The aim of this study is to explore, through rankings, tolerance towards morally debatable situations. The sample comprised 254 psychology college students (male and female) in Lima, 21-year old on average. The instrument used was the Morally Debatable Behaviors Scale. It is concluded that there is a similarity in the rankings by gender and age. In terms of religion, there are differences with the group of Evangelists, who are less tolerant than Catholics regarding pre-marital sexual relations. The impact of these results in understanding the tendency to action in terms of morality is discussed. / El objetivo del presente estudio es explorar la tolerancia ante situaciones moralmente controversiales mediante rankings. La muestra estuvo conformada por 254 estudiantes universitarios de la carrera de psicología, entre varones y mujeres, con una edad promedio de 21 años, de la ciudad de Lima. El instrumento utilizado fue la Escala de Comportamientos Morales Controvertidos. Se concluye que existe similitud, según género y edad, en relación a los rankings de los comportamientos morales controvertidos. En cuanto a la religión, se encontraron diferencias con el grupo evangélico, quienes resultan ser menos tolerantes en cuanto a las relaciones prematrimoniales; en comparación a los católicos. Discutimos el impacto de estos resultados en la comprensión de la tendencia a la acción en el plano de la moralidad.

This thesis deals with a finding ideal transformation which can model data well. We focus on transformations which we know up to a parametr. We need to estimate the parametr of the transformation. The main approach of study transformation is in linear regression and in nonparametric regression. In both cases we focus on estimating the transformation parametr and properties of this estimator such as consistency and asymptotic normality. We show in linear regression that the aprroach of least squares do not work properly. Instead of this we use a generalized moment method which can estimate parametr of transformation and also a regression coefficients. We show also a different solution for our problem in specific transformation called Box-Cox. For this situation we make a simulation study for estimators and standard deviations. The standard deviation are obtained by bootstrap method. In nonparametric regression we use profile likelihood to estimate transformation parametr. We also construct an estimator of density of error terms. In both cases we know the asymptotic distribution.

Profile monitoring is a relatively new approach in quality control best used when the process data follow a profile (or curve). The majority of previous studies in profile monitoring focused on the parametric modeling of either linear or nonlinear profiles under the assumption of the correct model specification. Our work considers those cases where the parametric model for the family of profiles is unknown or, at least uncertain. Consequently, we consider monitoring Poisson profiles via three methods, a nonparametric (NP) method using penalized splines, a nonparametric (NP) method using wavelets and a semi parametric (SP) procedure that combines both parametric and NP profile fits. Our simulation results show that SP method is robust to the common problem of model misspecification of the user's proposed parametric model. We also showed that Haar wavelets are a better choice than the penalized splines in situations where a sudden jump happens or the jump is edgy. In addition, we showed that the penalized splines are better than wavelets when the shape of the profiles are smooth. The proposed novel techniques have been applied to a real data set and compare with some state-of-the arts.

Doctor of Philosophy / Department of Statistics / Haiyan Wang / This thesis introduces a new partitioning algorithm to cluster variables in high dimensional low sample size (HDLSS) data and high dimensional longitudinal low sample size (HDLLSS) data. HDLSS data contain a large number of variables with small number of replications per variable, and HDLLSS data refer to HDLSS data observed over time. Clustering technique plays an important role in analyzing high dimensional low sample size data as is seen commonly in microarray experiment, mass spectrometry data, pattern recognition. Most current clustering algorithms for HDLSS and HDLLSS data are adaptations from traditional multivariate analysis, where the number of variables is not high and sample sizes are relatively large. Current algorithms show poor performance when applied to high dimensional data, especially in small sample size cases. In addition, available algorithms often exhibit poor clustering accuracy and stability for non-normal data. Simulations show that traditional clustering algorithms used in high dimensional data are not robust to monotone transformations. The proposed clustering algorithm PPCLUST is a powerful tool for clustering HDLSS data, which uses p-values from nonparametric rank tests of homogeneous distribution as a measure of similarity between groups of variables. Inherited from the robustness of rank procedure, the new algorithm is robust to outliers and invariant to monotone transformations of data. PPCLUSTEL is an extension of PPCLUST for clustering of HDLLSS data. A nonparametric test of no simple effect of group is developed and the p-value from the test is used as a measure of similarity between groups of variables. PPCLUST and PPCLUSTEL are able to cluster a large number of variables in the presence of very few replications and in case of PPCLUSTEL, the algorithm require neither a large number nor equally spaced time points. PPCLUST and PPCLUSTEL do not suffer from loss of power due to distributional assumptions, general multiple comparison problems and difficulty in controlling heterocedastic variances. Applications with available data from previous microarray studies show promising results and simulations studies reveal that the algorithm outperforms a series of benchmark algorithms applied to HDLSS data exhibiting high clustering accuracy and stability.

Doctor of Philosophy / Department of Statistics / Haiyan Wang / Various tests have been created to compare the means of two populations in many scenarios and applications. The two-sample t-test, Wilcoxon Rank-Sum Test and bootstrap-t test are commonly used methods. However, methods for skewed two-sample data set are not well studied. In this dissertation, several existing two sample tests were evaluated and four new tests were proposed to improve the test accuracy under moderate sample size and high population skewness. The proposed work starts with derivation of a first order Edgeworth expansion for the test statistic of the two sample t-test. Using this result, new two-sample tests based on Cornish Fisher expansion (TCF tests) were created for both cases of common variance and unequal variances. These tests can account for population skewness and give more accurate test results. We also developed three new tests based on three transformations (T[subscript i] test, i = 1; 2; 3) for the pooled case, which can be used to eliminate the skewness of the studentized statistic. In this dissertation, some theoretical properties of the newly proposed tests are presented. In particular, we derived the order of type I error rate accuracy of the pooled two-sample t-test based on normal approximation (TN test), the TCF and T[subscript i] tests. We proved that these tests give the same theoretical type I error rate under skewness. In addition, we derived the power function of the TCF and TN tests as a function of the population parameters. We also provided the detailed conditions under which the theoretical power of the two-sample TCF test is higher than the two-sample TN test. Results from extensive simulation studies and real data analysis were also presented in this dissertation. The empirical results further confirm our theoretical results. Comparing with commonly used two-sample parametric and nonparametric tests, our new tests (TCF and Ti) provide the same empirical type I error rate but higher power.

Dans cette thèse, on considère la modélisation de la loi jointe des statistiques d'ordre, c.à.d. des vecteurs aléatoires avec des composantes ordonnées presque sûrement. La première partie est dédiée à la modélisation probabiliste des statistiques d'ordre d'entropie maximale à marginales fixées. Les marginales étant fixées, la caractérisation de la loi jointe revient à considérer la copule associée. Dans le Chapitre 2, on présente un résultat auxiliaire sur les copules d'entropie maximale à diagonale fixée. Une condition nécessaire et suffisante est donnée pour l'existence d'une telle copule, ainsi qu'une formule explicite de sa densité et de son entropie. La solution du problème de maximisation d'entropie pour les statistiques d'ordre à marginales fixées est présentée dans le Chapitre 3. On donne des formules explicites pour sa copule et sa densité jointe. On applique le modèle obtenu pour modéliser des paramètres physiques dans le Chapitre 4.Dans la deuxième partie de la thèse, on étudie le problème d'estimation non-paramétrique des densités d'entropie maximale des statistiques d'ordre en distance de Kullback-Leibler. Le chapitre 5 décrit une méthode d'agrégation pour des densités de probabilité et des densités spectrales, basée sur une combinaison convexe de ses logarithmes, et montre des bornes optimales non-asymptotiques en déviation. Dans le Chapitre 6, on propose une méthode adaptative issue d'un modèle exponentiel log-additif pour estimer les densités considérées, et on démontre qu'elle atteint les vitesses connues minimax. L'application de cette méthode pour estimer des dimensions des défauts est présentée dans le Chapitre 7 / In this thesis we consider the modelling of the joint distribution of order statistics, i.e. random vectors with almost surely ordered components. The first part is dedicated to the probabilistic modelling of order statistics of maximal entropy with marginal constraints. Given the marginal constraints, the characterization of the joint distribution can be given by the associated copula. Chapter 2 presents an auxiliary result giving the maximum entropy copula with a fixed diagonal section. We give a necessary and sufficient condition for its existence, and derive an explicit formula for its density and entropy. Chapter 3 provides the solution for the maximum entropy problem for order statistics with marginal constraints by identifying the copula of the maximum entropy distribution. We give explicit formulas for the copula and the joint density. An application for modelling physical parameters is given in Chapter 4.In the second part of the thesis, we consider the problem of nonparametric estimation of maximum entropy densities of order statistics in Kullback-Leibler distance. Chapter 5 presents an aggregation method for probability density and spectral density estimation, based on the convex combination of the logarithms of these functions, and gives non-asymptotic bounds on the aggregation rate. In Chapter 6, we propose an adaptive estimation method based on a log-additive exponential model to estimate maximum entropy densities of order statistics which achieves the known minimax convergence rates. The method is applied to estimating flaw dimensions in Chapter 7

This thesis details several applications of Gaussian processes (GPs) for enhanced time series modeling. We first cover different approaches for using Gaussian processes in time series problems. These are extended to the state space approach to time series in two different problems. We also combine Gaussian processes and Bayesian online change point detection (BOCPD) to increase the generality of the Gaussian process time series methods. These methodologies are evaluated on predictive performance on six real world data sets, which include three environmental data sets, one financial, one biological, and one from industrial well drilling. Gaussian processes are capable of generalizing standard linear time series models. We cover two approaches: the Gaussian process time series model (GPTS) and the autoregressive Gaussian process (ARGP).We cover a variety of methods that greatly reduce the computational and memory complexity of Gaussian process approaches, which are generally cubic in computational complexity. Two different improvements to state space based approaches are covered. First, Gaussian process inference and learning (GPIL) generalizes linear dynamical systems (LDS), for which the Kalman filter is based, to general nonlinear systems for nonparametric system identification. Second, we address pathologies in the unscented Kalman filter (UKF).We use Gaussian process optimization (GPO) to learn UKF settings that minimize the potential for sigma point collapse. We show how to embed mentioned Gaussian process approaches to time series into a change point framework. Old data, from an old regime, that hinders predictive performance is automatically and elegantly phased out. The computational improvements for Gaussian process time series approaches are of even greater use in the change point framework. We also present a supervised framework learning a change point model when change point labels are available in training.

Nonparametric Bayesian inference has widespread applications in statistics and machine learning. In this thesis, we examine the most popular priors used in Bayesian non-parametric inference. The Dirichlet process and its extensions are priors on an infinite-dimensional space. Originally introduced by Ferguson (1983), its conjugacy property allows a tractable posterior inference which has lately given rise to a significant developments in applications related to machine learning. Another yet widespread prior used in nonparametric Bayesian inference is the Beta process and its extensions. It has originally been introduced by Hjort (1990) for applications in survival analysis. It is a prior on the space of cumulative hazard functions and it has recently been widely used as a prior on an infinite dimensional space for latent feature models. Our contribution in this thesis is to collect many diverse groups of nonparametric Bayesian tools and explore algorithms to sample from them. We also explore machinery behind the theory to apply and expose some distinguished features of these procedures. These tools can be used by practitioners in many applications.

This thesis analyses, derives and evaluates specification tests of Generalized Auto-Regressive Conditional Heteroskedasticity (GARCH) regression models, both univariate and multivariate. Of particular interest, in the first half of the thesis, is the derivation of robust test procedures designed to assess the Constant Conditional Correlation (CCC) assumption often employed in multivariate GARCH (MGARCH) models. New asymptotically valid conditional moment tests are proposed which are simple to construct, easily implementable following the full or partial Quasi Maximum Likelihood (QML) estimation and which are robust to non-normality. In doing so, a non-normality robust version of the Tse's (2000) LM test is provided. In addition, a new and easily programmable expressions of the expected Hessian matrix associated with the QMLE is obtained. The finite sample performances of these tests are investigated in an extensive Monte Carlo study, programmed in GAUSS.In the second half of the thesis, attention is devoted to nonparametric testing of GARCH regression models. First simultaneous consistent nonparametric tests of the conditional mean and conditional variance structure of univariate GARCH models are considered. The approach is developed from the Integrated Generalized Spectral (IGS) and Projected Integrated Conditional Moment (PICM) procedures proposed recently by Escanciano (2008 and 2009, respectively) for time series models. Extending Escanciano (2008), a new and simple wild bootstrap procedure is proposed to implement these tests. A Monte Carlo study compares the performance of these nonparametric tests and four parametric tests of nonlinearity and/or asymmetry under a wide range of alternatives. Although the proposed bootstrap scheme does not strictly satisfy the asymptotic requirements, the simulation results demonstrate its ability to control the size extremely well and therefore the power comparison seems justified. Furthermore, this suggests there may exist weaker conditions under which the tests are implementable. The simulation exercise also presents the new evidence of the effect of conditional mean misspecification on various parametric tests of conditional variance. The testing procedures are also illustrated with the help of the S&P 500 data. Finally the PICM and IGS approaches are extended to the MGARCH case. The procedure is illustrated with the help of a bivariate CCC-GARCH model, but can be generalized to other MGARCH specifications. Simulation exercise shows that these tests have satisfactory size and are robust to non-normality. The marginal mean and variance tests have excellent power; however the covariance marginal tests lack power for some alternatives.

In this thesis, we study the estimation of the nonlinear conditionalrisk-neutral density function (RND) in discrete time. Specifically, weevaluate the extent to which the estimated nonlinear conditional RNDvaluable insights to answer relevant economic questions regarding to optionpricing, the measurement of invertors' preferences and portfolio choice.We make use of large dataset of options contracts written on the S&P 500index from 1996 to 2011, to estimate the parameters of the conditional RNDfunctions by minimizing the squared option pricing errors delivered by thenonlinear models studied in the thesis.In the first essay, we show that a semi-nonparametric option pricing modelwith GARCH variance outperforms several benchmarks models in-sample andout-of-sample. In the second essay, we show that a simple two-state regimeswitching model in volatility is not able to fully account for the pricingkernel and the risk aversion puzzle; however, it provides a reasonablecharacterisation of the time-series properties of the estimated riskaversion.In the third essay, we evaluate linear stochastic discount factormodels using an out-of-sample financial metric. We find that multifactormodels outperform the CAPM when this metric is used, and that modelsproducing the best fit in-sample are also those exhibiting the bestperformance out-of-sample.