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A class of distributions and their application in Risk Theory

Author: JiaoShengHua
Tutor: YinChuanCun
School: Qufu Normal University
Course: Probability Theory and Mathematical Statistics
Keywords: Ruin Probability Interference model Heavy-tailed distribution Failure Rate Equilibrium distribution Brownian motion Levi process
CLC: O211.67
Type: Master's thesis
Year: 2005
Downloads: 75
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Abstract


This paper concentrates on the individual claims tailed discussion on the theory , discusses a category between heavy and light- tailed tailed distribution between the tail distribution . While taking into account the failure rate failure rate and balance the intrinsic link on this basis , to try to get a heavy-tailed distribution method for determining the nature and to promote relevant to the mixing distribution . In the first chapter , first introduced several risk models and related heavy-tailed basics, in this context, we focus on a special class of light -tailed distribution -S (γ) family of distributions , to obtain their relevant properties to determine , in general, a partial renewal risk model theorem, as well as in the case of bankruptcy of the interference estimates. 1.2.1 Theorem Let desired distribution of the limited distribution of F, G, and G, F ∈ L (γ), F ~ ω G is : (1) F ∈ S (γ) (?) G ∈ S (γ) ( 2) integral from 0 to ∞ (?) (y) dy (?) integral from 0 to ∞ (?) (y) dy lt; ∞, at this time there are Fe (x) ~ ω Ge (x) Theorem 1.2.2 Let the distribution F has finite expectation (?) r (x) = γ gt; 0, and there exists x 0 gt; 0, when (?) x ≥ x 0 when , r (x) ≤ γ, then F (?) S (γ) 1.2.3 Theorem Let the failure rate distribution F (?) r (x) = γ, r (x) ≥ γ, γ gt; 0, (?) x ∈ [0, ∞), then F ∈ S (γ) (?) G ∈ S * where (?) (x) = e γx ( ?) (x). Theorem 1.3.1 Let defined on [0, ∞) and having a limited distribution of expectations F, then F ∈ S (γ) is a necessary and sufficient condition G ∈ S (2γ). Wherein (?) (X) = e -γx (?) (X) satisfy the conditions set Theorem 1.3.2 ( 1.1.5 ) the distribution of the failure rate F r (x) exists , and r (x) ≥ γ, (?) x ∈ [0, ∞), if (?) x [r (x)-γ] lt; ∞, then F ∈ S (γ)

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CLC: > Mathematical sciences and chemical > Mathematics > Probability Theory and Mathematical Statistics > Theory of probability ( probability theory, probability theory ) > Random process > Expectations and Forecast
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