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Ratingsysteme sind ein zentraler Bestandteil der Kreditrisikomodellierung. Neben der Bonitätsbeurteilung auf der Ebene der Kreditnehmer und der Risikoquantifizierung auf der Ebene der Ratingklassen spielt dabei die Bildung der Ratingklassen eine wesentliche Rolle. Die Literatur zur Ratingklassenbildung setzt auf modellfreie, in gewisser Weise willkürliche Optimierungsverfahren. Ein Ziel der vorliegenden Arbeit ist es, stattdessen ein parametrisches statistisches Modell zur Bildung der Ratingklassen einzuführen. Ein geeignetes Modell ist im Bereich der Bruchpunktschätzung zu finden. Dieses Modell und die in der mathematischen Literatur vorgeschlagenen Parameter- und Intervallschätzer werden in der vorliegenden Arbeit dargestellt und gründlich diskutiert. Dabei wird Wert auf eine anwendungsnahe und anschauliche Formulierung der mathematisch-statistischen Sachverhalte gelegt. Anschließend wird die Methodik der Bruchpunktschätzung auf einen konkreten Datensatz angewendet und mit verschiedenen anderen Kriterien zur Ratingklassenbildung verglichen. Hier erweist sich die Bruchpunktschätzung als vorteilhaft. Aufbauend auf der empirischen Untersuchung wird abschließend weiterer Forschungsbedarf abgeleitet. Dazu werden insbesondere Konzepte für den Mehrklassenfall und für abhängige Daten entworfen.:1. Einleitung 2. Ratingsystem 3. Bruchpunktschätzung 4. Anwendung 5. Zusammenfassung und Ausblick / Rating systems are a key component of credit risk modeling. In addition to scoring at borrowers’ level and risk quantification at the level of rating classes, the formation of the rating classes plays a fundamental role. The literature on rating classification uses in a way arbitrary optimization methods. Therefore, one aim of this contribution is to introduce a parametric statistical model to form the rating classes. A suitable model can be found in the area of split-point estimation. This model and the proposed parameter and interval estimators are presented and thoroughly discussed. Here, emphasis is placed on an application-oriented and intuitive formulation of the mathematical and statistical issues. Subsequently, the methodology of split-point estimation is applied to a specific data set and compared with several other criteria for rating classification. Here, split-point estimation proves to be advantageous. Finally, further research questions are derived on the basis of the empirical study. In particular, concepts for the case of more than two classes and for dependent data are sketched.:1. Einleitung 2. Ratingsystem 3. Bruchpunktschätzung 4. Anwendung 5. Zusammenfassung und Ausblick

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When monitoring a binary process proportion p, it is usually assumed that the binary observations are independent. However, it is very common that the observations are correlated with p being the correlation between two successive observations. The first part of this research investigates the problem of monitoring p when the binary observations follow a first-order two-state Markov chain model with p remaining unchanged. A Markov Binary GLR (MBGLR) chart with an upper bound on the estimate of p is proposed to monitor a continuous stream of autocorrelated binary observations treating each observation as a sample of size n=1. The MBGLR chart with a large upper bound has good overall performance over a wide range of shifts. The MBGLR chart is optimized using the extra number of defectives (END) over a range of upper bounds for the MLE of p. The numerical results show that the optimized MBGLR chart has a smaller END than the optimized Markov binary CUSUM. The second part of this research develops a CUSUM-pp chart and a GLR-pp chart to monitor p and p simultaneously. The CUSUM-pp with two tuning parameters is designed to detect shifts in p and p when the shifted values are known. We apply two CUSUM-pp charts as a chart combination to detect increases in p and increases or decreases in p. The GLR-pp chart with an upper bound on the estimate of p, and an upper bound and a lower bound on the estimate of p works well when the shifts are unknown. We find that the GLR-pp chart has better overall performance. The last part of this research investigates the problem of monitoring p with p remains at the target value when the correlated binary observations are aggregated into samples with n>1. We assume that samples are independent and there is correlation between the observations in a sample. We proposed some GLR and CUSUM charts to monitor p and the performance of the charts are compared. The simulation results show MBNGLR has overall better performance than the other charts. / Ph. D.

When a system is monitored over time, changes can be discovered in the time series of monitored variables. Change Point Detection (CPD) aims at finding the time point where a change occurs in the monitored system. While CPD methods date back to the 1950’s with applications in quality control, few studies have been conducted on the impact of data dimensionality on CPD algorithms. This thesis intends to address this gap by examining five different algorithms using synthetic data that incorporates changes in mean, covariance, and frequency across dimensionalities up to 100. Additionally, the algorithms are evaluated on a collection of data sets originating from various domains. The studied methods are then assessed and ranked based on their performance on both synthetic and real data sets, to aid future users in selecting an appropriate CPD method. Finally, stock data from the 30 most traded companies on the Swedish stock market are collected to create a new CPD data set to which the CPD algorithms are applied. The changes of the monitored system that the CPD algorithms aim to detect are the changes in policy rate set by the Swedish central bank, Riksbank. The results of the thesis show that the dimensionality impacts the accuracy of the methods when noise is present and when the degree of mean or covariance change is small. Additionally, the application of the algorithms on real world data sets reveals large differences in performance between the studied methods, underlining the importance of comparison studies. Ultimately, the kernel based CPD method performed the best across the real world data set employed in the thesis. / När system övervakas över tid kan förändringar upptäckas i de uppmätade variablers tidsseriedata. Change Point Detection (CPD) syftar till att hitta tidpunkten då en förändring inträffar i det övervakade systemet’s tidseriedata. Medan CPD-metoder har sitt urspring i kvalitetskontroll under 1950-talet, har få studier undersökt datans dimensionalitets påverkan på CPD-algoritmer’s förmåga. Denna avhandling avser att fylla denna kunskapslucka genom att undersöka fem olika algoritmer med hjälp av syntetiska data som inkorporerar förändringar i medelvärde, kovarians och frekvens över dimensioner upp till 100. Dessutom jämförs algoritmerna med hjälp av en samling av data från olika domäner. De studerade metoderna bedöms och rangordnas sedan baserat på deras prestanda på både syntetiska och verkliga datauppsättningar för att hjälpa framtida användare att välja en lämplig CPD algoritm. Slutligen har aktiedata samlats från de 30 mest handlade företagen på den svenska aktiemarknaden för att skapa ett nytt data set. De förändringar i det övervakade systemet som CPD-algoritmerna syftar till att upptäcka är förändringarna i styrräntan som fastställs av Riksbanken. Resultaten av studien tyder på att dimensionaliteten påverkar förmågan hos algoritmerna att upptäcka förändringspunkterna när brus förekommer i datan och när graden av förändringen är liten. Dessutom avslöjar tillämpningen av algoritmerna på den verkliga datan stora skillnader i prestanda mellan de studerade metoderna, vilket understryker vikten av jämförelsestudier för att avslöja dessa skillnader. Slutligen presterade den kernel baserade CPD metoden bäst.

Semiparametric modeling is a hybrid of the parametric and nonparametric modelings where some function forms are known and others are unknown. In this dissertation, we have made several contributions to semiparametric modeling based on the single index model related to the following three topics: the first is to propose a model for detecting change points simultaneously with estimating the unknown function; the second is to develop two models for spatially correlated data; and the third is to further develop two models for spatially-temporally correlated data. To address the first topic, we propose a unified approach in its ability to simultaneously estimate the nonlinear relationship and change points. We propose a single index change point model as our unified approach by adjusting for several other covariates. We nonparametrically estimate the unknown function using kernel smoothing and also provide a permutation based testing procedure to detect multiple change points. We show the asymptotic properties of the permutation testing based procedure. The advantage of our approach is demonstrated using the mortality data of Seoul, Korea from January, 2000 to December, 2007. On the second topic, we propose two semiparametric single index models for spatially correlated data. One additively separates the nonparametric function and spatially correlated random effects, while the other does not separate the nonparametric function and spatially correlated random effects. We estimate these two models using two algorithms based on Markov Chain Expectation Maximization algorithm. Our approaches are compared using simulations, suggesting that the semiparametric single index nonadditive model provides more accurate estimates of spatial correlation. The advantage of our approach is demonstrated using the mortality data of six cities, Korea from January, 2000 to December, 2007. The third topic involves proposing two semiparametric single index models for spatially and temporally correlated data. Our first model has the nonparametric function which can separate from spatially and temporally correlated random effects. We refer it to "semiparametric spatio-temporal separable single index model (SSTS-SIM)", while the second model does not separate the nonparametric function from spatially correlated random effects but separates the time random effects. We refer our second model to "semiparametric nonseparable single index model (SSTN-SIM)". Two algorithms based on Markov Chain Expectation Maximization algorithm are introduced to simultaneously estimate parameters, spatial effects, and times effects. The proposed models are then applied to the mortality data of six major cities in Korea. Our results suggest that SSTN-SIM is more flexible than SSTS-SIM because it can estimate various nonparametric functions while SSTS-SIM enforces the similar nonparametric curves. SSTN-SIM also provides better estimation and prediction. / Ph. D.

Utförda hacker-attacker på moderna fordon belyser ett behov av snabb detektering av hot inom denna miljö, särskilt när det förekommer en trend inom denna industri där moderna fordon idag kan klassas som IoT-enheter. Det förekommer kända fall av attacker där en angripare förmår stoppa fordon i drift, eller ta bromsar ur funktion, och detta har påvisats ske fjärrstyrt. Denna studie undersöker detektion av utförda attacker, på en riktig bil, genom studie av CAN bus meddelanden. De två modellerna CUSUM, från området Change Point Detection, och Random Forests, från området maskininlärning, tillämpas på riktig datamängd, för att sedan jämföras på simulerad data sinsemellan. En ny hypotesdefinition introduceras vilket möjliggör att evalueringsmetoden Conditional expected delay kan nyttjas för fallet Random Forests, där resultat förmås jämföras med evalueringsresultat från CUSUM. Conditional expected delay har inte tidigare studerats för metod av maskininlärning. De båda metoderna evalueras också genom ROC-kurva. Sammantaget förmås de båda metoderna jämföras sinsemellan, med varandras etablerade evalueringsmetoder. Denna studie påvisar metod och hypotes för att brygga de två områdena change point detection och maskininlärning, för att evaluera de två enligt gemensamt motiverade parametervärden. / There are known hacker attacks which have been conducted on modern vehicles. These attacks illustrates a need for early threat detection in this environment. Development of security systems in this environment is of special interest due to the increasing interconnection of vehicles and their newfound classification as IoT devices. Known attacks, that have even been carried out remotely on modern vehicles, include attacks which allow a perpetrator to stop vehicles, or to disable brake mechanisms. This study examines the detection of attacks carried out on a real vehicle, by studying CAN bus messages. The two methods CUSUM, from the field of Change Point Detection, and Random Forests, from the field of Machine Learning, are both applied to real data, and then later comparably evaluated on simulated data. A new hypothesis defintion is introduced which allows for the evaluation method Conditional expected delay to be used in the case of Random Forests, where results may be compared to evaluation results from CUSUM. Conditional expected delay has not been studied in the machinelarning case before. Both methods are also evaluated by method of ROC curve. The combined hypothesis definition for the two separate fields, allow for a comparison between the two models, in regard to each other's established evaluation methods. This study present a method and hypothesis to bridge the two separate fields of study, change point detection, and machinelearning, to achieve a comparable evaluation between the two.

人口流動具有各種型態。其中包含了遷移、移動、以及通勤人口。在宏觀模型框架下，空間互動模型(簡稱SIM)對於測量人口流動扮演了重要的角色。距離遞減效應為空間互動模型中重要的因子。該效應描述了人口流動的頻率會隨著移動距離而逐漸下降。然而，從實證上，本研究發現人口流動與移動距離的函數，並非在距離上保有恆定的關係。 在本文中，我們提出了對此非恆定的距離遞減效應之修正方法。本修正法運用了轉折點模型的特點，套入了空間互動模型的距離函數上。本文首先運用了電腦模擬驗證了此方法的穩定性與有效性。接下來，研究將此方法應用在兩個人口流動資料。第一個是從台灣健保資料庫觀察出的民眾就醫地變化。健保資料庫包含了總人口的5%抽樣資料。由於在抽樣上瑕疵不大，因此健保抽樣資料具有了一定的代表性。第二個資料則是英國統計局所提供的人口遷移普查資料。在這兩個資料上，我們發現本研究所提修正法，相較於傳統的空間互動模型具有更好的模型配適表現。此改善程度在非都市地區尤其更為明顯。 / Population movement encompasses various forms, such as migration, mobility, and commuting. Spatial Interaction Model (SIM) serves as an important tool to calibrate these movements in the sense of macro modelling. One of the important features of this model is that the number of migrants often decays with the distance. However, we found that this is not always the case in practice and the decay pattern may change with distance. In this study, we propose a distanced-based modification to the SIM, via applying the techniques of change-point problem to construct distance functional form. Computer simulation is illustrated to validate the method and the empirical analysis of flow data from Taiwan’s National Health Insurance Research Database (NHIRD), and also England & Wales internal migration data also provides sound evidences to support the proposed approach. Note that the flow data from the NHIRD consists of a sample of about one million people and can be treated as a fine sample representative of Taiwan’s whole population (about 23 million people). Our results show that the modified approach is more adequate than the traditional SIM, especially for describing the movements of suburban areas in Taiwan.

Objective: We consider the following problem from credit risk modeling: Our sample (Xi; Yi), 1 < i < n, consists of pairs of variables. The first variable Xi measures the creditworthiness of individual i. The second variable Yi is the default indicator of individual i. It has two states: Yi = 1 indicates a default, Yi = 0 a non-default. A default occurs, if individual i cannot meet its contractual credit obligations, i. e. it cannot pay back its outstandings regularly. In afirst step, our objective is to estimate the threshold between good and bad creditworthiness in the sense of dividing the range of Xi into two rating classes: One class with good creditworthiness and a low probability of default and another class with bad creditworthiness and a high probability of default. Methods: Given observations of individual creditworthiness Xi and defaults Yi, the field of change point analysis provides a natural way to estimate the breakpoint between the rating classes. In order to account for dependency between the observations, the literature proposes a combination of three model classes: These are a breakpoint model, a linear one-factor model for the creditworthiness Xi, and a Bernoulli mixture model for the defaults Yi. We generalize the dependency structure further and use a generalized link between systematic factor and idiosyncratic factor of creditworthiness. So the systematic factor cannot only change the location, but also the form of the distribution of creditworthiness. Results: For the case of two rating classes, we propose several estimators for the breakpoint and for the default probabilities within the rating classes. We prove the strong consistency of these estimators in the given non-i.i.d. framework. The theoretical results are illustrated by a simulation study. Finally, we give an overview of research opportunities.