# Random Shifting and Scaling of Insurance Risks

^{*}

## Abstract

**:**

## 1. Introduction

- copula-based models (here one needs to be careful since dependence structures for infinite sequences are needed!);
- conditional dependence models;
- random scale models;
- transformation of simple independence models.

## 2. Credibility Premium in Random Shift Models

**X**is that $\mathit{X}|\mathbf{\Theta}=\mathit{\theta}$ follows a distribution function parametrized by

**θ**, say it follows $F(\mathit{x};\mathit{\theta})$. A direct way to formulate this model is via the random shift representation

**Y**has distribution function F and is independent of

**Θ**. If

**Θ**possesses a probability density function (pdf) h, then clearly

**X**also possesses a pdf given by

**E**$\left\{h\right(\mathit{x}-\mathit{Y}\left)\right\}$. Consequently, the Bayesian premium (under a ${L}_{2}$ loss function), when it exists, is given by

**Y**also possesses a pdf. Clearly, if $\mathit{Y}\stackrel{d}{=}-\mathit{Y}$ we have further

**Example 1.**

**ν**and covariance matrix A). Suppose further that $\Sigma +{\Sigma}_{0}$ is positive definite. It follows that $(\mathit{X},\mathbf{\Theta})\stackrel{d}{=}\mathit{Z}=(\mathbf{\Theta}+\mathit{Y},\mathbf{\Theta})$ with $\mathit{Y}\sim {\mathcal{N}}_{d}(0,\Sigma )$ independent of

**Θ**. Therefore, in the light of [1] the fact that

**Z**is normally distributed in ${\mathbb{R}}^{2d}$ implies that $\mathit{Y}\left|\right(\mathbf{\Theta}+\mathit{Y})=\mathit{x}$ is normally distributed with mean

## 3. Dirichlet Claim Sizes & Random Scaling

**O**. The reason for the name of ${L}_{p}$ Dirichlet random vector (and distribution) is that the angular component

**O**lives on the unit ${L}_{p}$-sphere of ${\mathbb{R}}^{d}$, i.e.,

**O**has the Dirichlet distribution on the unit simplex; see [17].

**Theorem 1.**

**Proof:**

**Corollary 2.**

## 4. Discussions and Extensions

**ν**. Further, denote by ${A}^{\top}$ the transpose of matrix A.

**U**; see, e.g., [20]. Consequently, since

**U**has components being symmetric about 0, we obtain the Bayesian premium formula

**E**$\left\{{R}_{x}\right\}<\infty $. In the special case that ${C}_{I,J}$ and ${C}_{J,I}$ have all entries equal to 0, and further

**O**defined in Equation (11) has components ${O}_{i},i\le d$ such that ${O}_{i}^{p}$ has beta distribution with parameters ${\alpha}_{i},{\sum}_{j\le d,j\ne i}{\alpha}_{j}$; see, e.g., [4]. In the special case that ${\alpha}_{i}=1/p$ for any $i\le d$, properties of

**O**and $\mathit{X}=R\mathit{O}$ with $R>0$ independent of

**O**are studied in [21]. Our result in Corollary 2 agrees with the finding of Theorem 4.4 in the aforementioned paper. Note that, for the case $p=2$, the corresponding result of Theorem 1 for spherically symmetric random sequences is well-known, see, e.g., [22,23].

**O**be given as in Equation (11). Further, let ${I}_{i},i\le d$ be independent Bernoulli random variables, defined on $(\Omega ,\mathcal{A},\mathbf{P})$ with $\mathit{P}\{{I}_{i}=1\}={q}_{i}=1-\mathit{P}\{{I}_{i}=-1\},{q}_{i}\in (0,1],i\le d$, which are further independent of the random vector $(R,\mathit{O})$. The random vector

**X**with stochastic representation

**X**has a centered Gaussian distribution with $N(0,1)$ independent components. The introduction of the weighted Dirichlet random vectors is important since it includes the normal distribution as a special case. In addition, weighted Dirichlet random vectors are suitable for modeling claim sizes in certain ruin models with double-sided jumps; see Example 4 in [4].

**Example 2.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- M. Denuit, J. Dhaene, M. Goovaerts, and R. Kass. Actuarial Theory for Dependent Risks: Measures, Orders and Models. Chichester, UK: John Wiley & Sons Ltd., 2006. [Google Scholar]
- A. Kume, and E. Hashorva. “Calculation of bayes premium for conditional elliptical risks.” Insur. Math. Econom. 51 (2012): 632–635. [Google Scholar] [CrossRef]
- L. Limin, and P. Dandan. “Central limit theorem and moderate deviations principle for dependent risks.” Appl. Math. Sci. 4 (2010): 1805–1809. [Google Scholar]
- C. Constantinescu, E. Hashorva, and L. Ji. “Archimedean copulas in finite and infinite dimensions—With application to ruin problems.” Insur. Math. Econom. 49 (2011): 487–495. [Google Scholar] [CrossRef]
- H. Albrecher, C. Constantinescu, and S. Loisel. “Explicit ruin formulas for models with dependence among risks.” Insur. Math. Econom. 48 (2011): 265–270. [Google Scholar] [CrossRef]
- A.V. Asimit, E. Furman, and R. Vernic. “On a multivariate Pareto distribution.” Insur. Math. Econom. 46 (2010): 308–316. [Google Scholar] [CrossRef]
- A.V. Asimit, R. Vernic, and R. Zitikis. “Evaluating Risk Measures and Capital Allocations Based on Multi-Losses Driven by a Heavy-Tailed Background Risk: The Multivariate Pareto-II Model.” Risks 1 (2013): 14–33. [Google Scholar] [CrossRef] [Green Version]
- A.V. Asimit, R. Vernic, and R. Zitikis. “Background Risk Models and Stepwise Portfolio Construction.” Available online: http://ssrn.com/abstract=2334967 (accessed on 3 October 2013).
- P. Embrechts, E. Hashorva, and T. Mikosch. “Aggregation of log-linear risks.” J. Appl. Probab., 2014, in press. [Google Scholar] [CrossRef]
- E.W. Frees, and E.A. Valdez. “Hierarchical insurance claims modeling.” J. Am. Stat. Assoc. 103 (2008): 1457–1469. [Google Scholar] [CrossRef]
- E. Furman. “On a multivariate gamma distribution.” Stat. Probab. Lett. 78 (2008): 2353–2360. [Google Scholar] [CrossRef]
- E. Hashorva, and D. Kortschak. “Tail asymptotics of random sum and maximum of log-normal risks.” Stat. Probab. Lett. 87 (2014): 167–174. [Google Scholar] [CrossRef] [Green Version]
- E.A. Valdez. “Tail conditional variance for elliptically contoured distributions.” Belg. Actuar. Bull. 5 (2005): 26–36. [Google Scholar]
- X. Yang, E.W. Frees, and Z. Zhang. “A generalized beta copula with applications in modeling multivariate long-tailed data.” Insur. Math. Econom. 49 (2011): 265–268. [Google Scholar] [CrossRef]
- Y. Yang, and E. Hashorva. “Extremes and products of multivariate AC-product risks.” Insur. Math. Econom. 52 (2013): 312–319. [Google Scholar] [CrossRef]
- F. Wu, E.A. Valdez, and M. Sherris. “Simulating from exchangeable Archimedean copulas.” Commun. Stat. 36 (2007): 1019–1034. [Google Scholar] [CrossRef]
- A.J. McNeil, and J. Nešlehová. “Multivariate Archimedean copulas, d-monotone functions and l
_{1}-norm symmetric distributions.” Ann. Stat. 37 (2009): 3059–3097. [Google Scholar] [CrossRef] - R. Durrett. Probability: Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics, 4th ed. Cambridge, UK: Cambridge University Press, 2010. [Google Scholar]
- M. Yor. “A note about Selberg’s integrals in relation with the beta-gamma algebra.” In Advances in Mathematical Finance. Boston, MA, USA: Birkhäuser Boston, 2007, pp. 49–58. [Google Scholar]
- S. Cambanis, S. Huang, and G. Simons. “On the theory of elliptically contoured distributions.” J. Multivar. Anal. 11 (1981): 368–385. [Google Scholar] [CrossRef]
- P.J. Szabłowski. “Uniform distributions on spheres in finite-dimensional L
_{α}and their generalizations.” J. Multivar. Anal. 64 (1998): 103–117. [Google Scholar] [CrossRef] - W. Bryc. Lecture Notes in Statistics. New York, NY, USA: Springer-Verlag, 1995, Volume 100. [Google Scholar]
- I.J. Schoenberg. “Metric spaces and completely monotone functions.” Ann. Math. 39 (1938): 811–841. [Google Scholar] [CrossRef]
- A.C. Cebrián, M. Denuit, and P. Lambert. “Analysis of bivariate tail dependence using extreme value copulas: An application to the soa medical large claims database.” Belg. Actuar. Bull. 3 (2003): 33–41. [Google Scholar]
- S.I. Resnick. Heavy-Tail Phenomena. Springer Series in Operations Research and Financial Engineering. New York, NY, USA: Springer, 2007. [Google Scholar]

© 2014 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Hashorva, E.; Ji, L.
Random Shifting and Scaling of Insurance Risks. *Risks* **2014**, *2*, 277-288.
https://doi.org/10.3390/risks2030277

**AMA Style**

Hashorva E, Ji L.
Random Shifting and Scaling of Insurance Risks. *Risks*. 2014; 2(3):277-288.
https://doi.org/10.3390/risks2030277

**Chicago/Turabian Style**

Hashorva, Enkelejd, and Lanpeng Ji.
2014. "Random Shifting and Scaling of Insurance Risks" *Risks* 2, no. 3: 277-288.
https://doi.org/10.3390/risks2030277