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For donor blood it is important not to contain viruses because this could lead to transmission of for example HIV. This is why a company like Sanquin conducts research into the removal or inactivation of viruses in donor blood in their production processes. In brief this research is done by testing deliberately infected material before and after a production step. The method used is called endpoint titration and involves diluting the starting material several times and taking a number of samples from every dilution. These samples are tested for the presence of virus. With these results the fraction of the original solution that gives a response with probability 1/2 is calculated. During this project, different methods have been explored to examine which method gives the best estimate for the amount of virus in blood.

Y-haplotype profiling presents special advantages for forensic casework, paternity and population studies. Y-chromosome DNA is male-specific and can sometimes be detected in stains where autosomal DNA cannot. When a suspect in a criminal case has the same Y-haplotype as found in a crime stain, what is the evidential value of the match? In such cases, knowing an estimate of the frequency of the suspect's haplotype in the population is of interest. Many solutions to find such an estimate have been proposed but never accepted by the entire forensic experts population. In this report we present a new method, called the Neighbor-Matching method, as well as a detailed comparison with the existing methods and a theoretical analysis of these methods. The Neighbor-Matching method uses a population evolution process to estimate which haplotypes genetically close to the suspect's haplotype, called neighbors, are expected to be in the population if the suspect's haplotype is common or rare. No assumptions are made as to the distribution of all haplotype frequencies in the population, and as much genetical information as possible is used to create the neighbors. After the analyze of the theoretical and numerical results, it appears that this new method is well adapted to the problem at hand and quite robust to changes in its parameters, so can be seen as reliable. However, the version presented in this report is a simplified prototype, and more work has to be done to create a program largely usable by forensic experts.

In forensic science it is common practice to work on problems where the likelihood ratio is based on observing single piece of evidence given two hypotheses. However, in a lot of cases, there is more than one piece of evidence. In this thesis three problems regarding combining DNA profiles are discussed. First, we derived a method to combine the evidential value of different partial Y-chromosomal DNA profiles of different stains. The method consists of finding a lower bound for the likelihood ratio when more than two propositions are compared and where we don't need the prior probabilities of the different propositions. Second, we made a simulation model to investigate the dendence of autosomal and Y chromosomal DNA profiles by assuming that everybody with the same Y-chromosomal DNA profile has a common ancestor and simulating the assignment of autosomal DNA profiles over different family tree structures. The results can be used in practice as scientific support for the assumption of independence between the Y and autosomal profile. Third, we developed a model to interpret (low-template) DNA profiles which is able to give the likelihood of observing the DNA profile given any allele-combination of the donor. This model assumes that a DNA profile is a result of a stochastic process where the input are the alleles of a possible donor. This model uses the information in peak heights without using any threshold. The model shows promising results in determining the combined evidential value of several low template DNA profiles that were obtained from the same stain.

Pairtrading is een financieel instrument waarvan beurshandelaren gebruik kunnen maken. Dit instrument kenmerkt zich door het openen of het sluiten van een aandelenpositie waarbij een tweetal aandelentransacties tegelijkertijd worden uitgevoerd. Deze transacties zijn tegengesteld in die zin dat het ene aandeel wordt gekocht en het (equivalent van het) andere aandeel wordt verkocht. In dit bacheloreindwerk doen we verslag van ons onderzoek naar de vraag of het mogelijk is om voor beurshandelaren een strategie te ontwikkelen waarmee zij met behulp van pairtrading winsten kunnen realiseren. Om deze vraag te beantwoorden hebben we een wiskundig model ontwikkeld. Dit model hebben we gebaseerd op de zogenoemde Brownse brug omdat deze brug bepaalde eigenschappen heeft die vergelijkbaar zijn met de verschilkoers van pairtrading. Vervolgens hebben we uit dit model een strategie afgeleid. Deze strategie hebben we tenslotte met beursdata getest. Bij de verificatietesten van ons model zijn we ervan uitgegaan dat de diverse essenti¨ele aannamen geldig zijn. Deze testen wijzen uit dat ons model goed werkt. Dat betekent dat onze strategie kan worden toegepast. Bij de validatietesten van ons model echter mogen we er a priori niet vanuit gaan dat deze aannamen geldig zijn. Die geldigheid moeten we - middels verantwoorde testen - vaststellen. Daarin zijn wel onvoldoende geslaagd. Daaruit mogen we echter niet de conclusie trekken dat deze aannamen niet geldig zijn. We hebben ze immers in overleg zorgvuldig en met overtuiging gekozen. Toch brengt die onzekerheid mee dat ons model en onze strategie nog niet succesvol door beurshandelaren kunnen worden ingezet. Daartoe zouden we methoden moeten ontwikkelen waarmee we onze aannamen kunnen testen op hun geldigheid. Daarnaast zouden we ook gebruik moeten kunnen maken van meer gecoïntegreerde paren. Tevens zouden we in staat moeten zijn over een langere periode testen uit te voeren. Tenslotte zouden we de schattingen van de variabelen a, μ, σ0 en σ1, moeten verbeteren. Deze werkzaamheden vallen evenwel buiten het bestek van dit bacheloreindwerk.

We show that, for a stationary version of Hammersley’s process, with Poisson sources on the positive x-axis and Poisson sinks on the positive y-axis, the variance of the length of a longest weakly North–East path L(t, t) from (0, 0) to (t, t) is equal to 2E(t − X(t))+, where X(t) is the location of a second class particle at time t . This implies that both E(t −X(t))+ and the variance of L(t, t) are of order t2/3. Proofs are based on the relation between the flux and the path of a second class particle, continuing the approach of Cator and Groeneboom [Ann. Probab. 33 (2005) 879–903].

The interplay between two-dimensional percolation growth models and one-dimensional particle processes has been a fruitful source of interesting mathematical phenomena. In this paper we develop a connection between the construction of Busemann functions in the Hammersley last-passage percolation model with i.i.d. random weights, and the existence, ergodicity and uniqueness of equilibrium (or timeinvariant) measures for the related (multi-class) interacting fluid system. As we shall see, in the classical Hammersley model, where each point has weight one, this approach brings a new and rather geometrical solution of the longest increasing subsequence problem, as well as a central limit theorem for the Busemann function.

In recent years a popular nonparametric model for coarsened data is an assumption on the coarsening mechanism called coarsening at random (CAR). It has been conjectured in several papers that this assumption cannot be tested by the data, that is, the assumption does not restrict the possible distributions of the data. In this paper we will show that this conjecture is not always true; an example will be current status data. We will also give conditions when the conjecture is true, and in doing so, we will introduce a generalized version of the CAR assumption. As an illustration, we retrieve the well-known result that the CAR assumption cannot be tested in the case of right-censored data.

The problem of estimating the center of symmetry of a symmetric signal in Gaussian white noise is considered. The underlying nuisance function f is not assumed to be differentiable, which makes a new point of view to the problem necessary. We investigate the well-known sieve maximum likelihood estimators based on the cumulated periodogram, and study minimax rates over classes of irregular functions. It is shown that if the class appropriately controls the growth to infinity of the Fisher information over the sieve, semiparametric fast rates of convergence are obtained. We prove a lower bound result which implies that these semiparametric rates are really slower than the parametric ones, contrary to the regular case. Our results also suggest that there may be room to improve on the popular cumulated periodogram estimator.

We show that, for a stationary version of Hammersley’s process, with Poisson “sources” on the positive x-axis, and Poisson “sinks” on the positive y-axis, an isolated second-class particle, located at the origin at time zero, moves asymptotically, with probability 1, along the characteristic of a conservation equation for Hammersley’s process. This allows us to show that Hammersley’s process without sinks or sources, as defined by Aldous and Diaconis [Probab. Theory Related Fields 10 (1995) 199–213] converges locally in distribution to a Poisson process, a result first proved in Aldous and Diaconis (1995) by using the ergodic decomposition theorem and a construction of Hammersley’s process as a one-dimensional point process, developing as a function of (continuous) time on the whole real line. As a corollary we get the result that EL(t, t)/t converges to 2, as t→∞, where L(t, t) is the length of a longest North-East path from (0, 0) to (t, t). The proofs of these facts need neither the ergodic decomposition theorem nor the subadditive ergodic theorem. We also prove a version of Burke’s theorem for the stationary process with sources and sinks and briefly discuss the relation of these results with the theory of longest increasing subsequences of random permutations.

Since mean-field approximations for susceptible-infected-susceptible (SIS) epidemics do not always predict the correct scaling of the epidemic threshold of the SIS metastable regime, we propose two novel approaches: (a) an ɛ-SIS generalized model and (b) a modified SIS model that prevents the epidemic from dying out (i.e., without the complicating absorbing SIS state). Both adaptations of the SIS model feature a precisely defined steady state (that corresponds to the SIS metastable state) and allow an exact analysis in the complete and star graph consisting of a central node and N leaves. The N-intertwined mean-field approximation (NIMFA) is shown to be nearly exact for the complete graph but less accurate to predict the correct scaling of the epidemic threshold τc in the star graph, which is found as τc=ατc(1), where α=√1/2logN+3/2loglogN and where τc(1)=1/√N<τc is the first-order epidemic threshold for the star in NIMFA and equal to the inverse of the spectral radius of the star's adjacency matrix.

Since the Susceptible-Infected-Susceptible (SIS) epidemic threshold is not precisely defined in spite of its practical importance, the classical SIS epidemic process has been generalized to the ɛ−SIS model, where a node possesses a self-infection rate ɛ, in addition to a link infection rate β and a curing rate δ. The exact Markov equations are derived, from which the steady state can be computed. The major advantage of the ɛ−SIS model is that its steady state is different from the absorbing (or overall-healthy state) and approximates, for a certain range of small ɛ>0, the in reality observed phase transition, also called the “metastable” state, that is characterized by the epidemic threshold. The exact steady-state analysis for the complete graph illustrates the effect of small ɛ and the quality of the first-order mean-field approximation, the N-intertwined model, proposed earlier. Apart from duality principles, often used in the mathematical literature, we present an exact recursion relation for the Markov infinitesimal generator.