Marie Kratz
ESSEC Business School
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Publication
Featured researches published by Marie Kratz.
Stochastic Models | 1996
Marie Kratz; Sidney I. Resnick
A common visual technique for assessing goodness of fit and estimating location and scale is the qq-plot. We apply this technique to data from a Pareto distribution and more generally to data generated by a distribution with a heavy tail. A procedure for assessing the presence of heavy tails and for estimating the parameter of regular variation is discussed which can supplement other standard techniques such as the Hill plot. Some examples are given using telecommunications data
Journal of Risk | 2015
Susanne Emmer; Marie Kratz; Dirk Tasche
Expected Shortfall (ES) has been widely accepted as a risk measure that is conceptually superior to Value-at-Risk (VaR). At the same time, however, it has been criticized for issues relating to backtesting. In particular, ES has been found not to be elicitable which means that backtesting for ES is less straight-forward than, e.g., backtesting for VaR. Expectiles have been suggested as potentially better alternatives to both ES and VaR. In this paper, we revisit commonly accepted desirable properties of risk measures like coherence, comonotonic additivity, robustness and elicitability. We check VaR, ES and Expectiles with regard to whether or not they enjoy these properties, with particular emphasis on Expectiles. We also consider their impact on capital allocation, an important issue in risk management. We find that, despite the caveats that apply to the estimation and backtesting of ES, it can be considered a good risk measure. In particular, there is no sufficient evidence to justify an all-inclusive replacement of ES by Expectiles in applications, especially as we provide an alternative way for backtesting of ES.
Probability Surveys | 2006
Marie Kratz
This paper presents a synthesis on the mathematical work done on level crossings of stationary Gaussian processes, with some exten- sions. The main results ((factorial) moments, representation into the Wiener Chaos, asymptotic results, rate of convergence, local time and number of crossings) are described, as well as the different approaches (normal com- parison method, Rice method, Stein-Chen method, a general m-dependent method) used to obtain them; these methods are also very useful in the general context of Gaussian fields. Finally some extensions (time occupa- tion functionals, number of maxima in an interval, process indexed by a bidimensional set) are proposed, illustrating the generality of the methods. A large inventory of papers and books on the subject ends the survey. AMS 2000 subject classifications: Primary 60G15; secondary 60G10, 60G12, 60G60, 60G70, 60F05. Keywords and phrases: (up) crossings, (non) central limit theorems, Gaussian processes/fields, Hermite polynomials, level curve, level function- als, local time, (factorial) moments, normal comparison method, number of maxima, Poisson convergence, rate of convergence, Rice method, sojourn, Wiener chaos.
Journal of Theoretical Probability | 2001
Marie Kratz; José R. León
We introduce a general method, which combines the one developed by authors in 1997 and one derived from the work of Malevich,(17) Cuzick(7) and mainly Berman,(3) to provide in an easy way a CLT for level functionals of a Gaussian process, as well as a CLT for the length of a level curve of a Gaussian field.
Stochastic Processes and their Applications | 1997
Marie Kratz; JoséR. León
We propose a new method to get the Hermite polynomial expansion of crossings of any level by a stationary Gaussian process, as well as the one of the number of maxima in an interval, under some assumptions on the spectral moments of the process.
Annals of Probability | 2006
Marie Kratz; José R. León
Cram er and Leadbetter introduced in 1967 the sucien t condition Z 0 r 00 (s) r 00 (0) s ds 0; to have a nite variance of the number of zeros of a centered stationary Gaussian process with twice dieren tiable covariance function r. This condition is known as the Geman condition, since Geman proved in 1972 that it was also a necessary condition. Up to now no such criterion was known for counts of crossings of a level other than the mean. This paper shows that the Geman condition is still sucien t and necessary to have a nite variance of the number of any xed level crossings. For the generalization to the number of a curve crossings, a condition on the curve has to be added to the Geman condition.
Statistics in Medicine | 2014
Armelle Guillou; Marie Kratz; Y. Le Strat
We propose a new method that could be part of a warning system for the early detection of time clusters applied to public health surveillance data. This method is based on the extreme value theory (EVT). To any new count of a particular infection reported to a surveillance system, we associate a return period that corresponds to the time that we expect to be able to see again such a level. If such a level is reached, an alarm is generated. Although standard EVT is only defined in the context of continuous observations, our approach allows to handle the case of discrete observations occurring in the public health surveillance framework. Moreover, it applies without any assumption on the underlying unknown distribution function. The performance of our method is assessed on an extensive simulation study and is illustrated on real data from Salmonella surveillance in France.
Advances in Applied Probability | 2011
Yann Demichel; Anne Estrade; Marie Kratz; Gennady Samorodnitsky
The modeling of random bi-phasic, or porous, media has been, and still is, under active investigation by mathematicians, physicists, and physicians. In this paper we consider a thresholded random process X as a source of the two phases. The intervals when X is in a given phase, named chords, are the subject of interest. We focus on the study of the tails of the chord length distribution functions. In the literature concerned with real data, different types of tail behavior have been reported, among them exponential-like or power-like decay. We look for the link between the dependence structure of the underlying thresholded process X and the rate of decay of the chord length distribution. When the process X is a stationary Gaussian process, we relate the latter to the rate at which the covariance function of X decays at large lags. We show that exponential, or nearly exponential, decay of the tail of the distribution of the chord lengths is very common, perhaps surprisingly so.
international conference on acoustics, speech, and signal processing | 2014
Nehla Debbabi; Marie Kratz
A Gauss-GPD hybrid model that links a Gaussian distribution to a Generalized Pareto Distribution (GPD) is considered for asymmetric heavy tailed data. The paper proposes a new un-supervised iterative algorithm to find successively the junction point between the two distributions and to estimate the hybrid model parameters. Simulation results show that this method provides a reliable position for the junction point, as well as an accurate estimation of the GPD parameters, which improves results when compared with other methods. Another advantage of this approach is that it can be adapted to any hybrid model.
arXiv: Probability | 2014
Meitner Cadena; Marie Kratz
We define a new class of positive and Lebesgue measurable functions in terms of their asymptotic behavior, which includes the class of regularly varying functions. We also characterize it by transformations, corresponding to generalized moments when these functions are random variables. We study the properties of this new class and discuss their applications to Extreme Value Theory.