Equilibrium Reinsurance Strategy and Mean Residual Life Function

In this paper,we analyze the relationship between the equilibrium reinsurance strategy and the tail of the distribution of the risk.Since Mean Residual Life(MRL)has a close relationship with the tail of the distribution,we consider two classes of risk distributions,Decreasing Mean Residual Life(DMRL)and Increasing Mean Residual Life(IMRL)distributions,which can be used to classify light-tailed and heavy-tailed distributions,respectively.We assume that the underlying risk process is modelled by the classical CramérLundberg model process.Under the mean-variance criterion,by solving the extended Hamilton-Jacobi-Bellman equation,we derive the equilibrium reinsurance strategy for the insurer and the reinsurer under DMRL and IMRL,respectively.Furthermore,we analyze how to choose the reinsurance premium to make the insurer and the reinsurer agree with the same reinsurance strategy.We find that under the case of DMRL,if the distribution and the risk aversions satisfy certain conditions,the insurer and the reinsurer can adopt a reinsurance premium to agree on a reinsurance strategy,and under the case of IMRL,the insurer and the reinsurer can only agree with each other that the insurer do not purchase the reinsurance.

A GMM approach in coupling internal data and external summary information with heterogeneous data populations

Because of advances in data collection and storage,statistical analysis in modern scientific research and practice now has opportunities to utilize external information such as summary statistics from similar studies.A likelihood approach based on a parametric model assumption has been developed in the literature to utilize external summary information when the populations for external and main internal data are assumed to be the same.In this article,we instead consider the generalized estimation equation(GEE)approach for statistical inference,which is semiparametric or nonparametric,and show how to utilize external summary information even when internal and external data populations are not the same.Our approach is coupling the internal data and external summary information to form additional estimation equations and then applying the generalized method of moments(GMM).We show that the proposed GMM estimator is asymptotically normal and,under some conditions,is more efficient than the GEE estimator without using external summary information.Estimators of the asymptotic covariance matrix of the GMM estimators are also proposed.Simulation results are obtained to confirm our theory and quantify the improvements by utilizing external data.An example is also included for illustration.

On the maximin distance properties of orthogonal designs via the rotation

Space-filling designs are widely used in computer experiments.They are frequently evaluated by the orthogonality and distance-related criteria.Rotating orthogonal arrays is an appealing approach to constructing orthogonal space-filling designs.An important issue that has been rarely addressed in the literature is the design selection for the initial orthogonal arrays.This paper studies the maximin L_(2)-distance properties of orthogonal designs generated by rotating two-level orthogonal arrays under three criteria.We provide theoretical justifications for the rotation method from a maximin distance perspective and further propose to select initial orthogonal arrays by the minimum G_(2)-aberration criterion.New infinite families of orthogonal or 3-orthogonal U-type designs,which also perform well under the maximin distance criterion,are obtained and tabulated.Examples are presented to show the effectiveness of the constructed designs for building statistical surrogate models.

Functional Multiple-Outcome Model in Application to Multivariate Growth Curves of Infant Data

Motivated by a medical study that attempts to analyze the relationship between growth curves and other variables and to measure the association among multiple growth curves,the authors develop a functional multiple-outcome model to decompose the total variation of multiple functional outcomes into variation explained by independent variables with time-varying coefficient functions,by latent factors and by noise.The latent factors are the hidden common factors that influence the multiple outcomes and are found through the combined functional principal component analysis approach.Through the coefficients of the latent factors one may further explore the association of the multiple outcomes.This method is applied to the multivariate growth data of infants in a real medical study in Shanghai and produces interpretable results.Convergence rates for the proposed estimates of the varying coefficient and covariance functions of the model are derived under mild conditions.

The abstract of doctoral dissertation‘nonlinear wavelet density estimation and hazard rate estimation with data missing at random’

In this thesis,we establish non-linear wavelet density estimators and studying the asymptotic properties of the estimators with data missing at random when covariates are present.The outstanding advantage of non-linear wavelet method is estimating the unsoothed functions,however,the classical kernel estimation cannot do this work.At the same time,we study the larger sample properties of the ISE for hazard rate estimator.

Analysis of Stock Splits Based on Risk Theory: Empirical Evidence from the Chinese Stock Markets

The paper first analyzes price change due to stock splits in Chinese stock markets,which shows stock prices typically go up for stock splits.Then theoretical analyses based on risk theory are presented to explain the reason,where the method comes from a new perspective and obtained theoretical conclusions show that stock splits typically make stock price go up if risk-compensation function is convex,and go down if risk-compensation function is concave.Stock prices typically go up for stock splits because risk-compensation functions are mainly convex.The obtained conclusions are consistent with the known results in the last three decades.

Novel loci and potential mechanisms of major depressive disorder,bipolar disorder,and schizophrenia

Different psychiatric disorders share genetic relationships and pleiotropic loci to certain extent.We integrated and analyzed datasets related to major depressive disorder(MDD),bipolar disorder(BIP),and schizophrenia(SCZ)from the Psychiatric Genomics Consortium using multitrait analysis of genome-wide association analysis(MTAG).MTAG significantly increased the effective sample size from 99,773 to 119,754 for MDD,from 909,061 to 1,450,972 for BIP,and from 856,677 to 940,613 for SCZ.We discovered 7,32,and 43 novel lead single nucleotide polymorphisms(SNPs)and 1,6,and 3 novel causal SNPs for MDD,BIP,and SCZ,respectively,after fine-mapping.We identified rs8039305 in the FURIN gene as a novel pleiotropic locus across the three disorders.We performed marker analysis of genomic annotation(MAGMA)and Hi-C-coupled MAGMA(H-MAGMA)based gene-set analysis and identified 101 genes associated with the three disorders,which were enriched in the regulation of postsynaptic membranes,postsynaptic membrane dopaminergic synapses,and Notch signaling pathway.Next,we performed Mendelian randomization analysis using different tools and detected a causal effect of BIP on SCZ.Overall,we demonstrated the usage of combined genome-wide association studies summary statistics for exploring potential novel mechanisms of the three psychiatric disorders,providing an alternative approach to integrate publicly available summary data.

Convergence rate of principal component analysis with local-linear smoother for functional data under a unified weighing scheme

The unified weighing scheme for the local-linear smoother in analysing functional data can deal with data that are dense,sparse or of neither type.In this paper,we focus on the convergence rate of functional principal component analysis using this method.Almost sure asymptotic consistency and rates of convergence for the estimators of eigenvalues and eigenfunctions have been established.We also provide the convergence rate of the variance estimation of the measurement error.Based on the results,the number of observations within each curve can be of any rate relative to the sample size,which is consistent with the earlier conclusions about the asymptotic properties of the mean and covariance estimators.

Regression models of Pearson correlation coefficient

We propose two simple regression models of Pearson correlation coefficient of two normal responses or binary responses to assess the effect of covariates of interest.Likelihood-based inference is established to estimate the regression coefficients,upon which bootstrap-based method is used to test the significance of covariates of interest.Simulation studies show the effectiveness of the method in terms of type-I error control,power performance in moderate sample size and robustness with respect to model mis-specification.We illustrate the application of the proposed method to some real data concerning health measurements.

Proportional Mean Residual Life Model with Varying Coefficients for Length-Biased and Right-Censored Data

Length-biased data are encountered in many fields,including economics,engineering and epidemiological cohort studies.There are two main challenges in the analysis of such data:the assumption of independent censoring is violated and the assumed model for the underlying population is no longer satisfied for the observed data.In this paper,a proportional mean residual life varyingcoefficient model for length-biased data is considered and a local pseudo likelihood method is proposed for estimating the coefficient functions in the model.Asymptotic properties are investigated for the proposed estimators.The finite sample performance of the proposed methodology is demonstrated by simulation studies.Finally,the method is applied to a real data set concerning the Academy Awards.

Partially Linear Single-Index Model in the Presence of Measurement Error

The partially linear single-index model(PLSIM) is a flexible and powerful model for analyzing the relationship between the response and the multivariate covariates. This paper considers the PLSIM with measurement error possibly in all the variables. The authors propose a new efficient estimation procedure based on the local linear smoothing and the simulation-extrapolation method,and further establish the asymptotic normality of the proposed estimators for both the index parameter and nonparametric link function. The authors also carry out extensive Monte Carlo simulation studies to evaluate the finite sample performance of the new method, and apply it to analyze the osteoporosis prevention data.

Integrated Square Error of Hazard Rate Estimation for Survival Data with Missing Censoring Indicators

The problem of hazard rate estimation under right-censored assumption has been investigated extensively.Integrated square error(ISE)of estimation is one of the most widely accepted measurements of the global performance for nonparametric kernel estimation.But there are no results available for ISE of hazard rate estimation under right-censored model with censoring indicators missing at random(MAR)so far.This paper constructs an imputation estimator of the hazard rate function and establish asymptotic normality of the ISE for the kernel hazard rate estimator with censoring indicators MAR.At the same time,an asymptotic representation of the mean integrated square error(MISE)is also presented.The finite sample behavior of the estimator is investigated via one simple simulation.

A Two-Step Test for the Two-Sample Problem of Processes of Ornstein-Uhlenbeck Type

The authors propose a two-step test for the two-sample problem of processes of OrnsteinUhlenbeck type. In the first step, the authors test the equality of correlation structures, based on the least square estimators of the correlation parameters, and the test statistic follows the standard normal distribution. If the null hypothesis is not rejected in the first step, the authors consider a second step to test the equality of marginal distributions, based on the weighted deviation of the empirical characteristic functions;the test statistic has a complicated asymptotic distribution, so that sequential bootstrap method is applied to reach a temporary decision. Simulation studies and real data analysis suggest that the proposed approach performs well in finite samples.

A review of distributed statistical inference

The rapid emergence of massive datasets in various fields poses a serious challenge to tra-ditional statistical methods.Meanwhile,it provides opportunities for researchers to develop novel algorithms.Inspired by the idea of divide-and-conquer,various distributed frameworks for statistical estimation and inference have been proposed.They were developed to deal with large-scale statistical optimization problems.This paper aims to provide a comprehensive review for related literature.It includes parametric models,nonparametric models,and other frequently used models.Their key ideas and theoretical properties are summarized.The trade-off between communication cost and estimate precision together with other concerns is discussed.

Sparse Estimation of High-Dimensional Inverse Covariance Matrices with Explicit Eigenvalue Constraints

Firstly,this paper proposes a generalized log-determinant optimization model with the purpose of estimating the high-dimensional sparse inverse covariance matrices.Under the normality assumption,the zero components in the inverse covariance matrices represent the conditional independence between pairs of variables given all the other variables.The generalized model considered in this study,because of the setting of the eigenvalue bounded constraints,covers a large number of existing estimators as special cases.Secondly,rather than directly tracking the challenging optimization problem,this paper uses a couple of alternating direction methods of multipliers(ADMM)to solve its dual model where 5 separable structures are contained.The first implemented algorithm is based on a single Gauss–Seidel iteration,but it does not necessarily converge theoretically.In contrast,the second algorithm employs the symmetric Gauss–Seidel(sGS)based ADMM which is equivalent to the 2-block iterative scheme from the latest sGS decomposition theorem.Finally,we do numerical simulations using the synthetic data and the real data set which show that both algorithms are very effective in estimating high-dimensional sparse inverse covariance matrix.

Optimal mean-variance reinsurance and investment strategy with constraints in a non-Markovian regime-switching model

This paper is devoted to study the proportional reinsurance/new business and investment problem under the mean-variance criterion in a continuous-time setting.The strategies are constrained in the non-negative cone and all coefficients in the model except the interest rate are stochastic processes adapted the filtration generated by a Markov chain.With the help of a backward stochastic differential equation driven by the Markov chain,we obtain the optimal strategy and optimal cost explicitly under this non-Markovian regime-switching model.The cases with one risky asset and Markov regime-switching model are considered as special cases.

Rejoinder on‘A review of distributed statistical inference’

We thank the editor,Professor Jun Shao,for orga-nizing this stimulating discussion.We are grateful to all discussants for their insightful comments on our review article on the distributed statistical inference.Due to the urgent need to process the datasets with massive sizes,various distributed computing methods have been proposed for the large-scale statistical prob-lems.Meanwhile,some important theoretical results were established.While we want to give a relatively comprehensive overview on this hot topic,there are still some important works that have been missed in our review written over a year ago.However,we are glad to see the discussants provide reviews of some new works and references.We hope that these discussions and our review would serve as a stimulus for further studies in this rapidly developing area.

Semiparametric quantile-difference estimation for length-biased and right-censored data

Prevalent cohort studies frequently involve length-biased and right-censored data, a fact that has drawn considerable attention in survival analysis. In this article, we consider survival data arising from lengthbiased sampling, and propose a new semiparametric-model-based approach to estimate quantile differences of failure time. We establish the asymptotic properties of our new estimators theoretically under mild technical conditions, and propose a resampling method for estimating their asymptotic variance. We then conduct simulations to evaluate the empirical performance and efficiency of the proposed estimators, and demonstrate their application by a real data analysis.

A Double Varying-coefficient Modeling Approach for Analyzing Longitudinal Observations

The identification of within-subject dependence is important for constructing efficient estimation in longitudinal data models.In this paper,we proposed a flexible way to study this dependence by using nonparametric regression models.Specifically,we considered the estimation of varying coefficient longitudinal data model with non-stationary varying coefficient autoregressive error process over observational time quantum.Based on spline approximation and local polynomial techniques,we proposed a two-stage nonparametric estimation for unknown functional coefficients and didn’t not drop any observations in a hybrid least square loss framework.Moreover,we showed that the estimated coefficient functions are asymptotically normal and derived the asymptotic biases and variances accordingly.Monte Carlo studies and two real applications were conducted for illustrating the performance of our proposed methods.

Semiparametric Likelihood-based Inference for Censored Data with Auxiliary Information from External Massive Data Sources

Published auxiliary information can be helpful in conducting statistical inference in a new study.In this paper,we synthesize the auxiliary information with semiparametric likelihood-based inference for censoring data with the total sample size is available.We express the auxiliary information as constraints on the regression coefficients and the covariate distribution,then use empirical likelihood method for general estimating equations to improve the efficiency of the interested parameters in the specified model.The consistency and asymptotic normality of the resulting regression parameter estimators established.Also numerical simulation and application with different supposed conditions show that the proposed method yields a substantial gain in efficiency of the interested parameters.