Abstract:The maximum entropy method was originally proposed as a variational technique to determine probability densities from the knowledge of a few expected values. The applications of the method beyond its original role in statistical physics are manifold. An interesting feature of the method is its potential to incorporate errors in the data. Here, we examine two possible ways of doing that. The two approaches have different intuitive interpretations, and one of them allows for error estimation. Our motivating example comes from the field of risk analysis, but the statement of the problem might as well come from any branch of applied sciences. We apply the methodology to a problem consisting of the determination of a probability density from a few values of its numerically-determined Laplace transform. This problem can be mapped onto a problem consisting of the determination of a probability density on [0, 1] from the knowledge of a few of its fractional moments up to some measurement errors stemming from insufficient data.
Here we present an application of two maxentropic procedures to determine the probability density distribution of compound sums of random variables, using only a finite number of empirically determined fractional moments. The two methods are the Standard method of Maximum Entropy (SME), and the method of Maximum Entropy in the Mean (MEM). We shall verify that the reconstructions obtained satisfy a variety of statistical quality criteria, and provide good estimations of VaR and TVaR, which are important measures for risk management purposes. We analyze the performance and robustness of these two procedures in several numerical examples, in which the frequency of losses is Poisson and the individual losses are lognormal random variables. As side product of the work, we obtain a rather accurate description of the density of the compound random variable. This is an extension of a previous application based on the Standard Maximum Entropy approach (SME) where the analytic form of the Laplace transform was available to a case in which only observed or simulated data is used.These approaches are also used to develop a procedure to determine the distribution 1 of the individual losses through the knowledge of the total loss. Then, in the case of having only historical total losses, it is possible to decompound or disaggregate the random sums in its frequency/severity distributions, through a probabilistic inverse problem.
In risk management the estimation of the distribution of random sums or collective models from historical data is not a trivial problem. This is due to problems related with scarcity of the data, asymmetries and heavy tails that makes difficult a good fit of the data to the most frequent distributions and existing methods.In this work we prove that the maximum entropy approach has important applications in risk management and Insurance Mathematics for the calculation of the density of aggregated risk events, and even for the calculation of the individual losses that come from the aggregated data, when the available information consists of an observed sample, which we usually do not have any information about the underlying process.From the knowledge of a few fractional moments, the Maxentropic methodologies provide an efficient methodology to determine densities when the data is scarce, or when the data presents correlation, large tails or multimodal characteristics. For this procedure, the input would be the sample moments E[e −αS ] = µ(α) or some interval that encloses the difference between the true value of µ(α) and the sample moments (for eight values of the Laplace transform), this interval would be related to the uncertainty (error) in the data, where the width of the interval may be adjusted by convenience. Through a simulation study we analyze the quality of the results, considering the differences with respect to the true density and in some cases the study of the size of the gradient and the time of convergence. We compare four different extensions of Maxentropic methodologies, the Standard Method of Maximum Entropy (SME), an extension of this methodology allows to incorporate additional information through a reference measure, called Method of Entropy in the Mean (MEM) and two extensions of the SME that allow introduce errors, called SME with errors or SMEE.Although our motivating example come from the field of Operational Risk analysis, the developed methodology may be applied to any branch of applied sciences.
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