# Abstracts Research Seminar Winter Term 2012/13

## Rob J Hyndman: Demographic forecasting using functional data analysis

Functional time series are curves that are observed sequentially in time. In demography, such data arise as the curves formed by annual death rates as a function of age or annual fertility rates as a function of age. I will discuss methods for describing, modelling and forecasting such functional time series data.

Challenges include:

* developing useful graphical tools (I will illustrate a functional version of the boxplot);

* dealing with outliers (e.g., death rates have outliers in years of wars or epidemics);

* cohort effects (how can we identify and allow for these in the forecasts);

* synergy between groups (e.g., we expect male and female mortality rates to evolve in a similar way in the future);

* deriving prediction intervals for forecasts;

* how to combine the mortality and fertility forecasts to obtain forecasts of the total population.

I will illustrate the ideas using data from Australia and France.

## Ludger Rüschendorf: Risk bounds, worst case dependence and optimal claims and contracts

This talk is concerned with the description of possible influence of positive dependence on the magnitude of risk in a portfolio vector. We discuss and review developments on the problem of risk bounds for several risk functionals related to various dependence orderings. In particular we consider the question, whether there exists for multivariate portfolios a universal worst case dependence structure as in the one-dimensional case. We characterize the worst case dependence structure and discuss several examples. Applications of the related ordering results are given to the determination of optimal claims and contracts.

## Wolfgang Härdle: Risk Patterns and Correlated Brain Activities. Multidimensional statistical analysis of fMRI data with application to risk patterns

(Authors: Alena Myšičková, Song Song, Piotr Majer, Peter N.C. Mohr, Hauke R. Heekeren, Wolfgang K. Härdle)

Decision making usually involves uncertainty and risk. Understanding which parts of the human brain are activated during decisions under risk and which neural processes underly (risky) investment decisions are important goals in neuroeconomics. Here, we reanalyze functional magnetic resonance imaging (fMRI) data on 17 subjects which were exposed to an investment decision task from Mohr et al. (2010b). We obtain a time series of three-dimensional images of the blood-oxygen-level dependent (BOLD) fMRI signals. Our goal is to capture the dynamic behavior of specific brain regions of all subjects in this high-dimensional time series data, by a exible factor approach resulting in a low dimensional representation. We apply a panel version of the dynamic semiparametric factor model (DSFM) presented in Park et al. (2009) and identify task-related activations in space and dynamics in time. Further, we classify the risk attitudes of all subjects based on the estimated lowdimensional time series. Our classification analysis successfully confirms the estimated risk attitudes derived directly from subjects' decision behavior.

## Andrea Riebler: Estimation and extrapolation of time trends in multivariate registry data using Bayesian age-period-cohort models

Age-period-cohort (APC) models are commonly used to analyze and project mortality or morbidity rates, in which effects related to the age of an individual, calendar time (period) and the generation (cohort) can reasonably be assumed to be present. Bayesian approaches facilitate estimation and improve predictions by assigning smoothing priors to age, period and cohort effects. A quirk of APC models is, however, the obvious linear dependence of age, period and cohort effects leading to a well-known identifiability problem. When rates are further stratified, for example, by countries, multivariate APC models can be used, where differences of stratum-specific effects are identifiable and interpretable as log relative risks. In this talk I will introduce the univariate APC model and illustrate the inherent identifiability problem before I will present novel methodology for statistical inference in multivariate Bayesian APC models. In particular, I will propose the use of correlated smoothing priors and correlated overdispersion parameters to capture the dependence present between multiple health outcomes. Compared to a model without correlation, the new approach may lead to more precise relative risk estimates. Furthermore, the imputation of missing data for one particular stratum may be improved, since the new approach takes advantage of the remaining strata if the corresponding observations are available there. Markov chain Monte Carlo and integrated nested Laplace approximations will be used for inference.

This is joint work with Leonhard Held and Havard Rue.

## Ralf Wunderlich: Optimal portfolio strategies under partial information with expert opinions

This talk considers optimal portfolio strategies for utility maximizing investors in a financial market with partial information on the drift. The drift is modelled by a continuous-time Markov chain with finitely many states which is not directly observable. Information on the drift is obtained from the observation of stock prices. Moreover, and this is the novel feature of the analysis, expert opinions are included in the analysis. This additional information we model by a marked point process with jump-size distribution depending on the current state of the hidden Markov chain. We derive the filtering equation for the return process and incorporate the filter into the state variables of the optimization problem. For this reformulated completely observable problem we investigate for the case of power utility the associated dynamic programming equation. Simplifying this equation by a change of measure leads to a new equation where the set of state variables is reduced to the filter variables.

## Karl Bang Christensen: Item response theory models for measuring level and change in latent variables

Latent variables are encountered in many areas of research. Common examples include intelligence, attitudes, and level of depression. Latent variables are not directly observable, but can be measured indirectly through the responses to a number of questions (items). Item response theory (IRT) models are statistical models that can be used to measure these latent variables. Application of IRT models can serve two overlapping purposes: (i) evaluation of the measurement instrument and (ii) modeling of the structure of the latent variables. For example we may use an IRT model to evaluate if a depression scale measures depression in the correct way and with the required amount of precision, but the model can also be used to compare two patient groups. A review of models and methods is offered and implementation in standard software like SAS and R is discussed. Furthermore applications to longitudinal data are discussed.

## Wolfgang Runggaldier: Variance reduction by conditioning in the pricing problem where the underlying is a continuous-time finite state Markov process

(Authors: Juan Miguel Montes, Valentina Prezioso, Wolfgang Runggaldier)

We consider a financial market model with a generic underlying security (may also be an interest rate) that evolves as a continuous-time finite-state Markov process with a given transition intensity matrix. If the intensity matrix is time inhomogeneous, the underlying is multivariate or the claim is path dependent an explicitly computable pricing formula is difficult to obtain and so a Monte-Carlo type approach may become appropriate. A plain MC approach may however lead to biased results. We show that one can analytically compute the expression for the price conditionally on the knowledge of the number of state transitions of underlying. With respect to a plain MC this fact allows us to propose a variance reduction method that is obtained by conditioning on the number of state transitions and simulating only the latter. In this way one can also considerably reduce the bias in plain MC. Numerical results and comparisons are also provided.

## Norman Verhelst: Profile Analysis: A generalization of DIF analysis

To investigate if some groups of testees are (dis)advantaged by a certain category of items, sometimes Differential Item Functioning (DIF) analysis is applied to all items of this category. It was hypothesized, for example, that younger testees might be disadvantaged by items belonging to the occupational domain in the Michigan English Test (MET). DIF analysis, however, showed unclear and non convincing results. It will be argued that DIF analysis in this context is not a very good method: it lacks clarity in the interpretation and statistical power in the applications.

A new method, called profile analysis, will be presented. In this method the focus is not on single items but on categories of items. For each category the sequence of observed scores (the observed profile) from a testee are compared to their expected value under the measurement model used (the expected profile). The difference between observed and expected profile is called the deviation profile. These deviation profiles are aggregated in each of two or more groups, and give rise to some statistical tests which show if different groups react differently to categories of items.

Profile analysis turns out to be statistically powerful and very flexible in its use: it is not restricted to two categories and the number of groups to be compared is unlimited. It is also shown that DIF analysis is a special case of profile analysis.