Parisian types of ruin probabilities for a class of dependent risk-reserve processes

Bladt, Mogens; Nielsen, Bo Friis; Peralta, Oscar

doi:10.1080/03461238.2018.1483420

Cited by 11 publications

(14 citation statements)

References 35 publications

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“…For further reading on IPH distributions and motivations for their use in modeling we refer to Albrecher & Bladt (2019). For the representation of some IPH distributions, we make use of functional calculus.…”

Section: Iph Distributionsmentioning

confidence: 99%

“…As illustrated in Albrecher & Bladt (2019), a number of IPH distributions can be expressed as classical distributions with matrix‐valued parameter. Important examples include the transformation

g (y) = β false(e^{y} - 1 false)

for

β > 0

in () leading to a matrix–Pareto distribution with density function and survival function,

f_{X} (x) = π {(\frac{x}{β} + 1)}^{T - I} t \frac{1}{β}, {\overline{F}}_{X} (x) = 1 - F_{X} (x) = π {(\frac{x}{β} + 1)}^{T} e,

respectively, as well as the matrix–Weibull distribution with density and survival function

\begin{align} f_{X} false(x false) = bold-italicπ e^{bold-italicT x^{β}} bold-italict β x^{β prefix- 1}, {\overset{F}{true}}_{X} false(x false) = bold-italicπ e^{bold-italicT x^{β}} bold-italice, \end{align}

obtained from

g (y) = y^{1 / β}

(

β > 0

), see Albrecher & Bladt (2019) for further details.…”

Section: Iph Distributionsmentioning

confidence: 99%

“…However, there is an important subclass for which the intensity matrices can be written as a constant matrix scaled by some real function. In this class all matrices commute, and it was shown in Albrecher & Bladt (2019) that along this way one in fact obtains, for instance, Pareto, Weibull and Generalized Extreme Value (GEV) distributions with matrix‐valued parameters. These distribution classes are all dense in the class of distributions on the positive halfline and inherit the computational advantages of the PH‐type class, but also provide excellent fits for heavy‐tailed data already for small dimensions, something that the original PH class could not achieve.…”

Section: Introductionmentioning

confidence: 99%

“…This is particularly the case for heavy‐tailed distributions, where the focus in modeling often lies on the tail of the distribution, and the latter is not well captured by the combination of exponential components of the PH construction. After some first amendment involving infinite‐dimensional matrices was suggested in Albrecher & Bladt (2015) and Bladt and Rojas‐Nandayapa (2017), recently a new way to circumvent this problem was proposed in Albrecher & Bladt (2019). Concretely, when the Markov jump process is allowed to be time‐inhomogeneous, one gains a lot of flexibility in terms of the structure of the individual components entering the matrix framework, which can reduce the complexity of appropriate fitting distributions drastically, in particular for distributions with heavy tails.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, Cai and Li (2005a) consider them for determining conditional tail expectations in risk management, Cai and Li (2005b) studied several types of ruin probabilities for a multivariate compound Poisson risk model when the claim size vector follows an MPH ∗ distribution, and Herbertsson (2011) used this class to model default contagion in credit risk. More recently, Albrecher & Bladt (2019) applied MPH ∗ distributions for the calculation of Parisian type ruin probabilities. In terms of fitting of the (time‐homogeneous) multivariate MPH ∗ distribution, Ahlström et al (1999) introduced an algorithm for a bivariate subclass of MPH ∗ , and an EM algorithm for parameter estimation in the general case was proposed in Breuer (2016).…”

Section: Introductionmentioning

confidence: 99%

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Fitting inhomogeneous phase‐type distributions to data: the univariate and the multivariate case

Albrecher

Bladt

Yslas

2020

Scandinavian J Statistics

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The class of inhomogeneous phase‐type distributions (IPH) was recently introduced in Albrecher & Bladt (2019) as an extension of the classical phase‐type (PH) distributions. Like PH distributions, the class of IPH is dense in the class of distributions on the positive halfline, but leads to more parsimonious models in the presence of heavy tails. In this paper we propose a fitting procedure for this class to given data. We furthermore consider an analogous extension of Kulkarni's multivariate PH class (Kulkarni, 1989) to the inhomogeneous framework and study parameter estimation for the resulting new and flexible class of multivariate distributions. As a by‐product, we amend a previously suggested fitting procedure for the homogeneous multivariate PH case and provide appropriate adaptations for censored data. The performance of the algorithms is illustrated in several numerical examples, both for simulated and real‐life insurance data.

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Section: Iph Distributionsmentioning

confidence: 99%