With the continuous expansion of data availability in many large-scale, complex, and networked systems, such as surveillance, security, the Internet, and finance, it becomes critical to advance the fundamental understanding of knowledge discovery and analysis from raw data to support decision-making processes. Existing knowledge discovery and data analyzing techniques have shown great success in many real-world applications such as applying Automatic Target Recognition (ATR) methods to detect targets of interest in imagery, drug activity prediction, computer vision recognition, and so on. Among these techniques, Multiple-Instance (MI) learning is different from standard classification since it uses a set of bags containing many instances as input. The instances in each bag are not labeled | instead the bags themselves are labeled. In this area many researchers have accomplished a lot of work and made a lot of progress. However, there still exist some areas which are not covered. In this thesis, we focus on two topics of MI learning: (1) Investigating the relationship between MI learning and other multiple pattern learning methods, which include multi-view learning, data fusion method and multi-kernel SVM. (2) Dealing with the class imbalance problem of MI learning. In the first topic, three different learning frameworks will be presented for general MI learning. The first uses multiple view approaches to deal with MI problem, the second is a data fusion framework, and the third framework, which is an extension of the first framework, uses multiple-kernel SVM. Experimental results show that the approaches presented work well on solving MI problem. The second topic is concerned with the imbalanced MI problem. Here we investigate the performance of learning algorithms in the presence of underrepresented data and severe class distribution skews. For this problem, we propose three solution frameworks: a data re-sampling framework, a cost-sensitive boosting framework and an adaptive instance-weighted boosting SVM (with the name IB_SVM) for MI learning. Experimental results - on both benchmark datasets and application datasets - show that the proposed frameworks are proved to be effective solutions for the imbalanced problem of MI learning.

Cette thèse se compose de trois essais empiriques qui contribuent à la littérature sur la political economy et économie du travail. Le premier chapitre analyse comment les effets du changement dans le degré de fédéralisme fiscal sur la composition de la dépense publique au niveau municipal dépend de la qualité des institutions ex-. Le deuxième chapitre se concentre sur la définition de la classe moyenne à l'aide de l'approche fondée sur les actifs et étudie comment, selon cette définition, les composantes du revenu et de la richesse se répartissent à travers les classes au fil du temps. Le troisième chapitre étudie plutôt comment la flexibilisation du marché du travail affecte le bien-être des travailleurs à l'entrée et au début de leur carrière. Les trois essais sont basés sur l'Italie pour faire la lumière sur trois sujets largement discutés tant au niveau académique que politique. / This thesis consists of three empirical essays that contribute to the literature on political economy and on labor economics. The first chapter analyzes how the effects of changes in the decree of fiscal federalism and mafia penetration on the composition of public expenditure at the municipal level depends on ex-ante institutional quality. The second chapter focus on the definition of the middle class using the asset based approach and studies how, according to this definition, income and wealth components distribute across classes over time. The third chapter, instead, studies how labor market flexibilisation affected workers well-being at entry and during the early career. All three essays are based on Italy to shed light on three topics widely discussed both at the academic and policy level.

The European Commission has foreseen a modal share of 30% by 2030 for rail freight transport. To achieve this increase in the modal share, enhanced reliability of rail freight services is required. Optimal functioning of shunting yards is one of the areas that can improve this reliability. Shunting yards are large areas allocated to reassemble freight trains for dispatching to new destinations. Their productivity has a direct impact on the overall performance of a rail freight network. Therefore, analysing and modelling of departure deviations from shunting yards are required to enhance the interactions between shunting yards and the network; this thesis contributes to this gap. Paper I investigates the probability and temporal distribution of departure deviations using a large data set comprising 250,000 departures over seven years from two main shunting yards (Malmö and Hallsberg) in Sweden. The probability distribution of departure deviations is found comparing four main distributions including the exponential, the log-normal, the gamma, and the Weibull according to the maximum likelihood estimates and the results of the Anderson-Darling goodness of fit test.  The log-normal and the gamma are shown the best fits for departure deviations: the former on delays, and the latter on early departures. In the temporal delay distribution, the weekly and monthly, but not yearly delayed departures are positively correlated with the network usage. However, for hourly delayed departures, a shunting yard involved with international traffic does not show any correlation between delayed departures and the network usage, whereas a domestic shunting yard shows a significant negative correlation between these two parameters.  The findings obtained from this thesis contribute to a better understanding of departure deviations from shunting yards, and can be applied in enhancing the operations and capacity utilization of shunting yards in future models. Papers II and III analyse the relationship between congestion in the arrival yard and departure delays using the same data set as paper I.  According to previous research, congestion plays an important role in shunting yard delays. With defining congestion as the number of arriving trains before departure time, paper II analyses this relationship limiting the arrivals and departures between the two shunting yards considering varying time periods before departure,whereas Paper III elaborates the analysis by defining congestion level in a fixed period of time before departure time including all arrivals and departures. Considering the data set used in the analysis, the results show that there is no significant relationship between the congestion in the arrival yard and departure delays of trains. It is possible that congestion may not impact the departure delays of trains, but it may impact the departure delays of wagons due to missed wagon connection or increasing wagon idle time, which can be explored with the availability of wagon connection data.  Additionally, future elaboration of congestion definition, covering congestion at the shunting yard level, may lead to further improved analyses. / Europeiska kommissionen har förutspått en markansandel på 30% framtill 2030 för järnvägstransporter av gods. För att uppnå denna ökning krävsökad tillförlitlighet hos järnvägstransporttjänster. Rangergodsbangårdars optimalafunktion är ett av de områden som kan förbättra denna tillförlitlighet.Rangergodsbangårdar stora områden som är avsedda för att koppla ihopgodståg för sändning till nya destinationer. Deras produktivitet har en direktinverkan på järnvägsnätets totala prestanda. Därför krävs analys och modelleringav avvikelser från dessa noder för att förbättra interaktionen mellanrangergodsbangårdar och järnvägsnätet. I papper I undersöks sannolikheten och den tidsmässiga fördelningen avavvikelser med hjälp av en stor datamängd som omfattar 250 000 avgångaröver sju år från två rangergodsbangårdar (Malmö och Hallsberg) i Sverige.Sannolikhetsdistributioner av avvikelser jämförs med fyra huvuddistributioner,exponentiell, log-normal, gamma och Weibull enligt de maximalasannolikhetsuppskattningarna och resultaten av Anderson-Darling godhetav passningstest. Log-normal och gamma visar sig passa bäst för avvikelser:den förstnämnda vid förseningar och den senare vid tidiga avgångar. I dentidsmässiga fördröjningsfördelningen är de veckovisa och månatliga men inteårliga försenade avgångarna positivt korrelerade med järnvägsnätets nyttjandegrad.För försenade avgångar per timme visar dock en rangergodsbangårdsom är inblandad i internationell trafik ingen korrelation mellan försenadeavgångar och järnvägsnätets nyttjandegrad, medan en inhemsk rangergodsbangårdvisaren signifikant negativ korrelation mellan dessa två parametrar.Resultaten från denna avhandling bidrar till en bättre förståelse av avvikelserfrån rangergodsbangårdar och kan användas för att förbättra drift och kapacitetsutnyttjandeav rangergodsbangårdar växelplatser i framtida modeller. Papper II och III analyserar förhållandet mellan trängsel i ankomstgårdenoch avgångsförseningar med hjälp av samma datamängd som i papperI. Enligt tidigare analyser spelar trängsel en viktig roll vid förseningar förrangergodsbangårdar. Trängsel definieras som antalet ankommande tåg föreavgångstid och papper II analyserar detta förhållande som begränsar ankomsteroch avgångar mellan de två rangergodsbangårdar med beaktande av olikatidsperioder före avgång, medan papper III utvecklar analysen genom attdefiniera trängselnivån under en fast tidsperiod före avgångstid inklusive allaankomster och avgångar. Med tanke på datamängden som användes i analysenvisar resultaten att det inte finns något signifikant samband mellan trängselni ankomstgården och tågens förseningar. Det är möjligt att trängsel kanskeinte påverkar tågens avgångsfördröjningar, men det kan påverka vagnarnasavgångsfördröjningar på grund av missad vagnanslutning eller öka vagnenstomgångstid, vilket kan undersökas med vid tillgång av vagnanslutningsdata.Dessutom kan framtida vidareutveckling av definitionen av trängsel som påen detaljerad nivå täcker rangergodsbangårdars alla delar, leda till ytterligareförbättrade analyser. / <p>QC 20201105</p> / Shift2Rail / FR8HUB

Dans le contexte des séries temporelles linéaires, on étudie les processus strictement stationnaires dits anticipatifs dépendant potentiellement de tous les termes d'une suite d'erreurs alpha-stables indépendantes et identiquement distribuées.On considère en premier lieu les processus autoregressifs (AR) et l'on montre que des moments conditionnels d'ordres plus élevés que les moments marginaux existent dès lors que le polynôme caractéristique admet au moins une racine à l'intérieur du cercle unité.Des formules fermées sont obtenues pour les moments d'ordre un et deux dans des cas particuliers.On montre que la méthode des moindres carrés permet d'estimer une représentation all-pass causale du processus dont la validité peut être vérifiée par un test de type portmanteau, et l'on propose une méthode fondée sur des propriétés d'extreme clustering pour retrouver la représentation AR originale.L'AR(1) stable anticipatif est étudié en détails dans le cadre des vecteurs stables bivariés et des formes fonctionnelles pour les quatre premiers moments conditionnels sont obtenues pour toute paramétrisation admissible.Lors des évènements extrêmes, il est montré que ces moments deviennent équivalents à ceux d'une distribution de Bernoulli chargeant deux évolutions futures opposées: accroissement exponentiel ou retour aux valeurs centrales.Des résultats parallèles sont obtenus pour l'analogue de l'AR(1) en temps continu, le processus d'Ornstein-Uhlenbeck stable anticipatif.Pour des moyennes mobiles alpha-stables infinies, la distribution conditionnelle des chemins futurs sachant la trajectoire passée est obtenue lors des évènements extrêmes par le biais d'une nouvelle représentation des vecteurs stables multivariés sur des cylindres unités relatifs à des semi-normes.Contrairement aux normes, ce type de représentation donne lieu à une propriété de variations régulières des queues de distribution utilisable dans un contexte de prévision, mais tout vecteur stable n'admet pas une telle représentation. Une caractérisation est donnée et l'on montre qu'un chemin fini de moyenne mobile alpha-stable sera représentable pourvu que le processus soit "suffisamment anticipatif".L'approche s'étend aux processus résultant de la combinaison linéaire de moyennes mobiles alpha-stables, et la distribution conditionnelle des chemins futurs s'interprète naturellement en termes de reconnaissance de formes. / In the framework of linear time series analysis, we study a class of so-called anticipative strictly stationary processes potentially depending on all the terms of an independent and identically distributed alpha-stable errors sequence.Focusing first on autoregressive (AR) processes, it is shown that higher order conditional moments than marginal ones exist provided the characteristic polynomials admits at least one root inside the unit circle. The forms of the first and second order moments are obtained in special cases.The least squares method is shown to provide a consistent estimator of an all-pass causal representation of the process, the validity of which can be tested by a portmanteau-type test. A method based on extreme residuals clustering is proposed to determine the original AR representation.The anticipative stable AR(1) is studied in details in the framework of bivariate alpha-stable random vectors and the functional forms of its first four conditional moments are obtained under any admissible parameterisation.It is shown that during extreme events, these moments become equivalent to those of a two-point distribution charging two polarly-opposite future paths: exponential growth or collapse.Parallel results are obtained for the continuous time counterpart of the AR(1), the anticipative stable Ornstein-Uhlenbeck process.For infinite alpha-stable moving averages, the conditional distribution of future paths given the observed past trajectory during extreme events is derived on the basis of a new representation of stable random vectors on unit cylinders relative to semi-norms.Contrary to the case of norms, such representation yield a multivariate regularly varying tails property appropriate for prediction purposes, but not all stable vectors admit such a representation.A characterisation is provided and it is shown that finite length paths of a stable moving average admit such representation provided the process is "anticipative enough".Processes resulting from the linear combination of stable moving averages are encompassed, and the conditional distribution has a natural interpretation in terms of pattern identification.

No description available.

Les modèles linéaires généralisés (GLMs) constituent l’une des classes de modèles les plus populaires en statistique. Cette classe contient une grande variété de modèles de régression fréquemment utilisés, tels que la régression linéaire normale, la régression logistique et les gamma GLMs. Dans les GLMs, la distribution de la variable de réponse définit une famille exponentielle. Un désavantage de ces modèles est qu’ils ne sont pas robustes par rapport aux valeurs aberrantes. Pour les modèles comme la régression linéaire normale et les gamma GLMs, la non-robustesse est une conséquence des ailes exponentielles des densités. La différence entre les tendances de l’ensemble des données et celles des valeurs aberrantes donne lieu à des inférences et des prédictions biaisées. A notre connaissance, il n’existe pas d’approche bayésienne robuste spécifique pour les GLMs. La méthode la plus populaire est fréquentiste ; c’est celle de Cantoni and Ronchetti (2001). Leur approche consiste à adapter les M-estimateurs robustes pour la régression linéaire au contexte des GLMs. Cependant, leur estimateur est dérivé d’une modification de la dérivée de la log-vraisemblance, au lieu d’une modification de la vraisemblance (comme avec les M-estimateurs robustes pour la régression linéaire). Par conséquent, il n’est pas possible d’établir une correspondance claire entre la fonction modifiée à optimiser et un modèle. Le fait de proposer un modèle robuste présente deux avantages. Premièrement, il permet de comprendre et d’interpréter la modélisation. Deuxièmement, il permet l’analyse fréquentiste et bayésienne. La méthode que nous proposons s’inspire des idées de la régression linéaire robuste bayésienne. Nous adaptons l’approche proposée par Gagnon et al. (2020), qui consiste à utiliser une distribution normale modifiée avec des ailes plus relevées pour le terme d’erreur. Dans notre contexte, la distribution de la variable de réponse est une version modifiée où la partie centrale de la densité est conservée telle quelle, tandis que les extrémités sont remplacées par des ailes log-Pareto, se comportant comme (1/|x|)(1/ log |x|)λ. Ce mémoire se concentre sur les gamma GLMs. La performance est mesurée à la fois théoriquement et empiriquement, avec une analyse des données sur les coûts hospitaliers. / Generalized linear models (GLMs) form one of the most popular classes of models in statistics. This class contains a large variety of commonly used regression models, such as normal linear regression, logistic regression and gamma GLMs. In GLMs, the response variable distribution defines an exponential family. A drawback of these models is that they are non-robust against outliers. For models like the normal linear regression and gamma GLMs, the non-robustness is a consequence of the exponential tails of the densities. The difference in trends in the bulk of the data and the outliers yields skewed inference and prediction. To our knowledge, there is no Bayesian robust approach specifically for GLMs. The most popular method is frequentist; it is that of Cantoni and Ronchetti (2001). Their approach is to adapt the robust M-estimators for linear regression to the context of GLMs. However, their estimator is derived from a modification of the derivative of the log-likelihood, instead of from a modification of the likelihood (as with robust M-estimators for linear regression). As a consequence, it is not possible to establish a clear correspondence between the modified function to optimize and a model. Having a robust model has two advantages. First, it allows for an understanding and an interpretation of the modelling. Second, it allows for both frequentist and Bayesian analysis. The method we propose is based on ideas from Bayesian robust linear regression. We adapt the approach proposed by Gagnon et al. (2020), which consists of using a modified normal distribution with heavier tails for the error term. In our context, the distribution of the response variable is a modified version where the central part of the density is kept as is, while the extremities are replaced by log-Pareto tails, behaving like (1/|x|)(1/ log |x|)λ. The focus of this thesis is on gamma GLMs. The performance is measured both theoretically and empirically, with an analysis of hospital costs data.

No / Characterising the future risk to biodiversity across multiple environmental drivers is fraught with uncertainty and is a major conservation challenge. Scenario planning - to identify robust decisions across a range of plausible futures - can aid biodiversity conservation when tactical decisions need to be made in the present-day, yet consequences are realised over many decades. Management responses to the impact of tree disease are an excellent candidate for scenario planning, because actions to reduce an imminent biodiversity threat need to be effective in the long-term by accounting for concomitant factors such as a changing climate. Lichen epiphytes were used to exemplify a guild sensitive to woodland change, such as a tree disease impact. Bioclimatic models for 382 British epiphytes were combined with species-specific tree association values, to explore scenarios of tree disease (ash dieback), climate change, and range-filling under a lower SO2 pollution regime, for northern Britain focussed on Scotland. Results indicated: 1. Exposure of lichen diversity to projected climate change is spatially structured and expected to be greater in continental northeast Scotland, compared to oceanic western Scotland. 2. Impact of tree disease showed analogous geographic trends, evidencing a critical interaction between the climatic and local ecological setting. On average, the loss of ash could have an effect on epiphyte assemblages comparable in magnitude to that of climate change under a 2080s high emissions scenario. 3. In general, tree disease impacts can be mitigated by increasing the diversity of substitute tree species within a stand, to generate complementarity among epiphyte communities. However, the effectiveness of alternate management scenarios varied locally between sites and temporally with the progression of climate change. Given this variability, scenario analysis is recommended to effectively manage for resilience, by scoping how local factors (e.g. managed woodland composition) can reduce epiphyte assemblage turnover beyond that uniquely associated with larger-scale environmental impacts. (C) 2014 Elsevier Ltd. All rights reserved.

Master of Science / Department of Statistics / Suzanne Dubnicka / To help with identifying distributions to effectively model univariate continuous data, the R function diagnostic is proposed. The function will aid in determining reasonable candidate distributions that the data may have come from. It uses a combination of the Pearson goodness of fit statistic, Anderson-Darling statistic, Lin’s concordance correlation between the theoretical quantiles and observed quantiles, and the maximum difference between the theoretical quantiles and the observed quantiles. The function generates reasonable candidate distributions, QQ plots, and histograms with superimposed density curves. When a simulation study was done, the function worked adequately; however, it was also found that many of the distributions look very similar if the parameters are chosen carefully. The function was then used to attempt to decipher which distribution could be used to model weekly grocery expenditures of a family household.

Thesis (PhD)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The challenge of conserving biodiversity is daunting. Despite some local conservation gains, most indicators of the condition of global biodiversity show declines since the 1970’s, while indicators of the threats to biodiversity all show increases. Humanity has in part responded to the global biodiversity extinction crisis by establishing protected areas (PA) and they are widely considered cornerstones of conservation. However, their efficacy in maintaining biodiversity is much debated. Previous studies have been unable to provide a general answer because of their typically restricted geographic and/or taxonomic focus, or qualitative approach. Using a global meta-analysis with 861 pairwise comparisons inside and outside PAs from 86 studies across five major taxon groups, I tested the hypothesis that PAs achieve significant conservation outcomes measured as higher biodiversity values compared with alternative land covers. I found that globally, PAs typically contain higher abundances of individual species, higher assemblage abundances and higher species richness. Variation in effect sizes among taxa nonetheless underscores that PA efficacy can be context specific. To examine factors driving the context specific nature of PA efficacy, an exact distance, timed point count methodology was used to assess PAs ecological effectiveness in terms of bird assemblages of the Kruger National Park (KNP), South Africa, in the Phalaborwa section. Bird assemblages inside the KNP were compared to matched sites in rural and urban land cover, as well as the connecting habitat matrix outside the KNP. Species richness and abundance were significantly lower inside the KNP compared to other land covers. However, the species assemblages are markedly different. The artificial addition of resources in an otherwise resources poor area, mainly in terms of gardening, provide suitable habitat for a range of species, consistent with the more individuals hypothesis. Large-bodied and ground nesting species are virtually absent outside the KNP. Thus species richness and abundance differences between land cover regions mask insidious changes in species traits. Nonetheless, not formally protected land can contribute positively to the regional biodiversity portfolio. Since an understanding of the mechanisms that structure species assemblages can aid in the consequences of anthropogenic drivers disentangling them, I describe and analyse the body size frequency distributions (BSFDs) of avian assemblages at several spatial scales in the Afrotropical biogeographic realm. I found that the African avifaunal continental BSFD is unimodal and right-skewed. African avifaunal BSFDs are quantitatively dissimilar to the African mammal BSFDs, which are bimodal at all spatial scales. Much of the change in median body size with spatial scale can be captured by a range-weighted null model, suggesting that differential turnover between smaller- and larger-bodied species might explain the shift in the central tendency of the BSFD. My results for the first time quantitatively demonstrate that PAs are a vital component of a global biodiversity conservation strategy. However, I also show that PA ecological effectiveness can be context specific, and understanding which species traits are at risk outside of PAs is critical to predicting their efficacy. / AFRIKAANSE OPSOMMING: Dit is ‘n uitnemende uitdaging om biodiversiteit te bewaar. Ten spyte van sommige sukses op ‘n lokale skaal, dui die meeste indikators aan dat die toestand van globale biodiveristeit afgeneem het vanaf die 1970’s, terwyl bedreigings daartoe toegeneem het. Die mensdom het gedeeltelik reageer op die biodiversiteits uitsterfings krisis deur die uiteensetting en instandhouding van bewaringsgebiede (BG). Hierdie metodiek word wêreldwyd geag as ‘n hoeksteen van bewaring. Die ekologiese doeltreffendheid van BG word egter baie debateer. Vorige studies was geografies beperk of het net op sekere takson groepe gefokus. Vorige studies is ook tipies statisties kwalitatief van aard. As gevolg daarvan het ek ‘n globale meta-analiese gebruik, wat bestaan het uit 861 gepaarde meetings vanaf 86 studies, oor vyf verskillende taksonomiese groepe. Ek het die hipotese getoets dat BG statisties beduidende bewaring laat gekiet in terme van hoër biodiveristeits waardes binnekant hul grense, in kontras met areas buite BG. Ek het gevind dat BG juis hoër waardes van hoër indivuduele spesies hoeveelhede, gemeenskaps hoeveelhede en spesiesrykheid bevat. Tog dui die variasie in effek grootte onder takson groepe aan dat BG effektiwiteit konteks spesifiek is. Aangesien BG effektiwiteit konteks spesifiek is, het ek verder ‘n tydstip, eksakte-afstand punttelling metodiek gevolg om die Kruger Nationale Park (KNP) in Suid Afrika, se ekologiese doeltreffendheid te bepaal. Ek het voëlgemeenskappe binne KNP vergelyk met voëlgemeenskappe buite KNP. Metings is buite die KNP gedoen in tuislande, die stadsgebied van Phalaborwa, asook in die habitat matriks wat die areas verbind. Beide spesiesrykheid en hoeveelheid is hoër buite KNP, maar die voëlgemeenskap struktuur tussen gebiede verskil noemenswaardig. Die uitbreiding van bronne ten opstigte van nesmaak en voeding (meestal deur tuinmaak), bied vir voëls goeie habitat in die stadsgebied, soos verwag kan word uit die meer individue hipotese. Hierdie veranderinge in spesiesrykheid tussen die verskillende areas versteek veranderinge in die spesies einskappe. Voëls met groot liggaamsmassa, veral die wat op die grond nes maak, kom in baie minder getalle voor buite KNP. Tog bied onbewaarde areas ‘n positiewe inpak tot die omgewing se biodiversiteit. ‘n Holistiese begrip van die meganismes wat spesies gemeenskappe struktureer kan help om die menslike invloed daarop uit te lig. Daarvolgens beskryf ek die liggaamsmassa frekwensie verspreiding (LMFV) oor verkillende skale in die Afrotropies biografiese streek. Ek het bevind dat Afrika se voëlgemeenskappe op die kontinentale skaal unimodaal en regs-geskewe is. Afrika se voël LMFV is beduidend verskillend van Afrika soogdier LMFV, wat bimodaal is op verskeie skale. Baie van die verandering in mediaan liggaams massa oor verskillende skale kan verduidelik word met ‘n verspreidings-aangepaste nul-model, wat voorstel dat die verskille in omset van voëls met of klein, of groot liggaamsmassa in die landskap die veranderings in LMFV verduidelik. My studie is die eerste van sy soort wat kwantitatief bepaal dat die gebruik van BG krities is tot ‘n globale bewaringsstrategie. Verder het ek bewys dat BG se ekologiese doeltreffendheid afhang van die kontkes op ‘n lokale skaal, en dat spesies eienskappe geïnkorporeer moet word om BG se doeltreffendheid te bepaal.

We show that finite de Finetti-type results may be viewed as results on the approximation of certain continuous functions of a parameter by a sequence of positive operators (Ln) . For distribtions that depend on a finite-dimensional statistic (Tn) only, Ln is the expectation operator of (Tn) under the extremal infinite exchangeable distributions. The rate of approximation of finite exchangeable distributions by mixtures of marginals of infinite exchangeable distributions is the rate of approximation of a single function of the parameter, namely the second indefinite integral of the Fisher information. Our results include a major part of what is known about finite de Finetti theorems. The theory is, however, not only valid for the case when the extremal infinite exchangeable distributions are products of identical distributions. It applies as well to Markov-exchangeable distributions or the linear model. Moreover, the metric is not restricted to the supremum norm. (author's abstract) / Series: Forschungsberichte / Institut für Statistik