Doctor of Philosophy / Department of Statistics / James Neill / The problem of testing for lack of fit in exponential family regression models is considered. Such nonlinear models are the natural extension of Normal nonlinear regression models and generalized linear models. As is usually the case, inadequately specified models have an adverse impact on statistical inference and scientific discovery. Models of interest are curved exponential families determined by a sequence of predictor settings and mean regression function, considered as a sub-manifold of the full exponential family. Constructed general alternative models are based on clusterings in the mean parameter components and allow likelihood ratio testing for lack of fit associated with the mean, equivalently natural parameter, for a proposed null model. A maximin clustering methodology is defined in this context to determine suitable clusterings for assessing lack of fit. In addition, a geometrically motivated goodness of fit test statistic for exponential family regression based on the information metric is introduced. This statistic is applied to the cases of logistic regression and Poisson regression, and in both cases it can be seen to be equal to a form of the Pearson chi[superscript]2 statistic. This same statement is true for multinomial regression. In addition, the problem of testing for equal means in a heteroscedastic Normal model is discussed. In particular, a saturated 3 parameter exponential family model is developed which allows for equal means testing with unequal variances. A simulation study was carried out for the logistic and Poisson regression models to investigate comparative performance of the likelihood ratio test, the deviance test and the goodness of fit test based on the information metric. For logistic regression, the Hosmer-Lemeshow test was also included in the simulations. Notably, the likelihood ratio test had comparable power with that of the Hosmer-Lemeshow test under both m- and n-asymptotics, with superior power for constructed alternatives. A distance function defined between densities and based on the information metric is also given. For logistic models, as the natural parameters go to plus or minus infinity, the densities become more and more deterministic and limits of this distance function are shown to play an important role in the lack of fit analysis. A further simulation study investigated the power of a likelihood ratio test and a geometrically derived test based on the information metric for testing equal means in heteroscedastic Normal models.

In this dissertation, we study etale correspondence of hyperbolic curves with unbounded dynamics. Mochizuki proved that over a field of characteristic 0, such curves are always Shimura curves. We explore variants of this question in positive characteristic, using graph theory, l-adic local systems, and Barsotti-Tate groups. Given a correspondence with unbounded dynamics, we construct an infinite graph with a large group of ”algebraic” automorphisms and roughly measures the ”generic dynamics” of the correspondence. We construct a specialization map to a graph representing the actual dynamics. Along the way, we formulate conjectures that etale correspondences with unbounded dynamics behave similarly to Hecke correspondences of Shimura curves. Using graph theory, we show that type (3,3) etale correspondences verify various parts of this philosophy. Key in the second half of this dissertation is a recent p-adic Langlands correspondence, due to Abe, which answers affirmatively the petites camarades conjecture of Deligne in the case of curves. This allows us the build a correspondence between rank 2 l-adic local systems with trivial determinant and Frobenius traces in Q and certain height 2, dimension 1 Barsotti-Tate groups. We formulate a conjecture on the fields of definitions of certain compatible systems of l-adic representations. Relatedly, we conjecture that the Barsotti-Tate groups over complete curves in positive characteristic may be ”algebraized” to abelian schemes.

This thesis will take a look at the roots of modern-day information geometry and some applications into statistical modeling. In order to truly grasp this field, we will first provide a basic and relevant introduction to differential geometry. This includes the basic concepts of manifolds as well as key properties and theorems. We will then explore exponential families with applications of probability distributions. Finally, we select a few time series models and derive the underlying geometries of their manifolds.

Wave propagation in unbounded media is a topic widely studied in different science and engineering fields. Global and local absorbing boundary conditions combined with the finite element method or the finite difference method are the usual numerical treatments. In this dissertation, an alternative is investigated based on the dynamic stiffness and the exponential window method in the space-wave number domain. Applying the exponential window in the space-wave number domain is equivalent to introducing fictitious damping into the system. The Discrete Fourier Transform employed in the dynamic stiffness can be properly performed in a damped system. An open boundary in space is thus created. Since the equation is solved by the finite difference formula in the time domain, this approach is in the time-wave number domain, which provides a complement for the original dynamic stiffness method, which is in the frequency-wave number domain. The approach is tested through different elasto-dynamic models that cover one-, two- and three-dimensional problems. The results from the proposed approach are compared with those from either analytical solutions or the finite element method. The comparison demonstrates the effectiveness of the approach. The incident waves can be efficiently absorbed regardless of incident angles and frequency contents. The approach proposed in this dissertation can be widely applied to the dynamics of railways, dams, tunnels, building and machine foundations, layered soil and composite materials.

Thesis (Ph.D.) -- McMaster University, 1999. / Includes bibliographical references (leaves 76-79). Also available via World Wide Web.

Thesis(M.S.)--Auburn University, 2004. / Abstract. Vita. Includes bibliographic references (p.71-72).

Thesis (doctoral)--Rheinischen Friedrich-Wilhelms-Universität, Bonn, 1980. / Includes bibliographical references (p. 252-258).

Thesis (doctoral)--Kath. Universiẗat, Eichstätt, 2003.

Made available in DSpace on 2014-06-12T15:53:07Z (GMT). No. of bitstreams: 2 arquivo4927_1.pdf: 1466209 bytes, checksum: 2dd8cd7b46b828a5aa1d2a3f50a6ebef (MD5) license.txt: 1748 bytes, checksum: 8a4605be74aa9ea9d79846c1fba20a33 (MD5) Previous issue date: 2001 / Conselho Nacional de Desenvolvimento Científico e Tecnológico / Cabral Mota, Alexandre; Cezar Alves Sampaio, Augusto. Model checking CSPZ: Techniques to overcome state explosion. 2001. Tese (Doutorado). Programa de Pós-Graduação em Ciência da Computação, Universidade Federal de Pernambuco, Recife, 2001.

Let Zp be the ring of p-adic integers and Ep be the map x-->exp(px) where exp denotes the exponential map determined by the usual power series. It defines an exponential ring (Zp, + , . , 0, 1, Ep). The goal of the thesis is to study the model theory of this structure. In particular, we are interested by the question of the decidability of this theory. The main theorem of the thesis is: Theorem: If the p-adic Schanuel's conjecture is true, then the theory of (Zp, + , . , 0, 1, Ep) is decidable. The proof involves: 1- A result of effective model-completeness (chapters 3 and 4): If F is a family of restricted analytic functions (i.e. power series with coefficients in the valuation ring and convergent on Zp) closed under decomposition functions and such that the set of terms in the language LF= (+, . , 0, 1, f; f in F) is closed under derivation, then we prove that the theory of Zp in the language LF is model-complete. And furthermore, if each term of LF has an effective Weierstrass bound, then the model-completeness is effective. 2- A resolution of the decision problem for existential formulas (assuming Schanuel's conjecture) in chapter 5. We also consider the problem of the decidability of the structure (Op, + , . , 0, 1, |, E_p) where Op denotes the valuation ring of Cp. We give a positive answer to this question assuming the p-adic Schanuel's conjecture.