Parameterschätzung in Verzweigungsprozessen mit Immigration

Im Verzweigungsprozess mit Immigration werden Schätzer für die erwartete Nachkommenzahl m eines Individuums und die erwartete Immigration λ pro Generation konstruiert. Sie sind nur aufgrund der beobachteten Populationsgröße einer jeden Generation konsistent, ohne Vorkenntnis darüber, ob der Prozess subkritisch (m<1), kritisch (m=1) oder superkritisch (m>1) ist. Im superkritischen Fall ist der Schätzer für λ jedoch nicht konsistent. Dies ist aber keine Einschränkung, denn es wird gezeigt, dass in diesem Fall kein konsistenter Schätzer für λ existiert. Des Weiteren werden Konvergenzgeschwindigkeit der Schätzer und asymptotische Verteilungen der Schätzfehler untersucht. Dabei werden die Fälle (m<1), (m>1) und (m=1) unterschieden, was gänzlich verschiedene Vorgehensweisen erfordert (Ergodizität, Martingalmethoden, Diffusionsapproximationen). Diese hier vorliegende Diplomarbeit orientiert sich an den Ideen und Ergebnissen von Wei und Winnicki (1989/90).

Ergodicity and regularity of invariant measure for branching Markov processes with immigration

In this thesis we consider systems of finitely many particles moving on paths given by a strong Markov process and undergoing branching and reproduction at random times. The branching rate of a particle, its number of offspring and their spatial distribution are allowed to depend on the particle's position and possibly on the configuration of coexisting particles. In addition there is immigration of new particles, with the rate of immigration and the distribution of immigrants possibly depending on the configuration of pre-existing particles as well. In the first two chapters of this work, we concentrate on the case that the joint motion of particles is governed by a diffusion with interacting components. The resulting process of particle configurations was studied by E. Löcherbach (2002, 2004) and is known as a branching diffusion with immigration (BDI). Chapter 1 contains a detailed introduction of the basic model assumptions, in particular an assumption of ergodicity which guarantees that the BDI process is positive Harris recurrent with finite invariant measure on the configuration space. This object and a closely related quantity, namely the invariant occupation measure on the single-particle space, are investigated in Chapter 2 where we study the problem of the existence of Lebesgue-densities with nice regularity properties. For example, it turns out that the existence of a continuous density for the invariant measure depends on the mechanism by which newborn particles are distributed in space, namely whether branching particles reproduce at their death position or their offspring are distributed according to an absolutely continuous transition kernel. In Chapter 3, we assume that the quantities defining the model depend only on the spatial position but not on the configuration of coexisting particles. In this framework (which was considered by Höpfner and Löcherbach (2005) in the special case that branching particles reproduce at their death position), the particle motions are independent, and we can allow for more general Markov processes instead of diffusions. The resulting configuration process is a branching Markov process in the sense introduced by Ikeda, Nagasawa and Watanabe (1968), complemented by an immigration mechanism. Generalizing results obtained by Höpfner and Löcherbach (2005), we give sufficient conditions for ergodicity in the sense of positive recurrence of the configuration process and finiteness of the invariant occupation measure in the case of general particle motions and offspring distributions.

Non-parametric estimation of the diffusion coefficient of a branching diffusion with immigration

Wir betrachten Systeme von endlich vielen Partikeln, wobei die Partikel sich unabhängig voneinander gemäß eindimensionaler Diffusionen [dX_t = b(X_t),dt + sigma(X_t),dW_t] bewegen. Die Partikel sterben mit positionsabhängigen Raten und hinterlassen eine zufällige Anzahl an Nachkommen, die sich gemäß eines Übergangskerns im Raum verteilen. Zudem immigrieren neue Partikel mit einer konstanten Rate. Ein Prozess mit diesen Eigenschaften wird Verzweigungsprozess mit Immigration genannt. Beobachten wir einen solchen Prozess zu diskreten Zeitpunkten, so ist zunächst nicht offensichtlich, welche diskret beobachteten Punkte zu welchem Pfad gehören. Daher entwickeln wir einen Algorithmus, um den zugrundeliegenden Pfad zu rekonstruieren. Mit Hilfe dieses Algorithmus konstruieren wir einen nichtparametrischen Schätzer für den quadrierten Diffusionskoeffizienten $sigma^2(cdot),$ wobei die Konstruktion im Wesentlichen auf dem Auffüllen eines klassischen Regressionsschemas beruht. Wir beweisen Konsistenz und einen zentralen Grenzwertsatz.

Pre-Columbian human occupation patterns in the eastern plains of the Llanos de Moxos, Bolivian Amazonia

Despite an increasing number of publications regarding the Pre-Columbian earthworks of the Llanos de Moxos, there have been no serious attempts to undertake a systematic survey of the archaeological remains of this lowland region in the Bolivian Amazon. Based on the GIS analysis of data gathered in the field and retrieved from satellite images, we discuss the spatial distribution of the Pre-Columbian settlements in a 4500 Km2 area of the Llanos de Moxos to the east of Trinidad, capital of the Beni Department, and their relationship with the geographical settings. Our findings shed new light on the prehistory of the region and bear important implications for our understanding of the impact of Pre-Columbian human occupation.

Ancient Calendars and Bolivian Modernity: Tiwanaku’s Gateway of the Sun, Arthur Posnansky, and the World Calendar Movement of the 1930s

Tourists to the archaeological site of Tiwanaku are presented with ancient calendars, of which the Gateway of the Sun is the most important, famous, and beautiful. Arthur Posnansky and other early 20th-century archaeologists claimed that its inscriptions constituted a written calendar. These claims were intimately connected to narratives of Tiwanaku as a central source of knowledge in both pre-Columbian times and the contemporary world. Posnansky presented his interpretation of Tiwanaku’s calendars as a response to the debates of the World Calendar Movement, which in the 1930s was attempting to rationalize the Gregorian calendar. In the Gateway, Posnansky found a uniquely Bolivian response to the international, North Atlantic-dominated scientific community’s search for a rational way to keep time in the world economy. Bolivian intellectuals merged their interest in the indigenous past with their concerns about the role of the modernist Bolivian state in the global system.

Mycobacterium tuberculosis transmission in a country with low tuberculosis incidence: role of immigration and HIV infection

Immigrants from high-burden countries and HIV-coinfected individuals are risk groups for tuberculosis (TB) in countries with low TB incidence. Therefore, we studied their role in transmission of Mycobacterium tuberculosis in Switzerland. We included all TB patients from the Swiss HIV Cohort and a sample of patients from the national TB registry. We identified molecular clusters by spoligotyping and mycobacterial interspersed repetitive-unit-variable-number tandem-repeat (MIRU-VNTR) analysis and used weighted logistic regression adjusted for age and sex to identify risk factors for clustering, taking sampling proportions into account. In total, we analyzed 520 TB cases diagnosed between 2000 and 2008; 401 were foreign born, and 113 were HIV coinfected. The Euro-American M. tuberculosis lineage dominated throughout the study period (378 strains; 72.7%), with no evidence for another lineage, such as the Beijing genotype, emerging. We identified 35 molecular clusters with 90 patients, indicating recent transmission; 31 clusters involved foreign-born patients, and 15 involved HIV-infected patients. Birth origin was not associated with clustering (adjusted odds ratio [aOR], 1.58; 95% confidence interval [CI], 0.73 to 3.43; P = 0.25, comparing Swiss-born with foreign-born patients), but clustering was reduced in HIV-infected patients (aOR, 0.49; 95% CI, 0.26 to 0.93; P = 0.030). Cavitary disease, male sex, and younger age were all associated with molecular clustering. In conclusion, most TB patients in Switzerland were foreign born, but transmission of M. tuberculosis was not more common among immigrants and was reduced in HIV-infected patients followed up in the national HIV cohort study. Continued access to health services and clinical follow-up will be essential to control TB in this population.