Pareto-optimal reinsurance for both the insurer and the reinsurer with general premium principles

Abstract

In this paper, we study Pareto-optimal reinsurance policies from the perspectives of an insurer and a reinsurer, assuming reinsurance premium principles satisfy risk loading and stop-loss ordering preserving. By geometric approach, we determine the forms of the optimal policies among two classes of ceded loss functions, the class of increasing convex ceded loss functions and the class that the constraints on both ceded and retained loss functions are relaxed to increasing functions. Then we demonstrate the applicability of our results by giving the parameters of the optimal ceded loss functions under Dutch premium principle and Wang’s premium principle.     

Risk minimization for an insurer with investment and reinsurance via g-expectation

Abstract

This paper is devoted to the study of a risk-based optimal investment and proportional reinsurance problem. The surplus process of the insurer and the risky asset process in the financial market are assumed to be general jump-diffusion processes. We use a convex risk measure generated by g-expectation to describe the risk of the terminal wealth with investment and reinsurance. Under the aim of minimizing the risk, the problem is solved by using techniques of stochastic maximum principles. Two interesting special cases are studied and the explicit expressions for optimal strategies and corresponding minimal risks are derived.     

Optimal impulse control for dividend and capital injection with proportional reinsurance and exponential premium principle

This article supposes that a large insurance company can control its surplus process by reinsurance, paying dividends, or injecting capitals. The exponential premium principle and proportional reinsurance are adopted in business activities. We investigate the general situation that the company needs to pay both proportional and fixed costs for dividends and capital injections. The object of the company is to determine an optimal joint reinsurance–dividend–capital injection strategy for maximizing the expected present value of dividends less capital injections until the time of bankruptcy. In both cases of non cheap and cheap reinsurance, we obtain the explicit solutions for value function and optimal strategy.     

Robust portfolio choice for a defined contribution pension plan with stochastic income and interest rate

This paper considers a robust portfolio choice problem for a defined contribution pension plan with stochastic income and stochastic interest rate. The investment objective of the pension plan is to maximize the expected utility of the wealth at the retirement time. We assume that the financial market consists of a stock, a zero-coupon bond and a risk-free asset. And the member of defined contribution pension plan is ambiguity-averse, which means that the member is uncertain about the expected return rate of the bond and stock. Meanwhile, the member's ambiguity-aversion level toward these two financial assets is quite different. The closed-form expressions of the robust optimal investment strategy and the corresponding value function are derived by adopting the stochastic dynamic programming approach. Furthermore, the sensitive analysis of model parameters on the optimal investment strategy are presented. We find that the member's aversion on model ambiguity increases her hedging demand and has remarkable impact on the optimal investment strategy. Moreover, we demonstrate that ignoring model uncertainty will lead to significant utility loss for the ambiguity-averse member, and the model uncertainty about the stock dynamics implies greater effect on the outcome of the investment than the bond.     

Optimal investment and premium control for insurers with ambiguity

Abstract

In this paper, we consider the optimal investment and premium control problem for insurers who worry about model ambiguity. Different from previous works, we assume that the insurer’s surplus process is described by a non-homogeneous compound Poisson model and the insurer has ambiguity on both the financial market and the insurance market. Our purpose is to find the impacts of model ambiguity on optimal policies. With the objective of maximizing the expected utility of terminal wealth, the closed-form solutions of the optimal investment and premium policies are obtained by solving HJB equations. Finally, numerical examples are also given to illustrate the results.     

Sharing beliefs and the absence of betting in the Choquet expected utility model

Choquet expected utility maximizers tend to behave in a more “cautious” way than Bayesian agents, i. e. expected utility maximizers. We illustrate this phenomenon in the particular case of betting behavior. Specifically, consider agents who are Choquet expected utility maximizers. Then, if the economy is large, Pareto optimal allocations provide full insurance if and only if the agents share at least on prior, i. e., if the intersection of the core of the capacities representing their beliefs is non empty. In the expected utility case, this is true only if they have a common prior. Received: July 2000; revised version: May 2001     

Robust Hotelling's T2 control charts under non-normality: the case of t-Student distribution

Traditional multivariate control charts are based upon the assumption that the observations follow a multivariate normal distribution. In many practical applications, however, this supposition may be difficult to verify. In this paper, we use control charts based on robust estimators of location and scale to improve the capability of detection observations out of control under non-normality in the presence of multiple outliers. Concretely, we use a simulation process to analyse the behaviour of the robust alternatives to Hotelling's T ², which use minimum volume ellipsoidal (MVE) and minimum covariance determinant (MCD) in the presence of observations with a Student's t-distribution. The results show that these robust control charts are good alternatives for small deviations from normality due to the fact that the percentage of out-of-control observations detected for these charts in the Phase II are higher.     

A characterization of the general mean

A characterization is given of the general mean which is (?X^p)^1/p or exp[log X] for X > 0 according as ρ ≠0 or ρ = 0, where ρ is an unspecified parameter. The approach is axiomatic. The general mean is defined as any quantity satisfying certain axioms which embrace the minimal requirements that any reasonable candidate for the role of mean or most typical value should satisfy. The quantity, indexed by p and given above, is uniquely implied by these axioms.     

Robust Bayesian hypothesis testing in the presence of nuisance parameters

Robust Bayesian testing of point null hypotheses is considered for problems involving the presence of nuisance parameters. The robust Bayesian approach seeks answers that hold for a range of prior distributions. Three techniques for handling the nuisance parameter are studied and compared. They are (i) utilize a noninformative prior to integrate out the nuisance parameter; (ii) utilize a test statistic whose distribution does not depend on the nuisance parameter; and (iii) use a class of prior distributions for the nuisance parameter. These approaches are studied in two examples, the univariate normal model with unknown mean and variance, and a multivariate normal example.     

Continuity of f-projections and applications to the iterative proportional fitting procedure*

This paper proves continuity of f-projections and the continuous dependence of the limit matrix of the iterative proportional fitting procedure (IPF procedure) on the given matrix as well as the given marginals under certain regularity constraints. For finite spaces, the concept of f-projections of finite measures on a compact and convex set is introduced and continuity of f-projections is proven. This result is applied to the IPF procedure. Given a nonnegative matrix as well as row and column marginals the IPF procedure generates a sequence of matrices, called the IPF sequence, by alternately fitting rows and columns to match their respective marginals. The procedure is equivalent to cyclic f-projections. If the IPF sequence converges, the application of the continuity of f-projections yields the continuous dependence of the limit matrix on the given matrix. By generalized convex programming and under some constraints, it is shown that the limit matrix of the IPF sequence continuously depends not only on the given matrix but also on the marginals.     

Robust estimation of the exponential mean parameter for small samples: Complete and censored data

This paper studies methods for simple estimation of the exponential mean parameter in small samples in the presence of outliers. Existing estimation methods are discussed. Adaptation of these methods to allow for Type I censoring is investigated. New robust procedures are proposed. A series of simulation experiments Indicate trimming provides significant protection against outliers while the premium is usually small when trimming uncontarninated samples. A linearly weighted mean is recommended for uncontarninated samples, both censored and complete. In larger samples, (n - 20), the proposed Huber-type estimator performs quite well in all situations of censoring and contarnination     

An application of a minimax Bayes rule and shrinkage estimators to the portofolio selection problem under the Bayesian approach

This paper shows that a minimax Bayes rule and shrinkage estimators can be effectively applied to portfolio selection under the Bayesian approach. Specifically, it is shown that the portfolio selection problem can result in a statistical decision problem in some situations. Following that, we present a method for solving a problem involved in portfolio selection under the Bayesian approach.     

Robust modelling of the relationship between CD4 and viral load for complex AIDS data

CD4 and viral load play important roles in HIV/AIDS studies, and the study of their relationship has received much attention with well-known results. However, AIDS datasets are often highly complex in the sense that they typically contain outliers, measurement errors, and missing data. These data complications can greatly affect statistical analysis results, but much of the literature fail to address these issues in data analysis. In this paper, we re-visit the important relationship between CD4 and viral load and propose methods which simultaneously address outliers, measurement errors, and missing data. We find that the strength of the relationship may be severely mis-estimated if measurement errors and outliers are ignored. The proposed methods are general and can be used in other settings, where jointly modelling several different types of longitudinal data is required in the presence of data complications.     

Robust estimation for non-homogeneous data and the selection of the optimal tuning parameter: the density power divergence approach

The density power divergence (DPD) measure, defined in terms of a single parameter α, has proved to be a popular tool in the area of robust estimation [1 A. Basu, I.R. Harris, N.L. Hjort and M.C. Jones, Robust and efficient estimation by minimizing a density power divergence, Biometrika 85 (1998), pp. 549–559. doi: 10.1093/biomet/85.3.549[Crossref], [Web of Science ®] , [Google Scholar]]. Recently, Ghosh and Basu [5 A. Ghosh and A. Basu, Robust estimation for independent non-homogeneous observations using density power divergence with applications to linear regression, Electron. J. Stat. 7 (2013), pp. 2420–2456. doi: 10.1214/13-EJS847[Crossref], [Web of Science ®] , [Google Scholar]] rigorously established the asymptotic properties of the MDPDEs in case of independent non-homogeneous observations. In this paper, we present an extensive numerical study to describe the performance of the method in the case of linear regression, the most common setup under the case of non-homogeneous data. In addition, we extend the existing methods for the selection of the optimal robustness tuning parameter from the case of independent and identically distributed (i.i.d.) data to the case of non-homogeneous observations. Proper selection of the tuning parameter is critical to the appropriateness of the resulting analysis. The selection of the optimal robustness tuning parameter is explored in the context of the linear regression problem with an extensive numerical study involving real and simulated data.     

A martingale approach to the problem of dimensionality and performance of bayesian classifiers: asymptotic results

A martingale approach to the problem of performance of Bayesian classifiers with increasing feature dimensionality is applied here . Martingale limit theorems are also used to demonstrate that the expected probability of correct classification tends monotonically to unity for two general classification problems.     

Inference Problems in the Analysis of the Relationship Between Alcohol Consumption and Coronary Heart Disease

Results concerning the dependence of life durations on alcohol use of individuals who die of coronary heart disease are brought into question when no account is taken of the mortality risks associated with other competing diseases. Procedures which restrict the analysis of coronary heart disease to respondents who die of this disease will produce biased estimates of the parameters of interest. Other dimensions of coronary heart disease are neglected by focusing on durations of life and well-constructed sample surveys can provide important new information on how alcohol consumption affects the risks of getting coronary heart disease.     

Recurrence formula on the joint distribution of sample skewness and kurtosis under normality

Abstract

Two recurrence relations with respect to sample size are given concerning the joint distribution of skewness and kurtosis of random observations from a normal population: one between the probability density functions and the other between the product moments. As a consequence, the latter yields a recurrence formula for the moments of sample kurtosis. The exact moments of Jarque-Bera statistic is also given.     

The Bayes rule of the variance parameter of the hierarchical normal and inverse gamma model under Stein's loss

For the variance parameter of the hierarchical normal and inverse gamma model, we analytically calculate the Bayes rule (estimator) with respect to a prior distribution IG (alpha, beta) under Stein's loss function. This estimator minimizes the posterior expected Stein's loss (PESL). We also analytically calculate the Bayes rule and the PESL under the squared error loss. Finally, the numerical simulations exemplify that the PESLs depend only on alpha and the number of observations. The Bayes rules and PESLs under Stein's loss are unanimously smaller than those under the squared error loss.     

Robust logistic regression modelling via the elastic net-type regularization and tuning parameter selection

The penalized logistic regression is a useful tool for classifying samples and feature selection. Although the methodology has been widely used in various fields of research, their performance takes a sudden turn for the worst in the presence of outlier, since the logistic regression is based on the maximum log-likelihood method which is sensitive to outliers. It implies that we cannot accurately classify samples and find important factors having crucial information for classification. To overcome the problem, we propose a robust penalized logistic regression based on a weighted likelihood methodology. We also derive an information criterion for choosing the tuning parameters, which is a vital matter in robust penalized logistic regression modelling in line with generalized information criteria. We demonstrate through Monte Carlo simulations and real-world example that the proposed robust modelling strategies perform well for sparse logistic regression modelling even in the presence of outliers.     

Effects of drift and noise on the optimal sliding window size for data stream regression models

The analysis of non stationary data streams requires a continuous adaption of the model to the relevant most recent data. This requires that changes in the data stream must be distinguished from noise. Many approaches are based on heuristic adaptation schemes. We analyze simple regression models to understand the joint effects of noise and concept drift and derive the optimal sliding window size for the regression models. Our theoretical analysis and simulations show that a near optimal window size can be crucial. Our models can be used as benchmarks for other models to see how they cope with noise and drift.