Emnet har til hensikt å dels presentere bruk av risikoanalyse knyttet opp mot konkrete prosjekter først og fremst fra petroleumsvirksomhet. Her vil en blant annet ta opp: Hendelses- og beslutningstrær, bruk av Monte Carlo simulering, integrert usikkerhetsanalyse. Dels vil en presentere noe av det teoretiske grunnlag for å utføre pålitelighetsanalyser av kompliserte systemer. Det vil lede frem til forskningsfronten i utvalgte delområder: Mål for den pålitelighetsmessige betydning av en komponent, multinær pålitelighetsteori der komponenter/systemer beskrives mer nyansert enn som funksjonerende eller ei, optimale vedlikeholds-, inspeksjons- og utskiftnings-strategier, teori i grenselandet pålitelighetsteori/grafteori.
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Emnet vil gi en innføring i finans og forsikring.
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Emnet gir en introduksjon til teorien omkring prising av opsjoner. Emnet bygger på grunnleggende sannsynlighetsregning og analyse, og egne seg for studenter som tar sikte på, eller arbeider med, en masteroppgave innen forsikring eller matematisk finans. Emnet er obligatorisk for aktuarer.
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En rekke praktiske eksempler ("cases") fra forsikring og finans vil bli diskutert. Det overordnete innhold er matematisk modellering som prosess, herunder det å sette opp en modell, benytte seg av erfaringsdata, benytte datamaskinen effektivt og til slutt presentere resultatene.
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I første del av kurset skal vi bli kjent med de grunnleggende prinsippene i moderne renteportføljeteori. Vi skal diskutere populære rentemodeller med spesiell vekt på statistisk dataanalyse og kalibrering. I andre del av kurset skal vi fokusere på generaliserte rentemodeller som er beskrevet av stokastiske partielle differentiallikninger eller evolusjonslikninger. Emnet egner seg for master- og bachelorstudenter som tar finans- eller forsikringsmatematikk.
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Emnet skal gi innsikt i statistisk problemformulering, modellbygging, analyse samt resultatvalidering. Sentralt er regresjon (dvs. empirisk bestemmelse av relasjoner mellom variable) og andre flervariable statistiske teknikker samt behandling av tidsavhengige data. Spesielt behandles lineær regresjon, variansanalyse, forsøksplanlegging, logistisk regresjon, levetidsanalyse og enkel tidsrekkeanalyse.
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The research topic can be from all areas of statistics.
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Important parts of statistical theory, along with many applications thereof, build on approximations to the most relevant distributions, and these approximations are valid when the sample size is large, e.g. compared to the dimension of underlying parameter vectors. The machinery of such larg-sample approximations is dealt with in this course, including the basics of probability theory; convergence in probability and in distribution; characteristic functions; central limit theorems; and the laws of large numbers. The theory is being illustrated by applications in various situations.
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The course provides a basis for general Bayesian theory and applications, involving the use of relevant methods for the formal combination of relevant prior knowledge (including expert opinions) and observed data. Bayes' formula, in various guises, lead to the appropriate posterior distributions for parameters of interest. The course will also go into empirical Bayes methodology, principles for decision taking, a comparison between non-Bayesian and Bayesian methods, applications in certain areas, and the use of simulation techniques.
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Modern data analysis refer to methods where fewer assumptions (such as a linear relation between response and explanatory variables) are made and where instead data determine the relation. Some keywords are nearest neighbor methods, kernel smoothing and generalized additive models. Statistical classification is problems where the response variable is a categorical variable("classes"). The course will present classical classification methods as well as more advanced methods based on modern regression methods. A central problem in the course is searching for structures in data, often referred to as "data mining" or "learning from data".
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Modern data analysis emphasizes multivariable techniques. In this course such techniques are introduced by first studying the multivariate normal distribution. Then Hotelling T2, MANOVA and multivariable regression is presented. Futhermore the course covers principal component analysis, factor analysis, canonical correlation analysis, discriminant analysis and cluster analysis.
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Statistical analysis of real world systems and models will typically require computer intensive methods. The course starts with a study of modern Monte Carlo methods, including Markov chain Monte Carlo, and variance reduction methods. Such methods are useful within Bayesian analysis and simulation based inference like bootstrapping and Monte Carlo tests. Maximization of likelihoods is another important numerical problem. The course covers several statistical optimization methods (such as Fisher-scoring and EM-algorithm) as well as general optimization methods.
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Estimation and testing of hypothesis with autoregressive processes and moving averages (i.e. ARMA-processes) and with stationary processes. Correlogram, periodogram, spectrum. State-space models (Kalman filter). Illustration on real data.
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Linear models include both regression and analysis of variance which are some of the most commonly applied statistical methods. Simple ANOVA uses fixed effects. A variation is random effects models where variance components are estimated. General linear models with fixed and random components. Illustrations and excercises with real data are integrated into the course.
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The course gives an introduction to the most important concepts and methods in survival and event history analysis. These methods have applications for instance in insurance, medicine and reliability. Topics that are presented are: censoring and truncation, non-parametric estimation of hazard and survival functions (Nelson-Aalen and Kaplan-Meier estimators), log-rank test and other non-parametric tests, parametric survival models, Cox` proportional hazards model, nested case-control and case-cohort studies, frailty models and marginal models.
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The course presents Bayesian time series analysis and dynamic linear models. Much of the content is technically common to Kalman filtering in cybernetics, but the advantage with this presentation is that all results can be argued for and interpreted probabilisticly. Special topics covered are: The generel dynamic linear model, discount factors to help assessing the signal variance, estimation of the observation variance, retrospective filtering, forecasts, model monitoring, intervention into the time series model due to new information, comparison with classical time series models.
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The course will give an overview of a number of different topics in modern extreme value theory including: Univariate extreme value theory, threshold models, point process characterization of extremes, maximum-likelihood estimation, peaks over thresholds, Hill-type estimation, multivariate extremes and extremes of processes, analysing heavy-tailed data in insurance and finance, Risk Management (enhancing Value-at-Risk). Software like Splus and R will be used.
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The course provides a general introduction to the problem area of constructing estimators for parameters in different classes of situations and models. Elements of general decision theory are also included, with loss, award, and risk functions. The topics discussed include optimal estimation for unbiased methods; equivariant estimators; minimaxity; admissibility; Bayes estimators; asymptotic optimality for likelihood based methods; and sequential estimation. The material is illustrated in various applications, e.g. to the class of exponential models and to transformation models.
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Principles of experimental design. Autual problems from agriculture and industry. 2n and 3n factorial experiments. Blocking and confounding. Incomplete blocks. Partial repetitions. Variance and bias with different choices of experimental design. Response surface methods.
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Processes in space and time. Statistical models and methods for spartially varying phenomena, that is, where the geographical position of measurements matters. Homogeneous/non homogeneous processes. Geostatistical interpolation methods; semi variogram; kriging (spatial prediction); estimation problems. Spatial point processes; simulation; spatial sampling strategies. Applications to ozone, climate data etc. Spatial models for grid data and inference in these (MRFs, pseudo MLE, Monte Carlo MLE, Bayesian methods). Extreme value methods- crossing limits for air pollution, changes in climate extremes. time series with long range dependency - trends in climate data.
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Traditional statistical analysis focuses on the learning of good techniques for analysing data from a given model. Such methods also typically assume that the model is correct. This course considers methods for selecting the most suitable model, from a list of candidate models, for given data. Model selection criteria, such as AIC, BIC and FIC, are developed and applied. The course also considers the statistical behaviour of these methods, as well as the consequences of applying traditional fixed-model techniques, when it is not given in advance which model is to be used.
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A somewhat ambitious statistical aim is to replace formulae with computer power. If one needs to estimate the standard deviation for the average statistic Xn, then one may of course use the explicit formula s/sqrt(n). An alternative is to simulate 1000 pseudo-realisations of Xn from pseudo-datasets with properties resembling the original dataset, e.g. via resampling, and then compute the empirical standard deviation for these1000 pseudo-averages. This turns out to be a fruitful idea with far-reaching consequences. Methods similar to the one sketched above are called bootstrapping, and generally involve simulations of pseudo-datasets from an estimated model. The power of the methods is that they may be applied with more or less the same ease in rather more complicated models (than the simple nonparametric one above), with arbitrary statistics (not only the Xn above), and with general and potentially complicated measures of spread (instead of merely sd{Xn}). They may in particular be used in situations where explicit formulae cannot be derived. The course focuses on the general theory for bootstrapping and jackknifing, for estimation and for construction of simulation based confidence intervals. Students need to apply both halves of their brains, as the theory is being used also with practical exercises and computers. The course aims at being practically useful.
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The course is built up by a sequence of short courses(1-3 days) that will be given by invited international researchers. The content of the short courses will depend on the current programme for such. A full course must contain at least eight teaching days. Each student can have different selections of short courses, but the selection needs to be approved by the teaching coordinator.
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The course will provide an introduction to finance, insurance and their practical interplay.
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Taking the course aims at training the student to carry out practical analysis of actuarial and financial risk, including the communicative aspect of clarifying assumptions and presenting conclusions in a clear fashion.
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