Time-delay systems: an overview of some recent advances and open problems

We compare quantitatively eight stochastic models explaining improvements in mortality rates in England &Wales and in the US. On the basis of the Bayes Information Criterion (BIC), we find that an extension of the Cairns, Blake & Dowd (2006b) model that incorporates the cohort effect fits the England & Wales data best, while for US data, the Renshaw & Haberman (2006) extension to the Lee & Carter (1992) model that also allows for a cohort effect provides the best fit. However, we identify problems with the robustness of parameter estimates of these models over different time periods. A different extension to the Cairns, Blake & Dowd (2006b) model that allows not only for a cohort effect, but also for a quadratic age effect, while ranking below the other models in terms of the BIC, exhibits parameter stability across different time periods for both data sets. This model also shows, for both data sets, that there have been approximately linear improvements over time in mortality rates at all ages, but that the improvements have been greater at lower ages than at higher ages, and that there are significant cohort effects.

A Quantitative Comparison of Stochastic Mortality Models Using Data from England & Wales and the United States

We provide a self-contained introduction to random matrices. While some applications are mentioned, our main emphasis is on three different approaches to random matrix models: the Coulomb gas method and its interpretation in terms of algebraic geometry, loop equations and their solution using topological recursion, orthogonal polynomials and their relation with integrable systems. Each approach provides its own definition of the spectral curve, a geometric object which encodes all the properties of a model. We also introduce the two peripheral subjects of counting polygonal surfaces, and computing angular integrals.

Random matrices

We compare quantitatively eight stochastic models explaining improvements in mortality rates in England and Wales and in the United States. On the basis of the Bayes Information Criterion (BIC), we find that, for higher ages, an extension of the Cairns-Blake-Dowd (CBD) model that incorporates a cohort effect fits the England and Wales males data best, while for U.S. males data, the Renshaw and Haberman (RH) extension to the Lee and Carter model that also allows for a cohort effect provides the best fit. However, we identify problems with the robustness of parameter estimates under the RH model, calling into question its suitability for forecasting. A different extension to the CBD model that allows not only for a cohort effect, but also for a quadratic age effect, while ranking below the other models in terms of the BIC, exhibits parameter stability across different time periods for both datasets. This model also shows, for both datasets, that there have been approximately linear improvements ove...

A Quantitative Comparison of Stochastic Mortality Models Using Data From England and Wales and the United States

The Algebraic Theory of Semigroups, Vol. I. By A. H. Clifford and G. B. Preston. Pp. xv + 224. $10.60. 1961. (American Mathematical Society, Providence.)

This paper first studies the performance of the Lee–Carter [J. Am. Stat. Assoc. 419 (87) (1992) 659–675] model for mortality forecasting on the Nordic countries. Three approaches for computing the model parameters are compared: Singular Value Decomposition, Weighted Least Square and Maximum Likelihood Estimation. Hypothetical projections are also made, based on variable period intervals. Secondly, the paper addresses an extension to the Lee–Carter method: a residual bootstrapped technique is used to construct confidence intervals for forecasted life expectancies. Uncertainties produced with this method incorporate the variability from all parameters in the model, while the original Lee–Carter method focuses on the variability in the time-varying parameter.

Evaluating and extending the Lee–Carter model for mortality forecasting: Bootstrap confidence interval

The approximation of delay systems is studied. A numerically well-behaved method for computing Hankel optima) rational approximations for delay systems is presented based on certain properties of Hankel matrices and the theory of Laguerre polynomials. The CF method of Trefethen and certain Pade approximations of delay systems are also considered. The importance of a certain Wiener algebra property in the analysis of the rational approximation of an important class of delay systems in the L∞ norm and the Hankel norm is shown, completing certain results presented in the literature for systems satisfying certain nuclearity or absolute continuity conditions. A case study and numerical comparison is presented for the approximation of the parametric family of first-order stable delay systems G(s) = exp ( —τs)/(Ts + 1). Numerical experience indicates that L°° optimized CF approximations based on short truncated Maclaurin series give, in general, somewhat smaller L∞ approximation errors than Hankel optim...

Approximation of delay systems—a case study

Probability Measures on Semigroups

The Allee effect is often modeled as a threshold level below which there is population extinction. A minimum population size is required for population persistence. The Allee effect has been primarily studied in deterministic models. In this investigation, we develop two discrete-time stochastic population models with an Allee effect. The stochastic models are discrete-time Markov chains with demographic stochasticity. They are based on two well-known deterministic discrete-time population models. It is shown that the conditional means of the stochastic models follow the solution dynamics of the underlying deterministic models. However, near the Allee threshold value, the dynamics of the deterministic and stochastic models differ significantly. The first model is based on the well-known Ricker population model with an Allee effect. The second model is more complex. The population is subdivided according to mating status of adults and is based on a model developed for the northern spotted owl. The dynamics...

Population extinction in discrete-time stochastic population models with an Allee effect

Two discrete population models, one with stochasticity in the carrying capacity and one with stochasticity in the per capita growth rate, are investigated. Conditions under which the corresponding Markov processes are null recurrent and positively recurrent are derived.

Göran Högnäs

Papers

Evaluating and extending the Lee–Carter model for mortality forecasting: Bootstrap confidence interval

Approximation of delay systems—a case study

Probability Measures on Semigroups

Population extinction in discrete-time stochastic population models with an Allee effect

Population models with environmental stochasticity