον εύχρηστων κατανομών δεν υπολογίζονται με αναλυτικό τρόπο για αυτό χρησιμοποιούνται φράγματα. Υπολογιστικά εύχρηστα φράγματα είναι τα τύπου Lundberg. Απαραίτητο εργαλείο για τον προσδιορισμό φραγμάτων είναι ο ρυθμός ακαριαίου θανάτου. / -

Cette thèse s'intéresse à l'utilisation de techniques numériques par approximation sous forme de séries et de techniques de simulation pour l'approximation de la queue de distribution de sommes de variables aléatoires à variations régulières. Le calcul de la probabilité que la somme soit plus grande qu'un seuil donné est important en gestion des risques. En particulier, ce calcul est utilisé pour définir le besoin en capital des sociétés d'assurances ou d'autres institutions financières. Le premier chapitre constitue l'introduction de la thèse. Il explique les principaux résultats et présente les outils mathématiques qui sont développés dans la thèse. Le second chapitre est basé sur le travail : ”Series expansions for the sum of the independent Pareto random variables”, article rédigé avec le Professeur Christian ROBERT, directeur de la thèse. Cet article est soumis à publication. Il propose un algorithme de calcul pour déterminer la queue de distribution d'une somme de variables aléatoires de type Pareto non nécessairement équidistribuées. Il propose une approximation sous forme de série de la fonction de survie de la somme. L'algorithme utilisé pour calculer l'approximation est simple, facile à implémenter, et offre de très bons résultats numériques. Le troisième chapitre de cette thèse est basée sur l'article : ”New efficient estimators in rare event simulation with heavy tails”, publié dans Journal of Computational and Applied Mathematics, et co-écrit avec le Professeur Christian ROBERT. Il s'intéresse à l'approximation par simulation de la probabilité que la somme de variables aléatoires indépendantes à variations régulières soit plus grande qu'un seuil élevé. Des estimateurs efficaces ont déjà été introduits dans la littérature associée à la simulation d'évènements rares. Nous proposons de nouvelles techniques de simulation qui sont plus efficaces que les méthodes précédemment proposées. Le quatrième chapitre poursuit l'analyse de la simulation d'évènements rares du type ”la somme est plus grande qu'un seuil”, mais cette fois-ci il s'intéresse à des situations où les variables aléatoires sont dépendantes. Il se focalise sur le cas où la dépendance est donnée par une copule archimédienne. Ce chapitre est basé sur l'article en relecture : ”Efficient simulation of tail probabilities of sums with heavy tailed random variables and Archimedean copulas”. Les équivalents asymptotiques de la probabilité de dépassement de seuil ne sont connus que dans des cas particuliers et ils fournissent en général des approximations très médiocres de la vraie valeur. Les techniques de simulation sont donc très appréciables pour obtenir rapidement des approximations précises. Nous proposons quatre estimateurs et quatre techniques de simulation associées. Nous montrons que les erreurs relatives sont asymptotiquement bornées pour presque tous les estimateurs. Les simulations montrent que certains estimateurs sont plus précis / This thesis aims to study computation and simulation methods to approximate tail distribution of the sums of regularly varying random variables. The paper proceeds as follows: The first chapter provides the general introduction of the thesis. The second chapter is essentially constituted by the article ”Series expansions for the sum of the independent Pareto random variables” which was co-written with Professor Christian ROBERT, actually submitted for publication. It deals with the problem of estimating tail distribution of the sum of independent Pareto variables. This problem has been studied for a long time but a complete solution has not yet been found. In this section, we acquire an exact formula, a series expansions, for the distribution of the sum of independent Pareto of non-integer tail indices. Not only is this formula simple and easy to apply but it also gives better numerical results than most of existing methods.The third chapter rests on the article ”New efficient estimators in rare event simulation with heavy tails”, co-written with Professor Christian ROBERT, currently published on ”Journal of Computational and Applied Mathematics 261, 39-47” in 2013. Practically, efficient estimation for tail distribution of the sum of i.i.d. regularly varying random variables is one of widely researched problems in rare event simulation. In this context, Asmussen and Kroese’s estimator has performed better than other works. This part will introduce a new way to approach the sum. Our obtained estimator is more efficient than Asmussen and Kroese’s estimator in the case of regularly varying tail. In other cases, combined with techniques of conditional Monte Carlo and importance sampling, our estimator is still better. In the fourth chapter, we continue to study the tail behavior of the sum of regularly varying variables, with additional assumption that the dependence follows an Archimedean copula or an Archimedean survival copula. This section hinges on the article ”Efficient simulation of tail probabilities of sums with heavy tailed random variables and Archimedean copulas” which is under consideration for being published. Almost all previous studies on this problem used asymptotic approaches which are hard to control the errors. Therefore, techniques of simulation to calculate the tail probability of the sum are presented. Though some of our estimators have bounded relative errors while the others do not, all of them give favorable numerical performances for such a challenging problem

This thesis reviewed various heavy tailed distributions and Extreme Value Theory (EVT) to estimate the catastrophic losses simulated from Florida Public Hurricane Loss Projection Model (FPHLPM). We have compared risk measures such as Probable Maximum Loss (PML) and Tail Value at Risk (TVaR) of the selected distributions with empirical estimation to capture the characteristics of the loss data as well as its tail distribution. Generalized Pareto Distribution (GPD) is the main focus for modeling the tail losses in this application. We found that the hurricane loss data generated from FPHLPM were consistent with historical losses and were not as heavy as expected. The tail of the stochastic annual maximum losses can be explained by an exponential distribution. This thesis also touched on the philosophical implication of small probability, high impact events such as Black Swan and discussed the limitations of quantifying catastrophic losses for future inference using statistical methods.

Many financial time series show heavy tail behavior. Such tail characteristic is important for risk management. In this research, we focus on the calculation of Value-at-Risk (VaR) for portfolios of financial assets. We consider nonlinear autoregressive models with heavy tail innovations to model the return. Predictive distribution of the return are used to compute the VaR of the portfolios of financial assets. Examples are also given to compare the VaR computed by our approach with those by other methods.

My dissertation is based on a unified self-consistent scaling framework which is consistent with key behavior exhibited by many spatially/temporally varying earth, environmental and other variables. This behavior includes tendency of increments to have symmetric, non-Gaussian frequency distributions characterized by heavy tails that often decay with lag; power-law scaling of sample structure functions (statistical moments of absolute increments) in midranges of lags, with breakdown in power-law scaling at small and/or large lags; linear relationships between log structure functions of successive orders at all lags, also known as extended self-similarity; and nonlinear scaling of structure function power-law exponents with function order. The major question we attempt to answer is: given data measured on a given support scale at various points throughout a 1D/2D/3D sampling domain, which appear to be statistically distributed and to scale in a manner consistent with that scaling framework, what can be said about the spatial statistics and scaling of its extreme values, on arbitrary separation or domain scales? To do so, we limit our investigation in 1D domain for simplicity and generate synthetic signals as samples from 1D sub-Gaussian random fields subordinated to truncated monofractal fractional Brownian motion (tfBm) or truncated fractional Gaussian noise (tfGn). Such sub-Gaussian fields are scale mixtures of stationary Gaussian fields with random variances that we model as being log-normal or Lévy α/2-stable. This novel interpretation of the data allows us to obtain maximum likelihood estimates of all parameters characterizing the underlying truncated sub-Gaussian fields. Based on synthetic data, we find these samples conform to the aforementioned scaling framework and confirm the effectiveness of generation schemes. We numerically investigate the manner in which variables, which scale according to the above scaling framework, behave at the tails of their distributions. Ours is the first study to explore the statistical scaling of extreme values, specifically peaks over thresholds or POTs, associated with such families of sub-Gaussian fields. Before closing this work, we apply and verify our analysis by investigating the scaling of statistics characterizing vertical increments in neutron porosity data, and POTs in absolute increments, from six deep boreholes in three different depositional environments.

We consider the distribution of the zero and typical cells of a (homogeneous) Poisson hyperplane tessellation. We give a direct proof adapted to our setting of the well known Complementary Theorem. We provide sharp bounds for the tail distribution of the number of facets. We also improve existing bounds for the tail distribution of size measurements of the cells, such as the volume or the mean width. We improve known results about the generalised D.G. Kendall's problem, which asks about the shape of large cells. We also show that cells with many facets cannot be close to a lower dimensional convex body. We tacle the much less study problem of the number of facets and the shape of small cells. In order to obtain the results above we also develop some purely geometric tools, in particular we give new results concerning the polytopal approximation of an elongated convex body.