Convex Bounds Approximation,Smooth Distribution Test And Related Actuarial Quantity Studies

This dissertation contains four parts.In the first part,we propose convex bound approximations for the sum of log-multivariate generalized hyperbolic random variables.We derive explicit formulas for the distributions of convex bounds and for the frequently-used risk measures such as Value-at-Risk,Conditional Tail Expectation and stop-loss premium.We present numerical results showing that such approximations are not only accurate but also robust.Moreover,we further prove that there exist asymptotic equivalences between the sum and its convex bounds.To further illustrate the potentials of the convex bound approximations,we provide an application to capital allocation.We show that our formulas can be easily applied to precisely approximate capital allocation rule based on the conditional tail expectation.This part is in chapter 2.The second part tests differences between non-smooth distributions.Economists often use kernel-based density tests to test for differences between unknown distributions.However,these tests assume that the underlying,unknown distributions are twice differentiable.Here we propose a kernel-based test for the difference between unknown distributions that does not require either distribution to be differentiable.We use this test in a replication study of Kumar and Russell(2002)to see if their kernel-based density tests are robust to the differentiability assumption.Despite concern in the applied literature over their potential mixing of distributions,at least in the construction of kernel-based density tests,these concerns appear unwarranted.This part is in chapter 3.In the third part,we exploit data from the China Household Finance Survey to examine the impact of changes in the minimum wage on employment and investment decisions.We are able to nonparametrically identify the average treatment effect on the treated via exogenous variation in the minimum wage across provinces.We find that changes in the minimum wage had no adverse effects on employment(in terms of days worked per month or hours worked per work day),but found evidence that changes in the minimum wage impacted the likelihood a family had a bank account,a family in a rural area owned their home and whether families(whose highest level of education was primary school)planned to purchase a home.This part is in chapter 4.The last part aims to derive the recursive formulas of the probabilities relate to some actuarial values in a time discrete interaction compound binomial risk model with a stochastic income process and time-correlated claims.In this model,we assume there are two classes of distribution of claims and allow the by-claims to delay.The stochastic income process in this paper is also a compound binomial process.In a fixed time horizon,this paper considers the probabilities that relate to the number of main claims occurred,the length of the longest continuous time periods of no claim,the length of the shortest continuous time periods of no claim,and the number of the continuous time periods of no claim.This part is in chapter 5.The first chapter provides a very brief introduction to the background information of this dissertation.The main creative points of this dissertation are1.In Chapter 2,we extend convex bounds approximation method to the MGH context and present two choices of the conditioning variable for the convex lower bound.The MGH distribution is a general probability distribution family,and plays an important role in various models in finance and insurance.Our results can provide a fast,accurate,and robust method to the evaluations of the interested quantities related to the sum of random variables under logMGH models.The content of this chapter has been published in Journal of Computational and Applied Mathematics.2.In Chapter 2,we prove there is an asymptotic equivalence between the sum and its convex bounds.By doing so,we underpin the well performance of the convex bound approximations.We present numerical examples to support our results and an application of our method to the capital allocation rule is also presented.3.In Chapter 3,we propose a smooth test base on the two-sample KolomogorovSmirnov(KS)test for tasting the difference of two unknown distribution.Compare to Li(1996)test,our test don’t assume the distribution function of the unknown distribution are second-order derivable which means it is more applicable in more situation.And in our performance test,our test outperform the popular Kolomogorov-Smirnov test in each case.The content of this chapter has not been published yet.4.In Chapter 3,we are able to replicate the density-based tests in Kumar and Russell(2002)as well as showcase our testing procedure with their data.It turn out the concerns regarding a potential mixture of distributions in their data appear unwarranted.5.In Chapter 4,we exploited a unique dataset that surveyed a representative sample of Chinese households.We were able to take arguably exogenous(to the families)shocks in minimum wages changes in order to examine average treatment effects on treated provinces.Similar to the literature,we did not find any significant impacts on employment as measured by days worked or hours worked per work day.At the mean time,we found significant impacts for the decision to open a bank account as well as the probability a family owned their home(in rural areas)and whether or not a family(with the highest level of education being primary school)were planning to purchase a home.The content of this chapter has been published in Journal of Risk and Financial Management.6.In Chapter 5,we add two kind of time correlated claims and a compound binomial stochastic income process to the classic compound binomial risk model.Under this model,we derive the recursive formulas and the initial condition of the probabilities relate to the four actuarial values that are rarely discussed in the literature.