In this thesis, we study re-weighting when estimating totals in survey sampling. The purpose of re-weighting is to adjust the structure of the sample in order to comply with the structure of the population (with respect to given auxiliary variables). We sum up some known results for methods of the traditional desin-based approach, more attention is given to the model-based approach. We generalize known asymptotic results in the model-based theory to a wider class of weighted estimators. Further, we propose a consistent estimator of asymptotic variance, which takes into consideration weights used in estimator of the total. This is in contrast to usually recommended variance estimators derived from the design-based approach. Moreover, the estimator is robust againts particular model misspecifications. In a simulation study, we investigate how the proposed estimator behaves in comparison with variance estimators which are usually recommended in the literature or used in practice. 1

abstract: Distributed inference has applications in fields as varied as source localization, evaluation of network quality, and remote monitoring of wildlife habitats. In this dissertation, distributed inference algorithms over multiple-access channels are considered. The performance of these algorithms and the effects of wireless communication channels on the performance are studied. In a first class of problems, distributed inference over fading Gaussian multiple-access channels with amplify-and-forward is considered. Sensors observe a phenomenon and transmit their observations using the amplify-and-forward scheme to a fusion center (FC). Distributed estimation is considered with a single antenna at the FC, where the performance is evaluated using the asymptotic variance of the estimator. The loss in performance due to varying assumptions on the limited amounts of channel information at the sensors is quantified. With multiple antennas at the FC, a distributed detection problem is also considered, where the error exponent is used to evaluate performance. It is shown that for zero-mean channels between the sensors and the FC when there is no channel information at the sensors, arbitrarily large gains in the error exponent can be obtained with sufficient increase in the number of antennas at the FC. In stark contrast, when there is channel information at the sensors, the gain in error exponent due to having multiple antennas at the FC is shown to be no more than a factor of 8/π for Rayleigh fading channels between the sensors and the FC, independent of the number of antennas at the FC, or correlation among noise samples across sensors. In a second class of problems, sensor observations are transmitted to the FC using constant-modulus phase modulation over Gaussian multiple-access-channels. The phase modulation scheme allows for constant transmit power and estimation of moments other than the mean with a single transmission from the sensors. Estimators are developed for the mean, variance and signal-to-noise ratio (SNR) of the sensor observations. The performance of these estimators is studied for different distributions of the observations. It is proved that the estimator of the mean is asymptotically efficient if and only if the distribution of the sensor observations is Gaussian. / Dissertation/Thesis / Ph.D. Electrical Engineering 2010

Available statistical methodologies focus more on accommodating continuous variables, however recently dealing with count data has received high interest in the statistical literature. In this dissertation, we propose some statistical approaches to investigate linear and nonlinear dependencies between two discrete random variables, or between a discrete and continuous random variables. Copula functions are powerful tools for modeling dependencies between random variables. We derive copula-based population version of Spearman’s rho when at least one of the marginal distribution is discrete. In each case, the functional relationship between Kendall’s tau and Spearman’s rho is obtained. The asymptotic distributions of the proposed estimators of these association measures are derived and their corresponding confidence intervals are constructed, and tests of independence are derived. Then, we propose a Bayesian copula factor autoregressive model for time series mixed data. This model assumes conditional independence and shares latent factors in both mixed-type response and multivariate predictor variables of the time series through a quadratic timeseries regression model. This model is able to reduce the dimensionality by accommodating latent factors in both response and predictor variables of the high-dimensional time series data. A semiparametric time series extended rank likelihood technique is applied to the marginal distributions to handle mixed-type predictors of the high-dimensional time series, which decreases the number of estimated parameters and provides an efficient computational algorithm. In order to update and compute the posterior distributions of the latent factors and other parameters of the models, we propose a naive Bayesian algorithm with Metropolis-Hasting and Forward Filtering Backward Sampling methods. We evaluate the performance of the proposed models and methods through simulation studies. Finally, each proposed model is applied to a real dataset.

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Le point de départ de cette thèse est la conjecture de Bieberbach : sa démonstration par De Branges utilise deux ingrédients, à savoir la théorie de Loewner des domaines plans croissants et une inégalité de Milin qui concerne les coefficients logarithmiques. Nous commençons par étudier les coefficients logarithmiques du whole-plane SLE en utilisant une méthode combinatoire, assistée par ordinateur. Nous retrouvons les résultats en utilisant une équation aux dérivées partielles analogue à celle obtenue par Beliaev et Smirnov. Nous généralisons ces résultats en définissant le spectre généralisé du whole-plane SLE, que nous calculons par la même méthode, à savoir en dérivant, par le calcul d’Itô, une EDP parabolique satisfaite par les quantités que nous moyennons. Cette famille à deux paramètres d’EDP admet une riche structure algébrique que nous étudions en détail. La dernière partie de la thèse concerne l’opérateur de Grunsky et ses généralisations. Plus expérimentale, nous y mettons à jour, grâce à un logiciel de calcul formel, une structure assez complexe dont nous avons commencé l’exploration. / The starting point of this thesis is Bieberbach’s conjecture: its proof, given by De Branges, uses two ingredients, namely Loewner’s theory of increasing plane domains and an inequality from Milin about the logarithmic coefficients. We start with a study of the logarithmic coefficients of the whole-plane SLE by using a combinatorial method, assisted by computer. We find the results by using a partial differential equation similar to that obtained by Beliaev and Smirnov. We generalize these results by defining the generalized spectrum of the whole-plane SLE, that we calculate by the same method, namely by deriving, thanks to Itô calculus, a parabolic PDE satisfied by the quantities of which we take the average. This two-parameter family of PDEs admits a rich algebraic structure that we study in detail. The last part of this thesis is about the Grunsky operator and its generalizations. In this part that is more experimental we update, thanks to a computer algebra system, a rather complex structure of which we began the exploration.

Soit f une instance du whole-plane $\SLE_\kappa$ : on sait que pour certaines valeurs de κ, p les moments dérivés $\mathbb{E}(\vert f'(z) \vert^p)$ peuvent être écrits sous une forme fermée, étude qui a permis de mettre au jour une nouvelle phase du spectre des moyennes intégrales. Le but de cette thèse est une étude des moments généralisés $\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}$ : cette étude permet de confirmer la structure algébrique riche du whole-plane SLE. On montre que les formes fermées des moments mixtes $\mathbb{E}\big(\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}\big)$ apparaissent sur une famille dénombrable de paraboles du plan (p, q), en étendant les équations de Beliaev-Smirnov à ce cas. Nous introduisons également le spectre généralisé β(p, q; κ), correspondant au comportement asymptotiques des moyennes intégrales mixtes. Le spectre généralisé moyen du whole-plane SLE prend quatre formes possibles, séparés par cinq séparatrices dans $\R^2$. Nous proposons également une approche semblable pour la dérivée Schwarziene S(f)(z) de l’application de SLE. Les calculs sur les équations de Beliaev-Smirnov d’une certaine générale forme de moment mène à une formulation explicite de $\mathbb{E}(S(f)(z))$ . Nous étudions finalement la variance asymptotique de McMullen et démontrons une relation entre la croissance infinitésimale du spectre de la moyenne intégrale et la variance asymptotique pour SLE₂. / Let f an instance of the whole-plane $\SLE_\kappa$ conformal map from the unit disk D to the slit plane: We know that for certain values of κ, p the derivative moments $\mathbb{E}(\vert f'(z) \vert^p)$ can be written in a closed form, study that has updated a new phase of the integral means spectrum. The goal of this thesis is a study on generalized moments $\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}$ : ΒββThis study permit confirm the rich algebraic structure of the whole-plane version of SLE. It will be showed that closed forms of the mixed moments E mixtes $\mathbb{E}\big(\frac{\vert f'(z) \vert^p}{\vert f(z) \vert^q}\big)$ can be obtained on a countable family of parabolas in the moment plane (p, q), by extending the so-called Beliaev–Smirnov equation to this case. We also introduce the generalized integral means spectrum, β(p, q; κ), corresponding to the singular behavior of the mixed moments. The average generalized spectrum of whole-plane SLE takes four possible forms, separated by five phase transition lines in $\R^2$. We also propose a similar approach for the Schwarzian derivative S(f)(z) of SLE maps. Computations on the Beliaev–Smirnov equation of a certain general form of moment lead to an explicit formula of $\mathbb{E}(S(f)(z))$ . We finally study the McMullen asymptotic variance and prove a relation between the infinitesimal growth of the integral mean spectrum and the asymptotic variance in an expectation sense for SLE₂.

Les méthodes de Monte Carlo par chaînes de Markov (MCMC) utilisent généralement des chaînes de Markov réversibles. Jusqu’à récemment, une grande partie de la recherche théorique sur les chaînes de Markov concernait ce type de chaînes, notamment les théorèmes de Peskun (1973) et de Tierney (1998) qui permettent d’ordonner les variances asymptotiques de deux estimateurs issus de chaînes réversibles différentes. Dans ce mémoire nous analysons des algorithmes simulants des chaînes qui ne respectent pas cette condition. Nous parlons alors de chaînes non réversibles. Expérimentalement, ces chaînes produisent souvent des estimateurs avec une variance asymptotique plus faible et/ou une convergence plus rapide. Nous présentons deux algorithmes, soit l’algorithme de marche aléatoire guidée (GRW) par Gustafson (1998) et l’algorithme de discrete bouncy particle sampler (DBPS) par Sherlock et Thiery (2017). Pour ces deux algorithmes, nous comparons expérimentalement la variance asymptotique d’un estimateur avec la variance asymptotique en utilisant l’algorithme de Metropolis-Hastings. Récemment, un cadre théorique a été introduit par Andrieu et Livingstone (2019) pour ordonner les variances asymptotiques d’une certaine classe de chaînes non réversibles. Nous présentons leur analyse de GRW. De plus, nous montrons que le DBPS est inclus dans ce cadre théorique. Nous démontrons que la variance asymptotique d’un estimateur peut théoriquement diminuer en ajoutant des propositions à cet algorithme. Finalement, nous proposons deux modifications au DBPS. Tout au long du mémoire, nous serons intéressés par des chaînes issues de propositions déterministes. Nous montrons comment construire l’algorithme du delayed rejection avec des fonctions déterministes et son équivalent dans le cadre de Andrieu et Livingstone (2019). / Markov chain Monte Carlo (MCMC) methods commonly use chains that respect the detailed balance condition. These chains are called reversible. Most of the theory developed for MCMC evolves around those particular chains. Peskun (1973) and Tierney (1998) provided useful theorems on the ordering of the asymptotic variances for two estimators produced by two different reversible chains. In this thesis, we are interested in non-reversible chains, which are chains that don’t respect the detailed balance condition. We present algorithms that simulate non-reversible chains, mainly the Guided Random Walk (GRW) by Gustafson (1998) and the Discrete Bouncy Particle Sampler (DBPS) by Sherlock and Thiery (2017). For both algorithms, we compare the asymptotic variance of estimators with the ones produced by the Metropolis- Hastings algorithm. We present a recent theoretical framework introduced by Andrieu and Livingstone (2019) and their analysis of the GRW. We then show that the DBPS is part of this framework and present an analysis on the asymptotic variance of estimators. Their main theorem can provide an ordering of the asymptotic variances of two estimators resulting from nonreversible chains. We show that an estimator could have a lower asymptotic variance by adding propositions to the DBPS. We then present empirical results of a modified DBPS. Through the thesis we will mostly be interested in chains that are produced by deterministic proposals. We show a general construction of the delayed rejection algorithm using deterministic proposals and one possible equivalent for non-reversible chains.