Modelling one-dimensional data can be performed by different wellknown ways. Modelling two-dimensional data is a more open question. There is no unique way to describe dependency of two dimensional data. In this thesis dependency is modelled by copulas. Insurance data from two different regions (Göinge and Kronoberg) in Southern Sweden is investigated. It is found that a suitable model is that marginal data are Normal Inverse Gaussian distributed and copula is a better dependence measure than the usual linear correlation together with Gaussian marginals.

Measuring, modeling, and managing risk has always been an important task for researchers. Many of the traditional assumptions relied on in risk research, such as the assumption of normality and single period optimization, have proven too restrictive and alternative methods have been developed. The objective of this dissertation is to explore and apply these tools to analyze geographical diversification. The first step to analyze geographical diversification is to understand how different climate and spatial variables impact yields. Yield dependencies for wheat, cotton, and sorghum are estimated using linear correlation and copulas. The copulas provide an alternative to linear correlation. The results of the different dependency estimations indicate that there is a significant difference between the results. The next step is to analyze geographical diversification in a portfolio setting. Traditional portfolio optimization has assumed that risk and dependence are symmetric. Using a single period model, an asymmetric risk measure, conditional value at risk, and asymmetric dependence measure, copulas, are implemented into the portfolio optimization model. The efficient frontiers under both symmetric and asymmetric assumptions show that ignoring the asymmetric nature of the data could lead to optimal portfolio allocations that could underestimate the actual risk exposure. The implication of these results provides researchers with more motivation to move beyond the standard assumptions of linear correlation and normality. Building on the single period problem, a multi-period portfolio model is formulated using discrete stochastic programming. One key in formulating a discrete stochastic program is the representation of uncertainty. Scenario generation is a method to obtain a discrete set of outcomes for the random variables. A moment matching routine is developed to capture the first four moments of the variables and the multivariate relationship is modeled using copulas. The results show that the moment matching routine closely captures the higher moments of the data. The results also indicate that there are possible gains from geographical diversification. Wealth levels increased for all three regions when production is diversified over the different regions. The optimal land allocation was dependent upon the base acreage assumption

This dissertation contains three essays. They are related to the exponential series estimation of copulas and the application of parametric copulas in financial econometrics. Chapter II proposes a multivariate exponential series estimator (ESE) to estimate copula density nonparametrically. The ESE attains the optimal rate of convergence for nonparametric density. More importantly, it overcomes the boundary bias of copula estimation. Extensive Monte Carlo studies show the proposed estimator outperforms kernel and log-spline estimators in copula estimation. Discussion is provided regarding application of the ESE copula to Asian stock returns during the Asian financial crisis. The ESE copula complements the existing nonparametric copula studies by providing an alternative dedicated to the tail dependence measure. Chapter III proposes a likelihood ratio statistic using a nonparametric exponential series approach. The order of the series is selected by Bayesian Information Criterion (BIC). I propose three further modifications on my test statistic: 1) instead of putting equal weight on the individual term of the exponential series, I consider geometric and exponential BIC average weights; 2) rather than using a nested sequence, I consider all subsets to select the optimal terms in the exponential series; 3) I estimate the likelihood ratio statistic using the likelihood cross-validation. The extensive Monte Carlo simulations show that the proposed tests enjoy good finite sample performances compared to the traditional methods such as the Anderson-Darling test. In addition, this data-driven method improves upon Neyman’s score test. I conclude that the exponential series likelihood ratio test can complement the Neyman’s score test. Chapter IV models and forecasts S&P500 index returns using the Copula-VAR approach. I compare the forecast performance of the Copula-VAR model with a classical VAR model and a univariate time series model. I use this approach to forecast S&P500 index returns. I apply a modified Diebold-Mariano test to test the equality of mean squared forecast errors and utilize a forecast encompassing test to evaluate forecasts. The findings suggest that allowing a more flexible specification in the error terms using copula tends improve the forecast accuracy. I also demonstrate combined forecasts improved forecasts accuracy over individual models.

Οι συζεύξεις εκφράζουν στην περίπτωση των διδιάστατων κατανομών τη συναρτησιακή σχέση της αθροιστικής συνάρτησης κατανομής μιας διδιάστατης κατανομής με τις αθροιστικές συναρτήσεις κατανομής των μονοδιάστατων περιθώριων κατανομών, όπου οι τελευταίες μας είναι πάντοτε γγωστές. Στην εργασία παρουσιάζονται κάποιες οικογένειες διδιάστατων κατανομών σε συνδυασμό με τα μέτρα συσχέτισης τους και ολοκληρώνεται με μια εφαρμογή. / -

Copulas allow a joint probability distribution to be decomposed such that the marginals inform us about how the data were generated, separately from the copula which fully captures the dependency structure between the variables. This is particularly useful when working with random variables which are both non-normal and possibly non-linearly correlated. However, when in addition, the dependence between these variables change in accordance with some underlying covariate, the model becomes significantly more complex. This research proposes using a Gaussian process conditional copula for this dependence modelling, focusing on time as the underlying covariate. Utilising a Bayesian non-parametric framework allows the simplifying assumptions often applied in conditional dependency computation to be relaxed, giving rise to a more flexible model. The importance of improving the accuracy of dependency modelling in applications such as finance, econometrics, insurance and meteorology is self-evident, considering the potential risks involved in erroneous estimation and prediction results. Including the underlying (conditional) variable reduces the chances of spurious dependence modelling. For our application, we include a textbook example on a simulated dataset, an analysis of the modelling performance of the different methods on four currency pairs from foreign exchange time series and lastly we investigate using copulas as a way to quantify the coupling efficiency between the solar wind and magnetosphere for the three known phases of geomagnetic storms. We find that the Student’s t Gaussian process conditional copula outperforms static copulas in terms of log-likelihood, and performs particularly well in capturing lower tail dependence. It further gives additional information about the temporal movement of the coupling between the two main variables, and shows potential for more accurate data imputation. / Mini Dissertation (MSc (Advanced Data Analytics))--University of Pretoria, 2020. / CSIR DSI-Interbursary Support Programme, UP Postgraduate Masters Coursework Bursary / Statistics / MSc (Advanced Data Analytics) / Unrestricted

Background: Copula methods have been proposed as a way of modeling dependence between random variables because it lies in the flexibility of the assumption on marginals. As previous authors stated, "A copula is a function which joins or “couples” a multivariate distribution function to its one-dimensional marginal distribution functions. Given that cost and effectiveness are often related to each other and therefore they show statistical dependence, the use of copulas to handle uncertainty caused by sampling variation could be potentially useful when cost-effectiveness analyses (CEA) are performed using patient-level data. The objective of this study was to empirically compare various copula distributions with two traditional methods, namely, the bootstrapping approach and the Bayesian approach assuming that incremental cost and LYs gained are bivariate normally distributed. Methods: The patient-level data from a previously published observational study were analyzed using four copula distributions: independent, Farlie-Gumbel-Morgenstern (FGM), Frank and Clayton copulas. Using the results from the traditional methods previously published, models were compared in terms of incremental cost, incremental life years (LYs) gained and the cost-effectiveness acceptability curves (CEACs) based on the net monetary benefit (NMB). Results: Using the traditional methods provided similar results. The most pronounced impact was the improvement in precision given that the confidence intervals were so much narrower for the copulas methods in comparison to the traditional methods. Consequently, the probability of being optimal derived from the Frank and Clayton copulas were close to 1.0 at a willingness to pay (𝜆) of CA$20,000. By contrast, the traditional methods were optimal for a 𝜆 of $100,000 CAD. Conclusions: The results of this study demonstreate the potential impact and importance of copulas in patient-level cost-effectiveness analysis. This approach could be particularly important in those situations where the data suggests some kind of dependence and some restrictions on the marginals, as observed in our case study. / Thesis / Master of Science (MSc)

Šio darbo tema yra jungčių (angl. copulas) panaudojimas ryšiams tarp daugiamačių atsitiktinių dydžių modeliuoti. Jungtis yra funkcija, kuri sujungia kelių atsitiktinių dydžių marginalinius skirstinius į bendrą daugiamatę funkciją. Jungties sąvoka pirmą kartą statistikoje įvesta 1959 m. Šiame darbe aprašomos pagrindinės jungčių savybės, keletas jungčių šeimų, išskiriant atskirą šeimą - Archimedo jungtis, taip pat priklausomumo matai tarp atsitiktinių dydžių. Vėliau tinkamos jungties pritaikymo turimam duomenų rinkiniui procedūra iliustruojama nagrinėjant transporto priemonių valdytojų civilinės atsakomybės privalomojo draudimo žalų ir išlaidų žaloms administruoti duomenis. / In this Master work the concept of copulas as a tool for modeling relationships among multivariate outcomes is introduced. A copula is a function that links univariate margins to their multivariate distribution. Copulas were introduced in 1959. The literature on the statistical properties and application of copulas has been developing rapidly in recent years. In this Master work basic properties of copulas are described, then several families of copulas and relationships to measures of dependences. Later procedure for selecting the parametric family of Archimedean copulas is illustrated by using Lithuanian Motor Third Party Liability insurance data losses and expenses. For these data it is shown how to fit copulas according to nonparametric procedure which was proposed by Genest and Rivest.

In this thesis, we investigate the advantages of using high-dimensional copula modeling to understand the riskiness of portfolio investments and to more realistically estimate future portfolio values. Our approach involves benchmarking some pre-determined fitted copulas to the 0.05-quantile, the Tail Conditional Expectation, and the probability of negative returns for each portfolio. We find that the two R-Vine copula models used in this study provide good estimations of the distribution of portfolio values for the 1-month time frame, the shortest we consider in this thesis, most probably due to their flexibility and ability to represet a diverse array of dependence structures. However, for longer time frames (1 year or more), the Clayton copula appears to be a more suitable model. It aligns more closely with market behaviour due to its capacity of capturing lower tail dependence. In conclusion, we argue that by employing the right copula model, in our case the Clayton copula, we obtain a more realistic view on the distribution of the future portfolio values. / I denna uppsats undersöker vi fördelarna med att använda högdimensionell copula-modellering för att förstå risken med portföljinvesteringar och för att på ett mer realistiskt sätt uppskatta framtida portföljvärden. Vårt tillvägagångssätt involverar benchmarking av de olika anpassade copulae till 0,05-kvantilen, den villkorliga svansförväntningen och sannolikheten för negativ avkastning för varje portfölj. Vi finner att de två R-Vine copula-modellerna som används i denna studie ger goda uppskattningar av fördelningen av portföljvärden för 1-månaders tidsram, den kortaste vi studerar, troligen på grund av deras flexibilitet och förmåga att representera en mångfald av beroendestrukturer. Men f ̈or l ̈angre tidsramar (1 år eller mer) verkar Clayton copula vara en mer lämplig modell. Det överensstämmer mer med marknadens beteende på grund av dess förmåga att fånga lägre svansberoende. Sammanfattningsvis hävdar vi att genom att använda rätt copula-modell, i vårt fall Clayton copula, får vi en mer realistisk fördelning av dynamiken i framtida portföljvärden.

Assuming a set of variates are independent and normally distributed is commonplace in statistics. In this thesis, we consider the consequences of these assumptions as they pertain to global sensitivity analysis. We begin by illustrating how the notion of sensitivity becomes distorted in the presence of codependent model inputs. This observation motivates us to develop a new methodology which accommodates for input codependencies. Our methodology can be summarized through three points: First, a new form of sensitivity is presented which performs as well as the classical form but can be obtained at a fraction of the computational cost. Second, we define a measure which quantifies the extent of distortion caused by codependent inputs. The third point is regarding the modelling of said codependencies. The multivariate normal distribution is a natural choice for modelling codependent inputs; however, our methodology uses a copula-based approach instead. Copulas are a contemporary strategy for constructing multivariate distributions whereby the marginal and joint behaviours are treated separately. As a result, a practitioner has more flexibility when modelling inputs. / Thesis (Master, Chemical Engineering) -- Queen's University, 2013-12-05 10:16:26.81

Credit risk models widely used in the financial market nowadays assume that losses are normally distributed and have linear dependence. Nevertheless it is well known that asset returns (loans included) are not normally distributed and present tail dependence. Therefore the traditional approaches are not able to capture possible stronger association among higher losses and tend to underestimate the probability of joint extreme losses. Copula functions are an alternative to overcome this drawback since they yield accurate dependence measures regardless of the distribution of the variables analysed. This technique was first applied to credit risk in 2000 but the studies in this field have been concentrated on corporate debt and derivatives. We filled this gap in the literature by employing copulas to estimate the dependence among consumer loans. In an empirical study based on a credit card portfolio of a large UK bank, we found evidence that standard models are misspecified as the dependence across default rates in the dataset is seldom expressed by the (Gaussian) copula implicit in those models. The comparison between estimations of joint high default rates from the conventional approach and from the best-fit copulas confirmed the superiority of the latter method. The initial investigation concerning pairs of credit segments was extended to groups of three segments with the purpose of accounting for potential heterogeneous dependence within the portfolio. To do so, we introduced vine copulas (combinations of bivariate copulas to form high-dimension copulas) to credit risk and the empirical estimations of simultaneous excessive defaults based on this technique were better than both the estimations from the pairwise copulas and from the conventional models. Another contribution of this work concerns the application of copulas to a method derived from the limited credit models: the calculation of the capital required to cover unexpected losses in financial institutions. Two models were proposed and, according to simulations, outperformed the current method (Basel) in most of the scenarios considered.