Modeling multi-country mortality dependence and its application in pricing survivor index swaps—A dynamic copula approach

This paper introduces mortality dependence in multi-country mortality modeling using a dynamic copula approach. Specifically, we use time-varying copula models to capture the mortality dependence structure across countries, examining both symmetric and asymmetric dependence structures. In addition, to capture the phenomenon of a heavy tail for the multi-country mortality index, we consider not only the setting of Gaussian innovations but also non-Gaussian innovations under the Lee–Carter framework model. As tests of the goodness of fit of different dynamic copula models, the pattern of mortality dependence, and the distribution of the innovations, we used empirical mortality data from Finland, France, the Netherlands, and Sweden. To understand the effect of mortality dependence on longevity derivatives, we also built a valuation framework for pricing a survivor index swap, then investigated the fair swap rates of a survivor swap numerically. We demonstrate that failing to consider the dynamic copula mortality model and non-Gaussian innovations would lead to serious underestimations of the swap rates and loss reserves.     

On stochastic management modeling with dependent variables

This paper presents an integrated framework for handling dependent random variables in a large class of stochastic management models, a class that includes stochastic break-even analysis and stochastic present-value analysis. We first demonstrate that the common approach of modeling dependent random variables is usually surprisingly inadequate, and a general “functional approach” is presented as a practical modeling alternative. Adopting this modeling approach, we then present a procedure for deriving the stochastic characteristics of the model's objective variable. In the context of stochastic breakeven analysis, this means determining the probabilities of achieving various profit levels and the expected utility of the stochastic profit. The procedure allows the model's random variables to assume diverse distribution and dependency forms, and the simplicity and reliability of the procedure is demonstrated by a numerical example.     

Bayesian reference analysis for Gaussian Markov random fields

Gaussian Markov random fields (GMRF) are important families of distributions for the modeling of spatial data and have been extensively used in different areas of spatial statistics such as disease mapping, image analysis and remote sensing. GMRFs have been used for the modeling of spatial data, both as models for the sampling distribution of the observed data and as models for the prior of latent processes/random effects; we consider mainly the former use of GMRFs. We study a large class of GMRF models that includes several models previously proposed in the literature. An objective Bayesian analysis is presented for the parameters of the above class of GMRFs, where explicit expressions for the Jeffreys (two versions) and reference priors are derived, and for each of these priors results on posterior propriety of the model parameters are established. We describe a simple MCMC algorithm for sampling from the posterior distribution of the model parameters, and study frequentist properties of the Bayesian inferences resulting from the use of these automatic priors. Finally, we illustrate the use of the proposed GMRF model and reference prior for studying the spatial variability of lip cancer cases in the districts of Scotland over the period 1975-1980.     

sppmix: Poisson point process modeling using normal mixture models

This paper describes the package sppmix for the statistical environment R. The sppmix package implements classes and methods for modeling spatial point patterns using inhomogeneous Poisson point processes, where the intensity surface is assumed to be a multiple of a finite additive mixture of normal components and the number of components is a finite, fixed or random integer. Extensions to the marked inhomogeneous Poisson point processes case are also presented. We provide an extensive suite of R functions that can be used to simulate, visualize and model point patterns, estimate the parameters of the models, assess convergence of the algorithms and perform model selection and checking in the proposed modeling context. In addition, several approaches have been implemented in order to handle the standard label switching issue which arises in any modeling approach involving mixture models. We adapt a hierarchical Bayesian framework in order to model the intensity surfaces and have implemented two major algorithms in order to estimate the parameters of the mixture models involved: the data augmentation and the birth–death Markov chain Monte Carlo (DAMCMC and BDMCMC). We used C++ (via the Rcpp package) in order to implement the most computationally intensive algorithms.     

A robust optimization approach to experimental design for model discrimination of dynamical systems

A high-ranking goal of interdisciplinary modeling approaches in science and engineering are quantitative prediction of system dynamics and model based optimization. Quantitative modeling has to be closely related to experimental investigations if the model is supposed to be used for mechanistic analysis and model predictions. Typically, before an appropriate model of an experimental system is found different hypothetical models might be reasonable and consistent with previous knowledge and available data. The parameters of the models up to an estimated confidence region are generally not known a priori. Therefore one has to incorporate possible parameter configurations of different models into a model discrimination algorithm which leads to the need for robustification. In this article we present a numerical algorithm which calculates a design of experiments allowing optimal discrimination of different hypothetic candidate models of a given dynamical system for the most inappropriate (worst case) parameter configurations within a parameter range. The design comprises initial values, system perturbations and the optimal placement of measurement time points, the number of measurements as well as the time points are subject to design. The statistical discrimination criterion is worked out rigorously for these settings, a derivation from the Kullback-Leibler divergence as optimization objective is presented for the case of discontinuous Heaviside-functions modeling the measurement decision which are replaced by continuous approximations during the optimization procedure. The resulting problem can be classified as a semi-infinite optimization problem which we solve in an outer approximations approach stabilized by a suggested homotopy strategy whose efficiency is demonstrated. We present the theoretical framework, algorithmic realization and numerical results.     

ESTIMATION OF SALMON ESCAPEMENT: MODELS WITH ENTRY,MORTALITY AND STOCHASTICITY

Understanding the dynamics of Prince William Sound pink salmon requires knowledge of the size of the spawning population in a stream over time. Periodic aerial surveys provide observations on the number of spawners, but the lack of daily observations requires a model to fill in the gaps. We develop a differential equation framework to represent the dynamics of escapement during the season. An exponential population growth model with a time-varying rate of growth is used for the number of spawners. The rate of growth consists of two primary components: the entry of salmon to the stream (escapement) and the mortality of spawners in the stream. The models for entry and mortality are also functions of time. The stochastic element of the model is based on a nonhomogeneous birth-and-death process which leads to a least squares estimation approach with either additive measurement or process errors. We illustrate the approach for a stream in Prince William Sound by fitting various models to observed spawner abundance, mortality counts from ground surveys and weir counts of the entry to the stream. We believe this approach could improve salmon escapement estimation, because the processes governing entry and mortality are explicitly considered.     

Intensity-based framework for surrender modeling in life insurance

In this paper, we propose an intensity-based framework for surrender modeling. We model the surrender decision under the assumption of stochastic intensity and use, for comparative purposes, the affine models of Vasicek and Cox–Ingersoll–Ross for deriving closed-form solutions of the policyholder’s probability of surrendering the policy. The introduction of a closed-form solution is an innovative aspect of the model we propose. We evaluate the impact of dynamic policyholders’ behavior modeling the dependence between interest rates and surrendering (affine dependence) with the assumption that mortality rates are independent of interest rates and surrendering. Finally, using experience-based decrement tables for both surrendering and mortality, we explain the calibration procedure for deriving our model’s parameters and report numerical results in terms of best estimate of liabilities for life insurance under Solvency II.     

Adaptive Bayesian Nonstationary Modeling for Large Spatial Datasets Using Covariance Approximations

Gaussian process models have been widely used in spatial statistics but face tremendous modeling and computational challenges for very large nonstationary spatial datasets. To address these challenges, we develop a Bayesian modeling approach using a nonstationary covariance function constructed based on adaptively selected partitions. The partitioned nonstationary class allows one to knit together local covariance parameters into a valid global nonstationary covariance for prediction, where the local covariance parameters are allowed to be estimated within each partition to reduce computational cost. To further facilitate the computations in local covariance estimation and global prediction, we use the full-scale covariance approximation (FSA) approach for the Bayesian inference of our model. One of our contributions is to model the partitions stochastically by embedding a modified treed partitioning process into the hierarchical models that leads to automated partitioning and substantial computational benefits. We illustrate the utility of our method with simulation studies and the global Total Ozone Matrix Spectrometer (TOMS) data. Supplementary materials for this article are available online.     

MODELING INDIVIDUAL TREE MORTALITY RATES USING MARGINAL AND RANDOM EFFECTS REGRESSION MODELS

Abstract Developing models to predict tree mortality using data from long‐term repeated measurement data sets can be difficult and challenging due to the nature of mortality as well as the effects of dependence on observations. Marginal (population‐averaged) generalized estimating equations (GEE) and random effects (subject‐specific) models offer two possible ways to overcome these effects. For this study, standard logistic, marginal logistic based on the GEE approach, and random logistic regression models were fitted and compared. In addition, four model evaluation statistics were calculated by means of K‐fold cross‐valuation. They include the mean prediction error, the mean absolute prediction error, the variance of prediction error, and the mean square error. Results from this study suggest that the random effects model produced the smallest evaluation statistics among the three models. Although marginal logistic regression accommodated for correlations between observations, it did not provide noticeable improvements of model performance compared to the standard logistic regression model that assumed impendence. This study indicates that the random effects model was able to increase the overall accuracy of mortality modeling. Moreover, it was able to ascertain correlation derived from the hierarchal data structure as well as serial correlation generated through repeated measurements.     

Modeling the spatiotemporal variations in brucellosis transmission

In this paper, we propose a nonlinear modeling framework to investigate the transmission dynamics of brucellosis, incorporating both the spatial and seasonal variations. The spatial modeling is based on a patch structure, and the seasonal impact is represented by utilizing time-periodic model parameters. We demonstrate this framework through a two-patch model and conduct detailed analysis, for the cases with and without seasonal oscillations, respectively. In particular, we establish the threshold dynamics results using the reproduction numbers defined under different scenarios. Our findings underscore the importance of including spatial and seasonal heterogeneities in the design of control strategies for brucellosis.     

Dynamic Bivariate Mortality Modelling

The dependence structure of the life statuses plays an important role in the valuation of life insurance products involving multiple lives. Although the mortality of individuals is well studied in the literature, their dependence remains a challenging field. In this paper, the main objective is to introduce a new approach for analyzing the mortality dependence between two individuals in a couple. It is intended to describe in a dynamic framework the joint mortality of married couples in terms of marginal mortality rates. The proposed framework is general and aims to capture, by adjusting some parametric form, the desired effect such as the “broken-heart syndrome”. To this end, we use a well-suited multiplicative decomposition, which will serve as a building block for the framework to relate the dependence structure and the marginals, and we make the link with existing practice of affine mortality models. Finally, given that the framework is general, we propose some illustrative examples and show how the underlying model captures the main stylized facts of bivariate mortality dynamics.

     

Kernel-based learning methods for preference aggregation

The mathematical representation of human preferences has been a subject of study for researchers in different fields. In multi-criteria decision making (MCDM) and fuzzy modeling, preference models are typically constructed by interacting with the human decision maker (DM). However, it is known that a DM often has difficulties to specify precise values for certain parameters of the model. He/she instead feels more comfortable to give holistic judgements for some of the alternatives. Inference and elicitation procedures then assist the DM to find a satisfactory model and to assess unjudged alternatives. In a related but more statistical way, machine learning algorithms can also infer preference models with similar setups and purposes, but here less interaction with the DM is required/allowed. In this article we discuss the main differences between both types of inference and, in particular, we present a hybrid approach that combines the best of both worlds. This approach consists of a very general kernel-based framework for constructing and inferring preference models. Additive models, for which interpretability is preserved, and utility models can be considered as special cases. Besides generality, important benefits of this approach are its robustness to noise and good scalability. We show in detail how this framework can be utilized to aggregate single-criterion outranking relations, resulting in a flexible class of preference models for which domain knowledge can be specified by a DM.      

Evolutionary dynamics of cancer:From epigenetic regulation to cell population dynamics——mathematical model framework,applications,and open problems

Predictive modeling of the evolutionary dynamics of cancer is a challenging issue in computational cancer biology. In this paper, we propose a general mathematical model framework for the evolutionary dynamics of cancer, including plasticity and heterogeneity in cancer cells. Cancer is a group of diseases involving abnormal cell growth, during which abnormal regulation of stem cell regeneration is essential for the dynamics of cancer development. In general, the dynamics of stem cell regeneration can be simplified as a G0 phase cell cycle model, which leads to a delay differentiation equation. When cell heterogeneity and plasticity are considered, we establish a differential-integral equation based on the random transition of epigenetic states of stem cells during cell division. The proposed model highlights cell heterogeneity and plasticity;connects the heterogeneity with cell-to-cell variance in cellular behaviors(for example, proliferation, apoptosis, and differentiation/senescence);and can be extended to include gene mutation-induced tumor development. Hybrid computational models are developed based on the mathematical model framework and are applied to the processes of inflammationinduced tumorigenesis and tumor relapse after chimeric antigen receptor(CAR)-T cell therapy. Finally, we propose several mathematical problems related to the proposed differential-integral equation. Solutions to these problems are crucial for understanding the evolutionary dynamics of cancer.     

Is the Home Equity Conversion Mortgage in the United States sustainable? Evidence from pricing mortgage insurance premiums and non-recourse provisions using the conditional Esscher transform

The purpose of this paper is to build a modeling and pricing framework to investigate the sustainability of the Home Equity Conversion Mortgage (HECM) program in the United States under realistic economic scenarios, i.e., whether the premium payments cover the fair premiums for the inherent risks in the HECM program. We note that earlier HECM models use static mortality tables, neglecting the dynamics of mortality rates and extreme mortality jumps. The earlier models also assume housing prices follow a geometric Brownian motion, which contradicts the fact that housing prices exhibit strong autocorrelation and varying volatility over time. To solve these problems, we propose a generalized Lee-Carter model with asymmetric jump effects to fit the mortality data, and model the house price index via an ARIMA-GARCH process. We then employ the conditional Esscher transform to price the non-recourse provision of reverse mortgages and compare it with the calculated mortgage insurance premiums. The HECM program turns out to be sustainable based on our model setup and parameter settings.     

Intelligent data analysis and model interpretation with spectral analysis fuzzy symbolic modeling

This paper proposes fuzzy symbolic modeling as a framework for intelligent data analysis and model interpretation in classification and regression problems. The fuzzy symbolic modeling approach is based on the eigenstructure analysis of the data similarity matrix to define the number of fuzzy rules in the model. Each fuzzy rule is associated with a symbol and is defined by a Gaussian membership function. The prototypes for the rules are computed by a clustering algorithm, and the model output parameters are computed as the solutions of a bounded quadratic optimization problem. In classification problems, the rules’ parameters are interpreted as the rules’ confidence. In regression problems, the rules’ parameters are used to derive rules’ confidences for classes that represent ranges of output variable values. The resulting model is evaluated based on a set of benchmark datasets for classification and regression problems. Nonparametric statistical tests were performed on the benchmark results, showing that the proposed approach produces compact fuzzy models with accuracy comparable to models produced by the standard modeling approaches. The resulting model is also exploited from the interpretability point of view, showing how the rule weights provide additional information to help in data and model understanding, such that it can be used as a decision support tool for the prediction of new data.     

Bayesian inference on partially linear mixed-effects joint models for longitudinal data with multiple features

The relationship between viral load and CD4 cell count is one of the interesting questions in AIDS research. Statistical models are powerful tools for clarifying this important problem. Partially linear mixed-effects (PLME) model which accounts for the unknown function of time effect is one of the important models for this purpose. Meanwhile, the mixed-effects modeling approach is suitable for the longitudinal data analysis. However, the complex process of data collection in clinical trials has made it impossible to rely on one particular model to address the issues. Asymmetric distribution, measurement error and left censoring are features commonly arisen in longitudinal studies. It is crucial to take into account these features in the modeling process to achieve reliable estimation and valid conclusion. In this article, we establish a joint model that accounts for all these features in the framework of PLME models. A Bayesian inferential procedure is proposed to estimate parameters in the joint model. A real data example is analyzed to demonstrate the proposed modeling approach for inference and the results are reported by comparing various scenarios-based models.     

Multi-population mortality models: A factor copula approach

Modeling mortality co-movements for multiple populations have significant implications for mortality/longevity risk management. A few two-population mortality models have been proposed to date. They are typically based on the assumption that the forecasted mortality experiences of two or more related populations converge in the long run. This assumption might be justified by the long-term mortality co-integration and thus be applicable to longevity risk modeling. However, it seems too strong to model the short-term mortality dependence. In this paper, we propose a two-stage procedure based on the time series analysis and a factor copula approach to model mortality dependence for multiple populations. In the first stage, we filter the mortality dynamics of each population using an ARMA–GARCH process with heavy-tailed innovations. In the second stage, we model the residual risk using a one-factor copula model that is widely applicable to high dimension data and very flexible in terms of model specification. We then illustrate how to use our mortality model and the maximum entropy approach for mortality risk pricing and hedging. Our model generates par spreads that are very close to the actual spreads of the Vita III mortality bond. We also propose a longevity trend bond and demonstrate how to use this bond to hedge residual longevity risk of an insurer with both annuity and life books of business.     

Robust additive Gaussian process models using reference priors and cut-off-designs

When powerful numerical tools like the finite element method encounter their limits for the evaluation of physical systems it is very common to use surrogate models as an approximation. There are many possible choices concerning the model approach, among which Gaussian process models are the most popular ones due to their clear statistical basis. A very desirable attribute of such surrogates is a high flexibility for making them applicable to a great class of underlying problems while obtaining interpretable results. To achieve this Gaussian processes are used as basis functions of an additive model in this work. Another important property of a surrogate is stability, which can be especially challenging when it comes to the estimation of the correlation parameters. To solve this we use a Bayesian approach where a reference prior is assigned to each component of the additive model assuring robust correlation matrices. Due to the additive structure of the model a simplified parameter estimation process is proposed that reduces the usually high-dimensional optimization problem to a few sub-routines of low dimension. Finally, we demonstrate this concept by modeling the magnetic field of a magnetic linear position detection system.     

Spatiotemporal Models for Gaussian Areal Data

We introduce a class of spatiotemporal models for Gaussian areal data. These models assume a latent random field process that evolves through time with random field convolutions; the convolving fields follow proper Gaussian Markov random field (PGMRF) processes. At each time, the latent random field process is linearly related to observations through an observational equation with errors that also follow a PGMRF. The use of PGMRF errors brings modeling and computational advantages. With respect to modeling, it allows more flexible model structures such as different but interacting temporal trends for each region, as well as distinct temporal gradients for each region. Computationally, building upon the fact that PGMRF errors have proper density functions, we have developed an efficient Bayesian estimation procedure based on Markov chain Monte Carlo with an embedded forward information filter backward sampler (FIFBS) algorithm. We show that, when compared with the traditional one-at-a-time Gibbs sampler, our novel FIFBS-based algorithm explores the posterior distribution much more efficiently. Finally, we have developed a simulation-based conditional Bayes factor suitable for the comparison of nonnested spatiotemporal models. An analysis of the number of homicides in Rio de Janeiro State illustrates the power of the proposed spatiotemporal framework.

Supplemental materials for this article are available online in the journal’s webpage.     

Inference for SDE Models via Approximate Bayesian Computation

Models defined by stochastic differential equations (SDEs) allow for the representation of random variability in dynamical systems. The relevance of this class of models is growing in many applied research areas and is already a standard tool to model, for example, financial, neuronal, and population growth dynamics. However, inference for multidimensional SDE models is still very challenging, both computationally and theoretically. Approximate Bayesian computation (ABC) allows to perform Bayesian inference for models which are sufficiently complex that the likelihood function is either analytically unavailable or computationally prohibitive to evaluate. A computationally efficient ABC-MCMC algorithm is proposed, halving the running time in our simulations. Focus here is on the case where the SDE describes latent dynamics in state-space models; however, the methodology is not limited to the state-space framework. We consider simulation studies for a pharmacokinetics/pharmacodynamics model and for stochastic chemical reactions and we provide a Matlab package that implements our ABC-MCMC algorithm.