A Class Of Distributions And Their Applications In Risk Theory

This dissertation is devoted to the development of ruin theory about the tailed distribution of individual claim.To begin with, we discuss theories of tailed- distribution between the heavy-tailed distribution and the light-tailed distribution. We also consider the relation between the failure rate and the equilibrium failure rate.we shall use these results to classify the relevant claim-size distributions. At last ,we do so in the mixture distributions.Some preliminaries are given in Chapter 1, we first introduce some risk models and heavy-tailed claim distributions. Then we study a special kind light-tailed distribution and get some base theories; a local limit theorem for the probability of ruin in the renewal risk model; ruin probability in perturbed risk models. Theorem1.2.1 If distributions F,G have finite mean and G, F ∈ L(γ), F ^ωG then Theorem 1.2.2 If the distribution F with finite mean, and lim r(x) = γ,γ > 0,when (?)x≥ x₀, x₀ > 0, r(x) ≤γ,then F∈S(r).Theorem 1.2.3 If the distribution F with lim r(x) = r, andr(x) ≥ γ,γ > 0,(?)x ∈ [0,∞) thenwhere G(x) = e^γxF(x).Theorem 1.3.1 If the distribution F with support on [0,∞), Thenwhere G(x) = e-"7IF(x).Theorem 1.3.2 Assume the distribution F with the failure rate r(x),and r(x) > 7, Va; G [0,00), if limsupx[r(x) - 7] < 00, then F G 5(7).1—yooTheorem 1.3.3 If the failure rate of F eventually decreases to 7,7 > 0, and Jo°°e?F{y)dy < oo,then(a) F G 5(7) ?=>■ /0°° exp{xr(x))F{x)dx = /0°° e?F(y)dy.(b) If the function x h exp(a;r(x)^5(x)) integrals on [0,00), then F G 5(7). Lemma 1.4.3 If F G 5(7),7 > 0, for every Z > 0, and n > 1 ,thenF*n{u, u + z) n(l - e"7z)F(u)( / . Theorem 1.4.1 If 7 G (0,oo),In Chapter 2, we mainly investigate the failure rate ,the equilibrium failure rate and get the relation between them , and get a way to classify the heavy -tailed distribution.At last,we study these in the mixture distribution and get some theorems. Theorem 2.2.1 If F has finite mean,(1) If F is IFR,then Re{x) > r(x),(2) If F is DFR,then Re(x) < r(x). Theorem 2.2.2 If F has finite mean,(1) If F is IFR, then re{x) >r{x).(2) If F is DFR, then re{x) < r{x).Theorem 2.2.3 Assume lim — 1jc' = c > 0, if rej(x) is the equilibrium failurerate of Fi,i = 1,2, and re\{x) ->? 7, then re2(x) ->■ 7.Theorem 2.3.1 If the distributions Fi(x),F2(x) have support on [0,00) andri(t),i = 1,2 are decreasing, then the equilibrium failure rate in (2.3.3) re(t) isdecreasing .Corollary 2.3.1 If the distributions^ (x), Fi(x) with support on [0,00) andn(t), i = 1,2,3...n are decreasing, then the equilibrium failure rate in(2.3.3) re(t)is decreasing.Corollary 2.3.2 If rei(t) = Jj^ -?? m € [0,oo],i = 1,2,3, ...n,t -> 00,Then re{t) -> mini 1, andrei(t) eventually decreases to ai, if rej(t),i = 1,2,3... are differentiable,(1) ^| = o(e"0,^<%-a1,(2) ^ = o(e^), ^ < min2