The success of the current Internet relies to a large extent on a cooperation between the users and the network. The network signals its current state to the users by marking or dropping packets. The users then strive to maximize the sending rate without causing network congestion. To achieve this, the users implement a flow-control algorithm that controls the rate at which data packets are sent into the Internet. More specifically, the Transmission Control Protocol (TCP) is used by the users to adjust the sending rate in response to changing network conditions. TCP uses the observation of packet loss events and estimates of the round trip time (RTT) to adjust its sending rate. In this thesis we investigate and propose stochastic models for TCP. The models are used to estimate network performance like throughput, link utilization, and packet loss rate. The first part of the thesis introduces the TCP protocol and contains an extensive TCP modeling survey that summarizes the most important TCP modeling work. Reviewed models are categorized as renewal theory models, fixed-point methods, fluid models, processor sharing models or control theoretic models. The merits of respective category is discussed and guidelines for which framework to use for future TCP modeling is given. The second part of the thesis contains six papers on TCP modeling. Within the renewal theory framework we propose single source TCP-Tahoe and TCP-NewReno models. We investigate the performance of these protocols in both a DropTail and a RED queuing environment. The aspects of TCP performance that are inherently depending on the actual implementation of the flow-control algorithm are singled out from what depends on the queuing environment. Using the fixed-point framework, we propose models that estimate packet loss rate and link utilization for a network with multiple TCP-Vegas, TCP-SACK and TCP-Reno on/off sources. The TCP-Vegas model is novel and is the first model capable of estimating the network's operating point for TCP-Vegas sources sending on/off traffic. All TCP and network models in the contributed research papers are validated via simulations with the network simulator ns-2. This thesis serves both as an introduction to TCP and as an extensive orientation about state of the art stochastic TCP models.

cDNA microarrays is one of the first high-throughput gene expression technologies that has emerged within molecular biology for the purpose of functional genomics. cDNA microarrays compare the gene expression levels between cell samples, for thousands of genes simultaneously. The microarray technology offers new challenges when it comes to data analysis, since the thousands of genes are examined in parallel, but with very few replicates, yielding noisy estimation of gene effects and variances. Although careful image analyses and normalisation of the data is applied, traditional methods for inference like the Student t or Fisher’s F-statistic fail to work. In this thesis, four papers on the topics of empirical Bayes and full Bayesian methods for two-channel microarray data (as e.g. cDNA) are presented. These contribute to proving that empirical Bayes methods are useful to overcome the specific data problems. The sample distributions of all the genes involved in a microarray experiment are summarized into prior distributions and improves the inference of each single gene. The first part of the thesis includes biological and statistical background of cDNA microarrays, with an overview of the different steps of two-channel microarray analysis, including experimental design, image analysis, normalisation, cluster analysis, discrimination and hypothesis testing. The second part of the thesis consists of the four papers. Paper I presents the empirical Bayes statistic B, which corresponds to a t-statistic. Paper II is based on a version of B that is extended for linear model effects. Paper III assesses the performance of empirical Bayes models by comparisons with full Bayes methods. Paper IV provides extensions of B to what corresponds to F-statistics.

<p>We investigate mathematical tools, Edgeworth series expansion and the saddle point method, which are approximation techniques that help us to estimate the distribution function for the standardized mean of independent identical distributed random variables where we will take into consideration the lattice case. Later on we will describe one important application for these mathematical tools where game developing companies can use them to reduce the amount of time needed to satisfy their standard requests before they approve any game</p>

<p>In regression with near collinear explanatory variables, the least squares predictor has large variance. Ordinary least squares regression (OLSR) often leads to unrealistic regression coefficients. Several regularized regression methods have been proposed as alternatives. Well-known are principal components regression (PCR), ridge regression (RR) and continuum regression (CR). The latter two involve a continuous metaparameter, offering additional flexibility.</p><p>For a univariate response variable, CR incorporates OLSR, PLSR, and PCR as special cases, for special values of the metaparameter. CR is also closely related to RR. However, CR can in fact yield regressors that vary discontinuously with the metaparameter. Thus, the relation between CR and RR is not always one-to-one. We develop a new class of regression methods, LSRR, essentially the same as CR, but without discontinuities, and prove that any optimization principle will yield a regressor proportional to a RR, provided only that the principle implies maximizing some function of the regressor's sample correlation coefficient and its sample variance. For a multivariate response vector we demonstrate that a number of well-established regression methods are related, in that they are special cases of basically one general procedure. We try a more general method based on this procedure, with two meta-parameters. In a simulation study we compare this method to ridge regression, multivariate PLSR and repeated univariate PLSR. For most types of data studied, all methods do approximately equally well. There are cases where RR and LSRR yield larger errors than the other methods, and we conclude that one-factor methods are not adequate for situations where more than one latent variable are needed to describe the data. Among those based on latent variables, none of the methods tried is superior to the others in any obvious way.</p>

<p>I denna avhandling studerar vi nödvändiga och tillräckliga villkor för svag konvergens av första-sällan-händelsetider för semi-Markovska processer.</p><p>I introduktionen ger vi nödvändiga grundläggande definitioner och beskrivningar av modeller som betraktas i avhandlingen, samt ger några exempel på situationer i vilka metoder av första-sällan-händelsetider kan vara lämpliga att använda. Dessutom analyserar vi publicerade resultat om asymptotiska problem för stokastiska funktionaler som definieras på semi-Markovska processer.</p><p>I artikel A betraktar vi första-sällan-händelsetider för semi-Markovska processer med en ändlig mängd av lägen. Vi ger också en sammanfattning av våra resultat om nödvändiga och tillräckliga villkor för svag konvergens, samt diskuterar möjliga tillämpningar inom aktuarie-området.</p><p>I artikel B redovisar vi i detalj de resultat som annonseras i artikel A och bevisen för dem. Vi ger också nödvändiga och tillräckliga villkor för svag konvergens av första-sällan-händelsetider för semi-Markovska processer med en ändlig mängd av lägen i ett icke-triangulärt tillstånd. Dessutom beskriver vi med hjälp av Laplacetransformationen klassen av alla möjliga gränsfördelningar.</p><p>I artikel C studerar vi villkor av svag konvergens av flöden av sällan-händelser i ett icke-triangulärt tillstånd. Vi formulerar nödvändiga och tillräckliga villkor för konvergens, och beskriver klassen av alla möjliga gränsflöden. Vi tillämpar också våra resultat i asymptotisk analys av icke-ruin-sannolikheten för störda riskprocesser.</p><p>I artikel D ger vi nödvändiga och tillräckliga villkor för svag konvergens av första-sällan-händelsetider för semi-Markovska rocesser med en ändlig mängd av lägen i ett triangulärt tillstånd, samt beskriver klassen av alla möjliga gränsfördelningar. Resultaten utvidgar slutsatser från artikel B till att gälla för ett allmänt triangulärt tillstånd.</p><p>I artikel E ger vi nödvändiga och tillräckliga villkor för svag konvergens av flöden av sällan-händelser för semi-Markovska processer i ett triangulärt tillstånd. Detta generaliserar resultaten från artikel C till att beskriva ett allmänt triangulärt tillstånd. Vidare ger vi tillämpningar av våra resultat på asymptotiska problem av störda riskprocesser och till kösystemen med snabb service.</p> / <p>In this thesis we study necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes, we describe the class of all possible limit distributions, and give the applications of the results to risk theory and queueing systems.</p><p>In paper <b>A</b>, we consider first-rare-event times for semi-Markov processes with a finite set of states, and give a summary of our results concerning necessary and sufficient conditions for weak convergence of first-rare-event times and their actuarial applications.</p><p>In paper <b>B</b>, we present in detail results announced in paper <b>A</b> as well as their proofs. We give necessary and sufficient conditions for weak convergence of first-rare-event times for semi-Markov processes with a finite set of states in non-triangular-array mode and describe the class of all possible limit distributions in terms of their Laplace transforms.</p><p>In paper <b>C</b>, we study the conditions for weak convergence for flows of rare events for semi-Markov processes with a finite set of states in non-triangular array mode. We formulate necessary and sufficient conditions of convergence and describe the class of all possible limit stochastic flows. In the second part of the paper, we apply our results to the asymptotical analysis of non-ruin probabilities for perturbed risk processes.</p><p>In paper <b>D</b>, we give necessary and sufficient conditions for the weak convergence of first-rare-event times for semi-Markov processes with a finite set of states in triangular array mode as well as describing the class of all possible limit distributions. The results of paper <b>D</b> extend results obtained in paper <b>B</b> to a general triangular array mode.</p><p>In paper <b>E</b>, we give the necessary and sufficient conditions for weak convergence for the flows of rare events for semi-Markov processes with a finite set of states in triangular array case. This paper generalizes results obtained in paper <b>C</b> to a general triangular array mode. In the second part of the paper, we present applications of our results to asymptotical problems of perturbed risk processes and to queueing systems with quick service</p>

<p>Industrial robots make up an important part in today’s industry and are assigned to a range of different tasks. Needless to say, businesses need to rely on their machine park to function as planned, avoiding stops in production due to machine failures. This is where fault detection methods play a very important part. In this thesis a specific fault detection method based on signal analysis will be considered. When testing a robot for fault(s), a specific test cycle (trajectory) is executed in order to be able to compare test data from different test occasions. Furthermore, different test cycles yield different measurements to analyse, which may affect the performance of the analysis. The question posed is: <em>Can we find an optimal test cycle so that the fault is best revealed in the test data?</em> The goal of this thesis is to, using regression analysis, investigate how the presently executed test cycle in a specific diagnosis method relates to the faults that are monitored (in this case a so called friction fault) and decide if a different one should be recommended. The data also includes representations of two disturbances.</p><p>The results from the regression show that the variation in the test quantities utilised in the diagnosis method are not explained by neither the friction fault or the test cycle. It showed that the disturbances had too large effect on the test quantities. This made it impossible to recommend a different (optimal) test cycle based on the analysis.</p>

<p>The distribution of asset returns may lead to structural breaks. Thesebreaks may result in changes of the optimal portfolio weights. For a port-folio investor, the ability of timely detection of any systematic changesin the optimal portfolio weights is of a great interest.In this master thesis work, the use of the Shewhart method, as amethod for detecting a sudden parameter change, the implied changein the multivariate portfolio weights and its performance is reviewed.</p><p> </p>

<p>This PhD thesis consists of an introduction and five papers, dealing with statistical methods in phylogenetics.</p><p>A phylogenetic tree describes the evolutionary relationships among species assuming that they share a common ancestor and that evolution takes place in a tree like manner. Our aim is to reconstruct the evolutionary relationships from aligned DNA sequences.</p><p>In the first two papers we investigate two measures of confidence for likelihood based methods, bootstrap frequencies with Maximum Likelihood (ML) and Bayesian posterior probabilities. We show that an earlier claimed approximate equivalence between them holds under certain conditions, but not in the current implementations of the two methods.</p><p>In the following two papers the divergence times of the internal nodes are considered. The ML estimate of the divergence time of the root is improved if longer sequences are analyzed or if more taxa are added. We show that the gain in precision is faster with longer sequences than with more taxa. We also show that the algorithm of the software package PATHd8 may give biased estimates if the global molecular clock is violated. A change of the algorithm to obtain unbiased estimates is therefore suggested.</p><p>The last paper deals with non-informative priors when using the Bayesian approach in phylogenetics. The term is not uniquely defined in the literature. We adopt the idea of data translated likelihoods and derive the so called Jeffreys' prior for branch lengths using Jukes Cantor model of evolution.</p>

This PhD thesis consists of an introduction and five papers, dealing with statistical methods in phylogenetics. A phylogenetic tree describes the evolutionary relationships among species assuming that they share a common ancestor and that evolution takes place in a tree like manner. Our aim is to reconstruct the evolutionary relationships from aligned DNA sequences. In the first two papers we investigate two measures of confidence for likelihood based methods, bootstrap frequencies with Maximum Likelihood (ML) and Bayesian posterior probabilities. We show that an earlier claimed approximate equivalence between them holds under certain conditions, but not in the current implementations of the two methods. In the following two papers the divergence times of the internal nodes are considered. The ML estimate of the divergence time of the root is improved if longer sequences are analyzed or if more taxa are added. We show that the gain in precision is faster with longer sequences than with more taxa. We also show that the algorithm of the software package PATHd8 may give biased estimates if the global molecular clock is violated. A change of the algorithm to obtain unbiased estimates is therefore suggested. The last paper deals with non-informative priors when using the Bayesian approach in phylogenetics. The term is not uniquely defined in the literature. We adopt the idea of data translated likelihoods and derive the so called Jeffreys' prior for branch lengths using Jukes Cantor model of evolution.

In regression with near collinear explanatory variables, the least squares predictor has large variance. Ordinary least squares regression (OLSR) often leads to unrealistic regression coefficients. Several regularized regression methods have been proposed as alternatives. Well-known are principal components regression (PCR), ridge regression (RR) and continuum regression (CR). The latter two involve a continuous metaparameter, offering additional flexibility. For a univariate response variable, CR incorporates OLSR, PLSR, and PCR as special cases, for special values of the metaparameter. CR is also closely related to RR. However, CR can in fact yield regressors that vary discontinuously with the metaparameter. Thus, the relation between CR and RR is not always one-to-one. We develop a new class of regression methods, LSRR, essentially the same as CR, but without discontinuities, and prove that any optimization principle will yield a regressor proportional to a RR, provided only that the principle implies maximizing some function of the regressor's sample correlation coefficient and its sample variance. For a multivariate response vector we demonstrate that a number of well-established regression methods are related, in that they are special cases of basically one general procedure. We try a more general method based on this procedure, with two meta-parameters. In a simulation study we compare this method to ridge regression, multivariate PLSR and repeated univariate PLSR. For most types of data studied, all methods do approximately equally well. There are cases where RR and LSRR yield larger errors than the other methods, and we conclude that one-factor methods are not adequate for situations where more than one latent variable are needed to describe the data. Among those based on latent variables, none of the methods tried is superior to the others in any obvious way.