Structural analysis of highway bridges using complicated 3-D FEMs to determine live load effects in bridge components is possible due to the readily available computational tools in design offices. However, building such complicated 3-D FEMs is tedious and time consuming. Accordingly, most design engineers prefer using simplified 2-D structural models of the bridge and live load distribution equations (LLDEs) available in current bridge design codes to determine live load effects in bridge components. Basically, the live load effect obtained from a 2-D model is multiplied by a factor obtained from the LLDE to calculate the actual live load effect in a 3-D structure. The LLDE available in current bridge design codes for jointed bridges were also used for the design of straight and skewed integral bridges by bridge engineers. As a result, these bridges are either designed conservatively leading to additional construction cost or unconservatively leading to unsafe bridge designs. Recently, LLDEs for integral bridges (IBs) with no skew are developed. To use these equations for skewed integral bridges (SIBs) a correction factor is needed to multiply these equations to include the effect of skew. Consequently, in this research study, skew correction factors for SIBs are developed. For this purpose, finite element models of 231 different three dimensional and corresponding two dimensional structural models of SIBs are built and analyzed under live load. The analyses results reveal that the effect of skew on the distribution of live load moment and shear is significant. It is also observed that skew generally tends to decrease live load effects in girders and substructure components of SIBs. Using the analyses results, analytical equations are developed via nonlinear regression techniques to include skew effects in the LLDEs developed for straight IBs. The developed skew correction factors are compared with FEAs results. This comparison revealed that the developed skew correction factors yield a reasonably good estimate of the reduction in live load effects due to the effect of skew.

My dissertation mainly focuses on the regularization of parameters in the multivariate linear regression under different assumptions on the distribution of the errors. It consists of two topics where we develop iterative procedures to construct sparse estimators for both the regression coefficient and scale matrices simultaneously, and a third topic where we develop a method for testing if the skewness parameter in the skew-normal distribution is parallel to one of the eigenvectors of the scale matrix. In the first project, we propose a robust procedure for constructing a sparse estimator of a multivariate regression coefficient matrix that accounts for the correlations of the response variables. Robustness to outliers is achieved using heavy-tailed t distributions for the multivariate response, and shrinkage is introduced by adding to the negative log-likelihood l1 penalties on the entries of both the regression coefficient matrix and the precision matrix of the responses. Taking advantage of the hierarchical representation of a multivariate t distribution as the scale mixture of normal distributions and the EM algorithm, the optimization problem is solved iteratively where at each EM iteration suitably modified multivariate regression with covariance estimation (MRCE) algorithms proposed by Rothman, Levina and Zhu are used. We propose two new optimization algorithms for the penalized likelihood, called MRCEI and MRCEII, which differ from MRCE in the way that the tuning parameters for the two matrices are selected. Estimating the degrees of freedom when penalizing the entries of the matrices presents new computational challenges. A simulation study and real data analysis demonstrate that the MRCEII, which selects the tuning parameter of the precision matrix of the multiple responses using the Cp criterion, generally does the best among all methods considered in terms of the prediction error, and MRCEI outperforms the MRCE methods when the regression coefficient matrix is less sparse. The second project is motivated by the existence of the skewness in the data for which the symmetric distribution assumption on the errors does not hold. We extend the procedure we have proposed to the case where the errors in the multivariate linear regression follow a multivariate skew-normal or skew-t distribution. Based on the convenient representation of skew-normal and skew-t as well as the EM algorithm, we develop an optimization algorithm, called MRST, to iteratively minimize the negative penalized log-likelihood. We also carry out a simulation study to assess the performance of the method and illustrate its application with one real data example. In the third project, we discuss the asymptotic distributions of the eigenvalues and eigenvectors for the MLE of the scale matrix in a multivariate skew-normal distribution. We propose a statistic for testing whether the skewness vector is proportional to one of the eigenvectors of the scale matrix based on the likelihood ratio. Under the alternative, the likelihood is maximized numerically with two different ways of parametrization for the scale matrix: Modified Cholesky Decomposition (MCD) and Givens Angle. We conduct a simulation study and show that the statistic obtained using Givens Angle parametrization performs well and is more reliable than that obtained using MCD.

If a known distribution on a real line is given, it can be wrapped on the circumference of a unit circle. This research entails the study of a univariate skew-normal distribution where the skew-normal distribution is generalised for the case of bimodality. Both the skew-normal and exible generalised skew-normal distributions are wrapped onto a unit circle, consequently referred to as a wrapped skew-normal and a wrapped exible generalised skew-normal distribution respectively. For each of these distributions a simulation study is conducted, where the performance of maximum likelihood estimation is evaluated. Skew scale mixtures of normal distributions with the wrapped version of these distributions are proposed and graphical representations are provided. These distributions are also compared in an application to wind direction data. / Dissertation (MSc)--University of Pretoria, 2019. / Statistics / MSc / Unrestricted

We define and study multivariate and matrix variate skew Pearson type VII and skew t-distributions. We derive the marginal and conditional distributions, the linear transformation, and the stochastic representations of the multivariate and matrix variate skew Pearson type VII distributions and skew t-distributions. Also, we study the limiting distributions.

Matrix variate skew elliptically contoured distributions generalize several classes of important distributions. This paper defines and explores matrix variate skew elliptically contoured distributions. In particular, we discuss the first two moments of the matrix variate skew elliptically contoured distributions.

Master of Science / Department of Statistics / Paul I. Nelson / Hotelling’s T2 test is the standard tool for inference about the mean of a multivariate normal population. However, this test may perform poorly when used on samples from multivariate distributions with highly skewed marginal distributions. The goal of our study was to investigate the type I error rate and power properties of Hotelling’s one sample test when sampling from a class of multivariate skew-normal (SN) distributions, which includes the multivariate normal distribution and, in addition to location and scale parameters, has a shape parameter to regulate skewness. Simulation results of tests carried out at nominal type I error rate 0.05 obtained from various levels of shape parameters, sample sizes, number of variables and fixed correlation matrix showed that Hotelling’s one sample test provides adequate control of type I error rates over the entire range of conditions studied. The test also produces suitable power levels for detecting departures from hypothesized values of a multivariate mean vector when data result from a random sample from a multivariate SN. The shape parameter of the SN family appears not to have much of an effect on the robustness of Hotelling’s test. However, surprisingly, it does have a positive impact on power.

This thesis examines how novel invariants called the "stability index" as proposed by Podvigina and Ashwin can be used to characterize the local geometry of riddled basins of attraction for both skew and non-skew product systems. In particular, it would be interesting to understand how the stability index behaves on the basin boundary between multiple basins of attraction. Then we can ask this question: How can we identify when a basin is riddled? To answer this, we present three models with the presence of riddled basins. In the first model, we present a skew product system of a simple example of a piecewise linear map. We prove that the riddled basin occurs within a certain range of parameter and calculate the stability index analytically for this map. Our results for the stability index at a point show that for Lebesgue almost all points in the map, the index is positive and for some points the index may be negative. We verify these results with our numerical computation for this index. We also make a corollary claiming that the formula for the stability index at a point can be expressed in terms of the stability index for an attractor and Lyapunov exponents for this map. This suggests that this index could be useful as a diagnostic tool to study bifurcation of the riddled basins of attraction. In the second model, we refer to a skew product map studied by Keller. Previously, Keller computed the stability index for an attractor in his map whereas in this thesis, we use an alternative way to compute the index; that is on the basins of attraction for Keller's map, found by inverting his map. Using the same map, we also verify maximum and minimum measures as obtained in his paper by studying Birkhoff averages on periodic points of Markov map in his system. We also conjecture result by Keller and Otani on the dimension of zero sets of invariant graph (i.e. basin boundary) that appears in Keller's map to a complete range of a parameter in the map. The last model is a non-skew product map which is also has a riddled basin. For this map, we compute the stability index for an attractor on the baseline of the map. The result indicates that the index is positive for Lebesgue almost all points whenever the riddled basin occurs.

The main purpose of this thesis is to propose the valuation method of PX index options. PX index consists of blue chip stocks traded on Prague Stock Exchange. There are traded a few futures contracts on PX index on Prague Stock Exchange. However, the options on PX index are traded neither on Prague Stock Exchange nor on the OTC market. It is reasonable to think that it is only question of time when the trading of these options will emerge and thus, it is highly relevant subject of research to propose the method for valuation of these options. The traditional Merton's approach for valuation of equity index options assumes constant volatility and constant risk free rate. This results in serious mispricing which can be easily seen when we compare market prices and Merton formula derived prices. Instead, this thesis releases the assumptions of constant risk free rate and constant volatility. Firstly, it is assumed that that the risk free rate is time dependent function based on current market expectations and secondly it is assumed that the volatility of underlying asset follows NGARCH-mean process. For the purpose of former, the validity of pure expectation theory assumption is made. This enables to employ the instantaneous forward rate curve estimation procedure. For the purpose of the latter, the locally risk-neutral valuation relationship is applied. The assumption of NGARCH-mean process is essential in an effort to capture usually observed patterns of volatility (volatility skews) whereas the assumption of time dependent risk free rate still moves the valuation option model closer to the reality. The author derives the expected path of risk free rate and estimates the parameters of NGARCH process. Subsequently, the empirical martingale Monte Carlo simulation is used to price the PX options with different moneyness and with different times to maturity. It is shown that this proposed model results in volatility pattern which is usually observed on developed markets and the author's results are in line with similar empirical studies testing the GARCH Option Pricing Theory. The author concludes that proposed valuation method superiors original Merton's model and thus is more appropriate for primary valuation of PX options.

O presente trabalho se propôs a estudar os esquemas estruturais alternativos para pontes esconsas e avaliar as modelagens matemáticas possíveis através de softwares especializados, como o SAP2000 e STRAP2010, a fim de identificar as melhores soluções e modelos para cada caso. O objetivo foi o de aprimorar os modelos de cálculo, especialmente no que diz respeito à previsão das reações de apoio e das forças cortantes, ponto mais delicado de obras esconsas. O estudo baseou-se em soluções de superestrutura em laje e em grelha, com ou sem transversinas, com esconsidade variável entre zero e sessenta graus, eixo longitudinal reto, modeladas por barras e elementos finitos. Não foi objeto deste estudo a influência da meso e da infraestrutura dessas pontes nos esforços na superestrutura. / The present work is proposed to study alternative structural schemes for skew bridges and to evaluate possible mathematical modeling through specialized software, such as SAP 2000 and STRAP2010, in order to identify the best solutions and models for each case. The aim was to improve the calculation models, especially with regard to the prediction of the support reactions and shear forces, most delicate point in skew bridges. The study was based on slab or grid, with or without transversal beams, superstructure solutions, with variable skew between zero and sixty degrees, straight longitudinal axis, modeled through bars and shell elements. It was not purpose of this study the influence of meso and infrastructure of the bridge on its superstructure.

In this paper, we prove that the joint distribution of random vectors Z 1 and Z 2 and the distribution of Z 2 are skew normal provided that Z 1 is skew normally distributed and Z 2 conditioning on Z 1 is distributed as closed skew normal. Also, we extend the main results to the matrix variate case.