Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing

Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing.Dependence structure, Extremal values, Copula modeling, Copula review

The analysis of competing risks data with a focus on estimation of cause-specific and subdistribution hazard ratios from a mixture model

Treatment efficacy in clinical trials is often assessed by time from treatment initiation to occurrence of a certain critical or beneficial event. In most cases the event of interest cannot be observed for all patients, as patients are only followed for a limited time or contact to patients is lost during their follow-up time. Therefore, certain methods were developed in the framework of the so called time-to-event or survival analysis, in order to obtain valid and consistent estimates in the presence of these "censored observations", using all available information. In classical event time analysis only one endpoint exists, as the death of a patient. As patients can die from different causes, in some clinical trials time to one out of two or more mutually exclusive types of event may be of interest. In many oncological studies, for example, time to cancer-specific death is considered as primary endpoint with deaths from other causes acting as so called competing risks. Different methods for data analysis in the competing risks framework were developed in recent years, which either focus on modelling the cause-specific or the subdistribution hazard rate or split the joint distribution of event times and event types into quantities, that can be estimated from observable data. In this work the analysis of event time data in the presence of competing risks is described, including the presentation and discussion of different regression approaches. A major topic of this work is the estimation of cause-specific and subdistribution hazard rates from a mixture model and a new approach using penalized B-splines (P-splines) for estimation of conditional hazard rates in a mixture model is proposed. In order to evaluate the behaviour of the new approach, a simulation study was conducted, using simulation techniques for competing risks data, which are described in detail in this work. The presented regression models were applied to data from a clinical cohort study investigating a risk stratification for cardiac mortality in patients, that survived a myocardial infarction. Finally, the use of the presented methods for event time analysis in the presence of competing risks and results obtained from the simulation study and the data analysis are discussed.Zur Beurteilung der Wirksamkeit von Behandlungen in klinischen Studien wird häufig die Zeit vom Beginn einer Behandlung bis zum Eintreten eines bestimmten kritischen oder erwünschten Ereignisses als Zielgröße verwendet. Da in vielen Fällen das entsprechende Ereignis nicht bei allen Patienten beobachtet werden kann, da z.B. Patienten nur für einen gewissen Zeitraum nachverfolgt werden können oder der Patientenkontakt in der Nachbeobachtungszeit abbricht, wurden im Rahmen der so genannten Ereigniszeit- bzw. Überlebenszeitanalyse Verfahren entwickelt, die bei Vorliegen dieser "zensierten Beobachtungen" konsistente Schätzer liefern und dabei die gesamte verfügbare Information verwenden. In der klassischen Ereigniszeitanalyse existiert nur ein möglicher Endpunkt, wie der Tod eines Patienten. Da Patienten jedoch an verschiedenen Ursachen versterben können, ist in manchen klinischen Studien die Zeit bis zu einem von zwei oder mehreren sich gegenseitig ausschließenden Ereignistypen von Interesse. So fungiert z.B. in vielen onkologischen Studien die Zeit bis zum tumor-bedingten Tod als primärer Endpunkt, wobei andere Todesursachen sogenannte konkurrierende Risiken ("Competing Risks") darstellen. In den letzten Jahren wurden mehrere Verfahren zur Datenanalyse bei Vorliegen konkurrierender Risiken entwickelt, bei denen entweder die ereignis-spezifische oder die Subdistribution-Hazardrate modelliert wird, oder bei denen die gemeinsame Verteilung von Ereigniszeiten und Ereignistypen als Produkt von Größen abgebildet wird, die aus den beobachtbaren Daten geschätzt werden können. In dieser Arbeit werden Methoden zur Analyse von Competing-Risks-Daten, einschließlich verschiedener Regressionsansätze, vorgestellt. Besonderes Augenmerk liegt auf der Schätzung der ereignis-spezifischen und Subdistribution-Hazardraten aus einem sogenannten Mixture Model. Diesbezüglich wird auch ein neuer Ansatz zur Schätzung der konditionalen Hazardraten in einem Mixture Model unter Verwendung penalisierter B-Spline-Funktionen (P-Splines) vorgestellt. Um die Eigenschaften des neuen Ansatzes zu untersuchen, wurde eine Simulationsstudie unter Einsatz verschiedener Simulationsstrategien für Competing-Risks-Daten, die in dieser Arbeit im Detail beschrieben werden, durchgeführt. Die Regressionsmodelle wurden auf Daten einer klinischen Kohortenstudie zur Evaluation einer Risikostratifizierung für Patienten, die einen Myokardinfarkt überlebt haben, angewandt. Abschließend werden die vorgestellten Methoden zur Analyse von Ereigniszeitdaten bei Vorliegen konkurrierender Risiken sowie die Ergebnisse der Simulationsstudie und der Datenanalyse diskutiert

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Principal component analysis and sparse polynomial chaos expansions for global sensitivity analysis and model calibration: application to urban drainage simulation

International audienceThis paper presents an efficient surrogate modeling strategy for the uncertainty quantification and Bayesian calibration of a hydrological model. In particular, a process-based dynamical urban drainage simulator that predicts the discharge from a catchment area during a precipitation event is considered. The goal is to perform a global sensitivity analysis and to identify the unknown model parameters as well as the measurement and prediction errors. These objectives can only be achieved by cheapening the incurred computational costs, that is, lowering the number of necessary model runs. With this in mind, a regularity-exploiting metamodeling technique is proposed that enables fast uncertainty quantification. Principal component analysis is used for output dimensionality reduction and sparse polynomial chaos expansions are used for the emulation of the reduced outputs. Sensitivity measures such as the Sobol indices are obtained directly from the expansion coefficients. Bayesian inference via Markov chain Monte Carlo posterior sampling is drastically accelerated

Methods for the Investigation of Spatial Clustering, With Epidemiological Applications

When analysing spatial data, it is often of interest to investigate whether or not the events under consideration show any tendency to form small aggregations, or clusters, that are unlikely to be the result of random variation. For example, the events might be the coordinates of the address at diagnosis of cases of a malignant disease, such as acute lymphoblastic leukaemia or non-Hodgkin's lymphoma. This thesis considers the usefulness of methods employing nonparametric kernel density estimation for the detection of clustering, as defined above, so that specific, and sometimes limiting, alternative hypotheses are not required, and the continuous spatial context of the problem is maintained. Two approaches, in particular, are considered; first, a generalisation of the Scan Statistic to two dimensions, with a correction for spatial heterogeneity under the null hypothesis, and secondly, a statistic measuring the squared difference between kernel estimates of the probability density functions of the principal events and a sample of controls. Chapter 1 establishes the background for this work, and identifies four different families of techniques that have been proposed, previously, for the study of clustering. Problems inherent in typical applications are discussed, and then used to motivate the approach taken subsequently. Chapter 2 describes the Scan Statistic for a one-dimensional problem, assuming that the distribution of events under the null hypothesis is uniform. A number of approximations to the statistic's distribution and methods of calculating critical values are compared, to enable significance testing to be carried out with minimum effort. A statistic based on the supremum of a kernel density estimate is also suggested, but an empirical study demonstrates that this has lower power than the Scan Statistic. Chapter 3 generalises the Scan Statistic to two dimensions and demonstrates empirically that existing bounds for the upper tail probability are not sufficiently sharp for significance testing purposes. As an aside, the chapter also describes a problem that can occur when a single pseudo-random number generator is used to produce parallel streams of uniform deviates. Chapter 4 investigates a method, suggested by Weinstock (1981), of correcting for a known, non-uniform null distribution when using the Scan Statistic in one dimension, and proposes that a kernel estimator replace the exact density, the estimate being calculated from a second set of (control) observations. The approach is generalised to two dimensions, and approximations are developed to simplify the computation required. However, simulation results indicate that the accuracy of these approximations is often poor, so an alternative implementation is suggested. For the case where two samples of observations are available, the events of interest and a group of control locations. Chapter 5 suggests the use of the integrated squared difference between the corresponding kernel density estimates as a measure of the departure of the events from null expectation. By exploiting its similarity to the integrated square error of a k.d.e., the statistic is shown to be asymptotically normal; the proof generalises a central limit theorem of Hall (1984) to the two-sample case. However, simulation results suggest that significance testing should use the bootstrap, since the exact distribution of the statistic appears to be noticeably skewed. A modified statistic, with the smoothing parameters of the two k.d.e.'s constrained to be equal and non-random, is also discussed, and shown, both asymptotically and empirically, to have greater power than the original. In Chapter 6, the two techniques are applied to the geographical distribution of cases of laryngeal cancer in South Lancashire for the period 1974 to 1983. The results are similar, for the most part, to a previous analysis of the data, described by Diggle (1990) and Diggle et al (1990). The differences in the two analyses appear to be attributable to the bias or variability of the k.d.e.'s required to calculate the integrated squared difference statistic, and the inaccuracy of the approximations used by the corrected Scan Statistic. Chapter 7 summarises the results obtained in the preceding sections, and considers the implications for further research of the observations made in Chapter 6 regarding the weaknesses of the two statistics. It also suggests extensions to the basic methodology presented here that would increase the range of problems to which the two methods could be applied

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science