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Main Directions in the Theory of Probability Metrics Read chapter Read first chapter

2013 | OriginalPaper | Chapter

17. Applications of Ideal Metrics for Sums of i.i.d. Random Variables to the Problems of Stability and Approximation in Risk Theory

Authors : Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi

Published in: The Methods of Distances in the Theory of Probability and Statistics

Publisher: Springer New York

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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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previous chapter Ideal Metrics and Rate of Convergence in the CLT for Random Motions
next chapter How Close Are the Individual and Collective Models in Risk Theory?
Footnotes
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
See Theorems 5.5.1 and 6.4.1 in Chaps.​ 5 and 6, respectively.
 
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
See Barlow and Proschan [1975] and Kalashnikov and Rachev [1988].
 
11
Apply Theorem 8.2.2 of Chap.​ 8 with \(c(x,y) = \vert x - y\vert \) and the representation (17.3.3). See also Remark 7.2.3.
 
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].
 
Literature
.
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 37:347–359MathSciNetMATHCrossRef
.
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
Metadata
Title
Applications of Ideal Metrics for Sums of i.i.d. Random Variables to the Problems of Stability and Approximation in Risk Theory
Authors
Svetlozar T. Rachev
Lev B. Klebanov
Stoyan V. Stoyanov
Frank J. Fabozzi
Copyright Year
2013
Publisher
Springer New York
Book
The Methods of Distances in the Theory of Probability and Statistics

Print ISBN: 978-1-4614-4868-6

Electronic ISBN: 978-1-4614-4869-3

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-4869-3_17