A solution to the ruin problem for Pareto distributions

Ramsay, Colin M.

doi:10.1016/s0167-6687(03)00147-1

Cited by 28 publications

(37 citation statements)

References 21 publications

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“…Pareto claims, ρ = 0.80 φ = 1.5 example because exact calculations of the ruin probabilities are available in this case, due to [17,18]. We apply the calibration procedure descrubed in Section 4 and approximate the distribution of X as a suitable mixture of scaled versions of Y .…”

Section: Note Thatmentioning

confidence: 99%

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Calculation of ruin probabilities for a dense class of heavy tailed distributions

Bladt¹,

Nielsen

Samorodnitsky

2014

Scandinavian Actuarial Journal

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Abstract. In this paper we propose a class of infinite-dimensional phase-type distributions with finitely many parameters as models for heavy tailed distributions. The class of finite-dimensional distributions is dense in the class of distributions on the positive reals and may hence approximate any such distribution. We prove that formulas from renewal theory, and with a particular attention to ruin probabilities, which are true for common phase-type distributions also hold true for the infinite-dimensional case. We provide algorithms for calculating functionals of interest such as the renewal density and the ruin probability. It might be of interest to approximate a given heavy-tailed distribution of some other type by a distribution from the class of infinite-dimensional phase-type distributions and to this end we provide a calibration procedure which works for the approximation of distributions with a slowly varying tail. An example from risk theory, comparing ruin probabilities for a classical risk process with Pareto distributed claim sizes, is presented and exact known ruin probabilities for the Pareto case are compared to the ones obtained by approximating by an infinite-dimensional hyper-exponential distribution.

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Section: Note Thatmentioning

confidence: 99%

“…This has turned out to be difficult. Practical computation algorithms have been developed for claim sizes with different versions of the pure Pareto distribution; see [17,18] and [1]. Our approach for calculating the ruin probability will apply to a wide range of levels of u and to a wide variety of claim size distributions.…”

Section: Introductionmentioning

confidence: 99%

Calculation of ruin probabilities for a dense class of heavy tailed distributions

Bladt¹,

Nielsen

Samorodnitsky

2014

Scandinavian Actuarial Journal

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“…The results show that the approximated ruin probabilities are remarkably close to the Ramsay value calculated using equation (20) of Ramsay (2003). Table 4.4: Theoretical error bounds, truncation error bounds, and total error bounds for approximation A.…”

Section: Numerical Studysupporting

confidence: 56%

“…The exact values of the ruin probability are given in Ramsay (2003), and are now considered a classical benchmark for comparison purposes.…”

Section: Numerical Studymentioning

confidence: 99%

Modelling heavy-tails with phase-type scale mixture distributions

Xie¹

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In this thesis, we present the new results found in exploring the asymptotic tail probability of the phase-type scale mixture distributions, and apply a subclass of the phase-type scale mixtures, the class of Erlangized scale mixture distributions, to the approximation of the probability of ruin.We consider the class of phase-type scale mixture distributions, which can be written as MellinStieltjes convolution Π G of a phase-type distribution G and a nonnegative but otherwise arbitrary distribution Π. Such a class can also be seen as the class of distributions of the product of two independent random variables: a phase-type random variable Y ∼ G and a nonnegative random variable S ∼ Π. We call S the scaling random variable and its corresponding distribution Π the scaling distribution. We investigate conditions for such a class of distributions to be either light-or heavy-tailed, we explore subexponentiality and determine their maximum domains of attraction.In this thesis, particular focus is on phase-type scale mixture distributions where the scaling distribution has discrete support -such a class has been recently used in risk applications to approximate heavy-tailed distributions. We extend an existing scheme for numerically calculating the probability of ruin of a classical Cramér-Lundberg reserve process having absolutely continuous but otherwise general claim size distributions. We employ a subclass of the phasetype scale mixtures, which is also a dense class of distributions that we denominate Erlangized scale mixtures (ESM). Such a class corresponds to nonnegative and absolutely continuous distributions which can be written as a Mellin-Stieltjes convolution Π G m of a discrete nonnegative distribution Π with an Erlang distribution G m . A distinctive feature of such a class is that it contains heavy-tailed distributions when the scaling distributions have unbounded support, and it provides sharp approximations to heavy-tailed distributions.We suggest a simple methodology for constructing a sequence of distributions having the form Π G m with the purpose of approximating the integrated tail distribution of the claim sizes.Then we adapt a recent result which delivers an explicit expression for the probability of ruin in the case that the claim size distribution is modelled as an Erlangized scale mixture. We provide simplified expressions for the approximation of the probability of ruin and construct explicit bounds for the error of approximation. We complement our results with a classical example where the claim sizes are heavy-tailed.ii

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“…Drȃgulescu and Yakovenko 2001;Kleiber and Kotz 2003;Clementi and Gallegati 2005;Klass et al 2006;Cowell and Flachaire 2007;Cowell and Victoria-Feser 2007;Ogwang 2011;Alfons et al 2013). 1 However, the model is also heavily used in several other areas of economics to model the right-hand tails of fluctuations in stock prices (Lauridsen 2000;Gabaix et al 2003Gabaix et al , 2006Balakrishnan et al 2008), exchange rates (Wagner and Marsh 2005), firm sizes (Axtell 2001;Luttmer 2007), city sizes (Soo 2005), countries' interactions in international trade (Hinloopen and van Marrewijk 2012), CEO compensation (Gabaix and Landier 2008), supply of regulations (Mulligan and Shleifer 2005), tourist visits (Ulubaşoglu and Hazari 2004), claims in actuarial problems (Ramsay 2003), macroeconomic disasters (Barro and Jin 2011), and macroeconomic fluctuations (Gaffeo et al 2003). In addition, Pareto distribution appears widely in physics, biology, earth and planetary sciences, computer science, and in other disciplines (Newman 2005).…”

Section: Introductionmentioning

confidence: 99%

Robust estimation of the Pareto tail index: a Monte Carlo analysis

Brzeziński

2015

Empir Econ

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The Pareto distribution is often used in many areas of economics to model the right tail of heavy-tailed distributions. However, the standard method of estimating the shape parameter (the Pareto tail index) of this distribution-the maximum likelihood estimator (MLE), also known as the Hill estimator-is non-robust, in the sense that it is very sensitive to extreme observations, data contamination or model deviation. In recent years, a number of robust estimators for the Pareto tail index have been proposed, which correct the deficiency of the MLE. However, little is known about the performance of these estimators in small-sample setting, which often occurs in practice. This paper investigates the small-sample properties of the most popular robust estimators for the Pareto tail index, including the optimal B-robust estimator (Victoria-Feser and Ronchetti in Can J Stat 22:247-258, 1994), the weighted maximum likelihood estimator (Dupuis and Victoria-Feser in Can J Stat 34:639-658, 2006), the generalized median estimator (Brazauskas and Serfling in Extremes 3:231-249, 2001), the partial density component estimator (Vandewalle et al. in Comput Stat Data Anal 51:6252-6268, 2007), and the probability integral transform statistic estimator (PITSE) (Finkelstein et al. in N Am Actuar J 10:1-10, 2006). Monte Carlo simulations show that the PITSE offers the desired compromise between ease of use and power to protect against outliers in the small-sample setting.

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A solution to the ruin problem for Pareto distributions

Cited by 28 publications

References 21 publications

Calculation of ruin probabilities for a dense class of heavy tailed distributions

Calculation of ruin probabilities for a dense class of heavy tailed distributions

Modelling heavy-tails with phase-type scale mixture distributions

Robust estimation of the Pareto tail index: a Monte Carlo analysis

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