Abstract
In this chapter, we present applications of ideal probability metrics to insurance risk theory. First, we describe and analyze themathematical framework.
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Notes
- 1.
As before, we will write μ(X, Y), ν(X, Y), τ(X, Y) instead of μ(F X , F Y ), ν(F X , F Y ), τ(F X , F Y ).
- 2.
See Case D in Sect. 4.4 of Chap. 4.
- 3.
- 4.
See Definition 15.3.1 in Chap. 15.
- 5.
More specifically, see Lemma 18.2.2.
- 6.
Indeed, if (17.3.10) fails for some \(j = 0, 1,\ldots ,m\), then \(\boldsymbol \zeta _{m,p}(X_{1},\widetilde{X}_{1}) \geq \) \(\mathop{\sup}\limits_{c>0}\vert E(cX_{1}^{j} - x\widetilde{X}_{1}^{j})\vert = +\infty \).
- 7.
An upper bound for \(\boldsymbol \zeta _{m,p}(X_{1},\widetilde{X}_{1})\) in terms of \(\boldsymbol \kappa _{r}\) (\(r = m + 1/p\)) is given by Lemma 15.3.6.
- 8.
See Kalashnikov and Rachev [1988, Theorem 3.10.2].
- 9.
See Sect. 15.2 in Chap. 15.
- 10.
- 11.
- 12.
Apply Theorem 8.2.2 of Chap. 8 with \(c(x,y) = \vert x - y{\vert }^{2}\).
- 13.
See Kalashnikov and Rachev [1988, Lemma 4.2.1].
- 14.
See Basu and Ebrahimi [1985] and the references therein for testing whether \(F_{W_{1}}\) belongs to the aging classes.
- 15.
See Teugels [1985].
- 16.
See Theorem 17.3.1.
- 17.
See Kalashnikov and Rachev [1988, Theorems 4.9.7 and 4.9.8].
References
Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing: probability models. Holt, Rinehart, and Winston, New York
Basu AP, Ebrahimi N (1985) Testing whether survival function is harmonic new better than used in expectation. Ann Inst Stat Math 37:347–359
Kalashnikov VV, Rachev ST (1988) Mathematical methods for construction of stochastic queueing models. Nauka, Moscow (in Russian) [English transl., (1990) Wadsworth, Brooks–Cole, Pacific Grove, CA]
Teugels JL (1985) Selected topics in insurance mathematics. Katholieke Universiteit Leuven, Leuven
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Rachev, S.T., Klebanov, L.B., Stoyanov, S.V., Fabozzi, F.J. (2013). Applications of Ideal Metrics for Sums of i.i.d. Random Variables to the Problems of Stability and Approximation in Risk Theory. In: The Methods of Distances in the Theory of Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4869-3_17
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