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The field of statistics is becoming increasingly more important as the amount of data in the world grows. This thesis studies the Growth Curve model in multivariate statistics which is a model that is not widely used. One difference compared with the linear model is that the Maximum Likelihood Estimators are more complicated. That makes it more difficult to use and to interpret which may be a reason for its not so widespread use. From this perspective this thesis will compare the traditional mean estimator for the Growth Curve model with the unweighted mean estimator. The unweighted mean estimator is simpler than the regular MLE. It will be proven that the unweighted estimator is in fact the MLE under certain conditions and examples when this occurs will be discussed. In a more general setting this thesis will present conditions when the un-weighted estimator has a smaller covariance matrix than the MLEs and also present confidence intervals and hypothesis testing based on these inequalities.

<p>In this thesis, a MISO wireless communication system having even transmitter antennas and a single receiver antenna is considered, where neither the transmitter nor the receiver knows channel state information. Particularly when the number of transmitter antennas is two, a novel concept called a uniquely factorable constellation pair (UFCP) is first proposed for the systematic design of a noncoherent full diversity collaborative unitary space-time block code by normalizing two Alamouti codes. It is proved that such a unitary UFCP code assures the unique identification of both channel coefficients and transmitted signals in a noise-free case as well as full diversity for the noncoherent maximum likelihood (ML) receiver in a noise case. To further improve error performance, an optimal unitary UFCP code is designed by appropriately and uniquely factorizing a pair of energy-efficient cross quadrature amplitude modulation (QAM) constellations to maximize the coding gain subject to a transmission bit rate constraint. After a deep investigation of the fractional coding gain function, a technical approach developed in this thesis to maximizing the coding gain is to carefully design an energy scale to compress the first three largest energy points in the corner of the QAM constellations in the denominator of the objective as well as carefully design a constellation triple forming two UFCPs, with one collaborating with the other two so as to make the accumulated minimum Euclidean distance along the two transmitter antennas in the numerator of the objective as large as possible and at the same time, to avoid as many corner points of the QAM constellations with the largest energy as possible to achieve the minimum of the numerator. In other words, the optimal coding gain is attained by intelligent constellations collaboration and efficient energy compression. Another contribution of this thesis is to generalize the design for the two transmitter antennas into that of the noncoherent system with any even number of transmitter antennas. Using the Alamouti coding scheme and the Toeplitz matrix structure, a novel noncoherent nonunitary space-time block code, which is called an Alamoutibased Toeplitz space-time block code, is proposed. By the fundamentals of Galois theory and algebraic number theory, two important properties on the two Alamouti codes generated from a pair of coprime phase shift keying (PSK) constellations, i.e., the uniqueness of factorization itself and the shift-invariant uniqueness of factorization, are first revealed and rigorously proved. Then, it is further shown that it is these two kinds of the unique factorizations that enable the unique blind identification of both the channel coefficients and the transmitted signals by only processing two block received signals as well as noncoherent full diversity with a generalized likelihood ratio test (GLRT) receiver. In addition, a full diversity unitary code design is also proposed by simply applying the QR decomposition to the full diversity nonunitary Alamoutibased Toeplitz space-time block code. Computer simulations demonstrate that error performance of both optimal unitary UFCP code and Alamouti-based Toeplitz code presented in this thesis outperform those of the differential code and the SNR-efficient training code, which is the best code in current literatures for the system.</p> / Master of Applied Science (MASc)

This thesis addresses a number of problems all related to parameter estimation in sensor array processing. The unifying theme is that some of these parameters are known before the measurements are acquired. We thus study how to improve the estimation of the unknown parameters by incorporating the knowledge of the known parameters; exploiting this knowledge successfully has the potential to dramatically improve the accuracy of the estimates. For covariance matrix estimation, we exploit that the true covariance matrix is Kronecker and Toeplitz structured. We then devise a method to ascertain that the estimates possess this structure. Additionally, we can show that our proposed estimator has better performance than the state-of-art when the number of samples is low, and that it is also efficient in the sense that the estimates have Cram\'er-Rao lower Bound (CRB) equivalent variance. In the direction of arrival (DOA) scenario, there are different types of prior information; first, we study the case when the location of some of the emitters in the scene is known. We then turn to cases with additional prior information, i.e.~when it is known that some (or all) of the source signals are uncorrelated. As it turns out, knowledge of some DOA combined with this latter form of prior knowledge is especially beneficial, giving estimators that are dramatically more accurate than the state-of-art. We also derive the corresponding CRBs, and show that under quite mild assumptions, the estimators are efficient. Finally, we also investigate the frequency estimation scenario, where the data is a one-dimensional temporal sequence which we model as a spatial multi-sensor response. The line-frequency estimation problem is studied when some of the frequencies are known; through experimental data we show that our approach can be beneficial. The second frequency estimation paper explores the analysis of pulse spin-locking data sequences, which are encountered in nuclear resonance experiments. By introducing a novel modeling technique for such data, we develop a method for estimating the interesting parameters of the model. The technique is significantly faster than previously available methods, and provides accurate estimation results. / Denna doktorsavhandling behandlar parameterestimeringsproblem inom flerkanals-signalbehandling. Den gemensamma förutsättningen för dessa problem är att det finns information om de sökta parametrarna redan innan data analyseras; tanken är att på ett så finurligt sätt som möjligt använda denna kunskap för att förbättra skattningarna av de okända parametrarna. I en uppsats studeras kovariansmatrisskattning när det är känt att den sanna kovariansmatrisen har Kronecker- och Toeplitz-struktur. Baserat på denna kunskap utvecklar vi en metod som säkerställer att även skattningarna har denna struktur, och vi kan visa att den föreslagna skattaren har bättre prestanda än existerande metoder. Vi kan också visa att skattarens varians når Cram\'er-Rao-gränsen (CRB). Vi studerar vidare olika sorters förhandskunskap i riktningsbestämningsscenariot: först i det fall då riktningarna till ett antal av sändarna är kända. Sedan undersöker vi fallet då vi även vet något om kovariansen mellan de mottagna signalerna, nämligen att vissa (eller alla) signaler är okorrelerade. Det visar sig att just kombinationen av förkunskap om både korrelation och riktning är speciellt betydelsefull, och genom att utnyttja denna kunskap på rätt sätt kan vi skapa skattare som är mycket noggrannare än tidigare möjligt. Vi härleder även CRB för fall med denna förhandskunskap, och vi kan visa att de föreslagna skattarna är effektiva. Slutligen behandlar vi även frekvensskattning. I detta problem är data en en-dimensionell temporal sekvens som vi modellerar som en spatiell fler-kanalssignal. Fördelen med denna modelleringsstrategi är att vi kan använda liknande metoder i estimatorerna som vid sensor-signalbehandlingsproblemen. Vi utnyttjar återigen förhandskunskap om källsignalerna: i ett av bidragen är antagandet att vissa frekvenser är kända, och vi modifierar en existerande metod för att ta hänsyn till denna kunskap. Genom att tillämpa den föreslagna metoden på experimentell data visar vi metodens användbarhet. Det andra bidraget inom detta område studerar data som erhålls från exempelvis experiment inom kärnmagnetisk resonans. Vi introducerar en ny modelleringsmetod för sådan data och utvecklar en algoritm för att skatta de önskade parametrarna i denna modell. Vår algoritm är betydligt snabbare än existerande metoder, och skattningarna är tillräckligt noggranna för typiska tillämpningar. / <p>QC 20131115</p>

Markov-type inequalities provide upper bounds on the norm of the (higher order) derivative of an algebraic polynomial in terms of the norm of the polynomial itself. The present thesis considers the cases in which the norms are of the Laguerre, Gegenbauer, or Hermite type, with respective weights chosen differently on both sides of the inequality. An answer is given to the question on the best constant so that such an inequality is valid for every polynomial of degree at most n. The demanded best constant turns out to be the operator norm of the differential operator. The latter conicides with the tractable spectral norm of its matrix representation in an appropriate set of orthonormal bases. The methods to determine these norms vary tremendously, depending on the difference of the parameters accompanying the weights. Up to a very small gap in the parameter range, asymptotics for the best constant in each of the aforementioned cases are given. / Markovungleichungen liefern obere Schranken an die Norm einer (höheren) Ableitung eines algebraischen Polynoms in Bezug auf die Norm des Polynoms selbst. Diese vorliegende Arbeit betrachtet den Fall, dass die Normen vom Laguerre-, Gegenbauer- oder Hermitetyp sind, wobei die entsprechenden Gewichte auf beiden Seiten unterschiedlich gewählt werden. Es wird die kleinste Konstante bestimmt, sodass diese Ungleichung für jedes Polynom vom Grad höchstens n erfüllt ist. Die gesuchte kleinste Konstante kann als die Operatornorm des Differentialoperators dargestellt werden. Diese fällt aber mit der Spektralnorm der Matrixdarstellung in einem Paar geeignet gewählter Orthonormalbasen zusammen und kann daher gut behandelt werden. Zur Abschätzung dieser Normen kommen verschiedene Methoden zum Einsatz, die durch die Differenz der in den Gewichten auftretenden Parameter bestimmt werden. Bis auch eine kleine Lücke im Parameterbereich wird das asymptotische Verhalten der kleinsten Konstanten in jedem der betrachteten Fälle ermittelt.

Dans cette thèse, nous prouvons des résultats de théorie spectrale, directe et inverse, dans la limite semi-classique, pour les opérateurs de Toeplitz autoadjoints sur les surfaces. Pour les opérateurs pseudo-différentiels, les résultats en question sont déjà connus, et il est naturel de vouloir les étendre aux opérateurs de Toeplitz. Les conditions de Bohr-Sommerfeld usuelles, qui caractérisent les valeurs propres proches d'une valeur régulière du symbole principal, ont été obtenues il y a quelques années seulement pour les opérateurs de Toeplitz. Notre contribution consiste en l'extension de ces conditions près de valeurs critiques non dégénérées. Nous traitons le cas d'une valeur critique elliptique à l'aide d'une technique de forme normale ; l'opérateur modèle est la réalisation de l'oscillateur harmonique sur l'espace de Bargmann, dont le spectre est bien connu. Dans le cas d'une valeur critique hyperbolique, la forme normale ne suffit plus et nous complétons l'étude en faisant appel à des arguments dus à Colin de Verdière et Parisse, à qui l'on doit le résultat analogue dans le cas pseudo-différentiel. Enfin, nous établissons un résultat de théorie spectrale inverse pour les opérateurs de Toeplitz autoadjoints sur les surfaces ; plus précisément, nous montrons que sous certaines hypothèses génériques, la connaissance du spectre à l'ordre deux dans la limite semi-classique permet de retrouver le symbole principal à symplectomorphisme près. Ce résultat s'appuie en grande partie sur l'écriture des règles de Bohr-Sommerfeld.

Markov-type inequalities provide upper bounds on the norm of the (higher order) derivative of an algebraic polynomial in terms of the norm of the polynomial itself. The present thesis considers the cases in which the norms are of the Laguerre, Gegenbauer, or Hermite type, with respective weights chosen differently on both sides of the inequality. An answer is given to the question on the best constant so that such an inequality is valid for every polynomial of degree at most n. The demanded best constant turns out to be the operator norm of the differential operator. The latter conicides with the tractable spectral norm of its matrix representation in an appropriate set of orthonormal bases. The methods to determine these norms vary tremendously, depending on the difference of the parameters accompanying the weights. Up to a very small gap in the parameter range, asymptotics for the best constant in each of the aforementioned cases are given. / Markovungleichungen liefern obere Schranken an die Norm einer (höheren) Ableitung eines algebraischen Polynoms in Bezug auf die Norm des Polynoms selbst. Diese vorliegende Arbeit betrachtet den Fall, dass die Normen vom Laguerre-, Gegenbauer- oder Hermitetyp sind, wobei die entsprechenden Gewichte auf beiden Seiten unterschiedlich gewählt werden. Es wird die kleinste Konstante bestimmt, sodass diese Ungleichung für jedes Polynom vom Grad höchstens n erfüllt ist. Die gesuchte kleinste Konstante kann als die Operatornorm des Differentialoperators dargestellt werden. Diese fällt aber mit der Spektralnorm der Matrixdarstellung in einem Paar geeignet gewählter Orthonormalbasen zusammen und kann daher gut behandelt werden. Zur Abschätzung dieser Normen kommen verschiedene Methoden zum Einsatz, die durch die Differenz der in den Gewichten auftretenden Parameter bestimmt werden. Bis auch eine kleine Lücke im Parameterbereich wird das asymptotische Verhalten der kleinsten Konstanten in jedem der betrachteten Fälle ermittelt.

No description available.

The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwell’s equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwell’s equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.