Os modelos geoestatísticos bivariados denem funções aleatórias para dois processos estocásticos com localizações espaciais conhecidas. Pode-se adotar a suposição da existência de um campo aleatório gaussiano latente para cada variável aleatória. A suposição de gaussianidade do processo latente é conveniente para inferências sobre parâmetros do modelo e para obtenção de predições espaciais, uma vez que a distribuição de probabilidade conjunta para um conjunto de pontos do processo latente é também gaussiana. A matriz de covariância dessa distribuição deve ser positiva denida e possuir a estrutura de variabilidade espacial entre e dentre os atributos. Gelfand et al. (2004) e Diggle e Ribeiro Jr. (2007) propuseram estratégias para estruturar essa matriz, porém não existem muitos relatos sobre o uso e avaliações comparativas entre essas abordagens. Neste trabalho foi conduzido um estudo de simulação de modelos geoestatísticos bivariados em conjunto com estimação por máxima verossimilhança e krigagem ordinária, sob diferentes congurações amostrais de localizações espaciais. Também foram utilizados dados provenientes da análise de solo de uma propriedade agrícola com 51,8ha de área, onde foram amostradas 67 localizações georeferenciadas. Foram utilizados os valores mensurados de pH e da saturação por bases do solo, que foram submetidas à análise descritiva espacial, modelagens geoestatísticas univariadas, bivariadas e predições espaciais. Para vericar vantagens quanto à adoção de modelos univariados ou bivariados, a amostra da saturação por bases, que possui coleta mais dispendiosa, foi dividida em uma subamostra de modelagem e uma subamostra de controle. A primeira foi utilizada para fazer a modelagem geoestatística e a segunda foi utilizada para comparar as precisões das predições espaciais nas localizações omitidas no processo de modelagem. / Bivariate geostatistical models dene random functions for two stochastic processes with known spatial locations. Existence of a Gaussian random elds can be assumed for each latent random variable. This Gaussianity assumption for the latent process is a convenient one for the inferences on the model parameters and for spatial predictions once the joint distribution for a set of points is multivariate normal. The covariance matrix of this distribution should be positivede nite and to have the spatial variability structure between and among the attributes. Gelfand et al. (2004) and Diggle e Ribeiro Jr. (2007) suggested strategies for structuring this matrix, however there are few reports on comparing approaches. This work reports on a simulation study of bivariate models together with maximum likelihood estimators and spatial prediction under dierent sets of sampling locations space. Soil sample data from a eld with 51.8 hectares is also analyzed with the two soil attributes observed at 67 spatial locations. Data on pH and base saturation were submitted to spatial descriptive analysis, univariate and bivariate modeling and spatial prediction. To check for advantages of the adoption of univariate or bivariate models, the sample of the more expensive variable was divided into a modeling and testing subsamples. The rst was used to t geostatistical models, and the second was used to compare the spatial prediction precisions in the locations not used in the modeling process.

by Tang Chi Nang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 104-[109]). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Gaussian elimination --- p.2 / Chapter 1.1.1 --- Numerical stability --- p.2 / Chapter 1.2 --- Block Gaussian elimination --- p.3 / Chapter 1.2.1 --- Numerical stability --- p.4 / Chapter 1.3 --- Elimination graph --- p.4 / Chapter 1.4 --- Elimination ordering --- p.5 / Chapter 1.5 --- Computation and storage requirement --- p.6 / Chapter 1.6 --- Outline of the thesis --- p.7 / Chapter 2 --- Weighted graph elimination --- p.8 / Chapter 2.1 --- Weighted elimination graph --- p.8 / Chapter 2.2 --- Sparse Gaussian elimination --- p.9 / Chapter 2.3 --- Computation and storage requirement --- p.12 / Chapter 2.3.1 --- Computation requirement --- p.12 / Chapter 2.3.2 --- Storage requirement --- p.14 / Chapter 2.4 --- Elimination ordering --- p.15 / Chapter 2.5 --- Repeated structure --- p.18 / Chapter 3 --- Main theory --- p.21 / Chapter 3.1 --- Motivation --- p.21 / Chapter 3.2 --- Notations --- p.22 / Chapter 3.2.1 --- Connectivity --- p.23 / Chapter 3.2.2 --- Separator --- p.23 / Chapter 3.2.3 --- Equivalence --- p.24 / Chapter 3.3 --- Repetition separator --- p.25 / Chapter 3.4 --- Repetition elimination process --- p.30 / Chapter 3.5 --- Multiple Separators --- p.32 / Chapter 3.6 --- Feasibility --- p.33 / Chapter 3.6.1 --- Two-separator case --- p.34 / Chapter 3.6.2 --- General case --- p.39 / Chapter 3.6.3 --- Successive repetition elimination process (SREP) --- p.41 / Chapter 3.7 --- Generalized repetition elimination process --- p.42 / Chapter 3.7.1 --- Extra edges --- p.42 / Chapter 3.7.2 --- Acyclic edges --- p.43 / Chapter 3.7.3 --- Generalized repetition separator --- p.45 / Chapter 4 --- Application in queueing analysis --- p.52 / Chapter 4.1 --- Markov Chain Reduction Principle --- p.54 / Chapter 4.1.1 --- Numerical stability --- p.57 / Chapter 4.2 --- Multi-class MMPP/M/1/L queue --- p.57 / Chapter 4.2.1 --- Single-class case (QBD case) --- p.58 / Chapter 4.2.2 --- Preemptive LCFS case --- p.63 / Chapter 4.2.3 --- Non-preemptive LCFS case --- p.70 / Chapter 4.2.4 --- FCFS case --- p.72 / Chapter 4.2.5 --- Extension to phase type service time --- p.77 / Chapter 4.3 --- 2-class priority system --- p.77 / Chapter 5 --- Choosing the right algorithm --- p.85 / Chapter 5.1 --- MMPP/M/1/L system with bursty arrival --- p.86 / Chapter 5.1.1 --- Algorithm Comparison --- p.89 / Chapter 5.1.2 --- Numerical Examples --- p.90 / Chapter 5.2 2 --- -class priority system --- p.90 / Chapter 5.2.1 --- Algorithm Comparison --- p.95 / Chapter 5.2.2 --- Numerical Examples --- p.95 / Chapter 5.3 --- Conclusion --- p.95 / Chapter 6 --- Conclusion --- p.98 / Chapter 6.1 --- Further research --- p.99 / Chapter A --- List of frequently-used notations --- p.101 / Chapter A.l --- System of equations and Digraph --- p.101 / Chapter A.2 --- General-purpose functions --- p.102 / Chapter A.3 --- Single repetition separator --- p.102 / Chapter A.4 --- Sequence of repetition separators --- p.103 / Chapter A.5 --- Compatibility --- p.103 / Bibliography --- p.104

Probabilistic machine learning and circular statistics—the branch of statistics concerned with data as angles and directions—are two research communities that have grown mostly in isolation from one another. On the one hand, probabilistic machine learning community has developed powerful frameworks for problems whose data lives on Euclidean spaces, such as Gaussian Processes, but have generally neglected other topologies studied by circular statistics. On the other hand, the approximate inference frameworks from probabilistic machine learning have only recently started to the circular statistics landscape. This thesis intends to redress the gap between these two fields by contributing to both fields with models and approximate inference algorithms. In particular, we introduce the multivariate Generalised von Mises distribution (mGvM), which allows the use of kernels in circular statistics akin to Gaussian Processes, and an augmented representation. These models account for a vast number of applications comprising both latent variable modelling and regression of circular data. Then, we propose methods to conduct approximate inference on these models. In particular, we investigate the use of Variational Inference, Expectation Propagation and Markov chain Monte Carlo methods. The variational inference route taken was a mean field approach to efficiently leverage the mGvM tractable conditionals and create a baseline for comparison with other methods. Then, an Expectation Propagation approach is presented drawing on the Expectation Consistent Framework for Ising models and connecting the approximations used to the augmented model presented. In the final MCMC chapter, efficient Gibbs and Hamiltonian Monte Carlo samplers are derived for the mGvM and the augmented model.

Machine learning is a branch of artificial intelligence that uses data to automatically build inferences and models designed to generalise and make predictions. In this thesis, the use of machine learning in materials science is explored, for two different problems: the optimisation of gallium nitride optoelectronic devices, and the prediction of material failure in the setting of laboratory earthquakes. Light emitting diodes based on III-nitrides quantum wells have become ubiquitous as a light source, owing to their direct band-gap that covers UV, visible and infra-red light, and their very high quantum efficiency. This efficiency originates from most electronic transitions across the band-gap leading to the emission of a photon. At high currents however this efficiency sharply drops. In chapters 3 and 4 simulations are shown to provide an explanation for experimental results, shedding a new light on this drop of efficiency. Chapter 3 provides a simple and yet accurate model that explains the experimentally observed beneficial effect that silicon doping has on light emitting diodes. Chapter 4 provides a model for the experimentally observed detrimental effect that certain V-shaped defects have on light emitting diodes. These results pave the way for the association of simulations to detailed multi-microscopy. In the following chapters 5 to 7, it is shown that machine learning can leverage the use of device simulations, by replacing in a targeted and efficient way the very labour intensive tasks of making sure the numerical parameters of the simulations lead to convergence, and that the physical parameters reproduce experimental results. It is then shown that machine learning coupled with simulations can find optimal light emitting diodes structures, that have a greatly enhanced theoretical efficiency. These results demonstrate the power of machine learning for leveraging and automatising the exploration of device structures in simulations. Material failure is a very broad problem encountered in a variety of fields, ranging from engineering to Earth sciences. The phenomenon stems from complex and multi-scale physics, and failure experiments can provide a wealth of data that can be exploited by machine learning. In chapter 8 it is shown that by recording the acoustic waves emitted during the failure of a laboratory fault, an accurate predictive model can be built. The machine learning algorithm that is used retains the link with the physics of the experiment, and a new signal is thus discovered in the sound emitted by the fault. This new signal announces an upcoming laboratory earthquake, and is a signature of the stress state of the material. These results show that machine learning can help discover new signals in experiments where the amount of data is very large, and demonstrate a new method for the prediction of material failure.

Tang Him John. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (leaves 76-80). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgment --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Background --- p.2 / Chapter 1.1.1 --- Linear Time Series --- p.2 / Chapter 1.1.2 --- Mixture Models --- p.3 / Chapter 1.1.3 --- EM algorithm --- p.6 / Chapter 1.1.4 --- Model Selection --- p.6 / Chapter 1.2 --- Main Objectives --- p.7 / Chapter 1.3 --- Outline of this thesis --- p.7 / Chapter 2 --- Finite Gaussian Mixture ARMA-GARCH Model --- p.9 / Chapter 2.1 --- Introduction --- p.9 / Chapter 2.1.1 --- "AR, MA, and ARMA" --- p.10 / Chapter 2.1.2 --- Stationarity --- p.11 / Chapter 2.1.3 --- ARCH and GARCH --- p.12 / Chapter 2.1.4 --- Gaussian mixture --- p.13 / Chapter 2.1.5 --- EM and GEM algorithms --- p.14 / Chapter 2.2 --- Finite Gaussian Mixture ARMA-GARCH Model --- p.16 / Chapter 2.3 --- Estimation of Gaussian mixture ARMA-GARCH model --- p.17 / Chapter 2.3.1 --- Autocorrelation and Stationarity --- p.20 / Chapter 2.3.2 --- Model Selection --- p.24 / Chapter 2.4 --- Experiments: First Step Prediction --- p.26 / Chapter 2.5 --- Chapter Summary --- p.28 / Chapter 2.6 --- Notations and Terminologies --- p.30 / Chapter 2.6.1 --- White Noise Time Series --- p.30 / Chapter 2.6.2 --- Lag Operator --- p.30 / Chapter 2.6.3 --- Covariance Stationarity --- p.31 / Chapter 2.6.4 --- Wold's Theorem --- p.31 / Chapter 2.6.5 --- Multivariate Gaussian Density function --- p.32 / Chapter 3 --- Finite Mixture-of-Expert ARMA-GARCH Model --- p.33 / Chapter 3.1 --- Introduction --- p.33 / Chapter 3.1.1 --- Mixture-of-Expert --- p.34 / Chapter 3.1.2 --- Alternative Mixture-of-Expert --- p.35 / Chapter 3.2 --- ARMA-GARCH Finite Mixture-of-Expert Model --- p.36 / Chapter 3.3 --- Estimation of Mixture-of-Expert ARMA-GARCH Model --- p.37 / Chapter 3.3.1 --- Model Selection --- p.38 / Chapter 3.4 --- Experiments: First Step Prediction --- p.41 / Chapter 3.5 --- Second Step and Third Step Prediction --- p.44 / Chapter 3.5.1 --- Calculating Second Step Prediction --- p.44 / Chapter 3.5.2 --- Calculating Third Step Prediction --- p.45 / Chapter 3.5.3 --- Experiments: Second Step and Third Step Prediction . --- p.46 / Chapter 3.6 --- Comparison with Other Models --- p.50 / Chapter 3.7 --- Chapter Summary --- p.57 / Chapter 4 --- Stable Estimation Algorithms --- p.58 / Chapter 4.1 --- Stable AR(1) estimation algorithm --- p.59 / Chapter 4.2 --- Stable AR(2) Estimation Algorithm --- p.60 / Chapter 4.2.1 --- Real p1 and p2 --- p.61 / Chapter 4.2.2 --- Complex p1 and p2 --- p.61 / Chapter 4.2.3 --- Experiments for AR(2) --- p.63 / Chapter 4.3 --- Experiment with Real Data --- p.64 / Chapter 4.4 --- Chapter Summary --- p.65 / Chapter 5 --- Conclusion --- p.66 / Chapter 5.1 --- Further Research --- p.69 / Chapter A --- Equation Derivation --- p.70 / Chapter A.1 --- First Derivatives for Gaussian Mixture ARMA-GARCH Esti- mation --- p.70 / Chapter A.2 --- First Derivatives for Mixture-of-Expert ARMA-GARCH Esti- mation --- p.71 / Chapter A.3 --- First Derivatives for BYY Harmony Function --- p.72 / Chapter A.4 --- First Derivatives for stable estimation algorithms --- p.73 / Chapter A.4.1 --- AR(1) --- p.74 / Chapter A.4.2 --- AR(2) --- p.74 / Bibliography --- p.80

Sen Sui. / Thesis submitted in: August 2003. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 88-92). / Abstracts in English and Chinese. / Abstract --- p.ii / Acknowledgement --- p.iv / Dedication --- p.v / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Motivation and Objective --- p.1 / Chapter 1.2 --- Contributions --- p.4 / Chapter 1.3 --- Thesis Organization --- p.5 / Chapter 2 --- Background of Risk Management --- p.7 / Chapter 2.1 --- Measuring Return --- p.8 / Chapter 2.2 --- Objectives of Risk Measurement --- p.11 / Chapter 2.3 --- Simple Statistics for Measurement of Risk --- p.15 / Chapter 2.4 --- Methods for Value-at-Risk Measurement --- p.16 / Chapter 2.5 --- Conditional VaR --- p.18 / Chapter 2.6 --- Portfolio VaR Methods --- p.18 / Chapter 2.7 --- Coherent Risk Measure --- p.20 / Chapter 2.8 --- Summary --- p.22 / Chapter 3 --- Selection of Independent Factors for VaR Computation --- p.23 / Chapter 3.1 --- Mixture Convolution Approach Restated --- p.24 / Chapter 3.2 --- Procedure for Selection and Evaluation --- p.26 / Chapter 3.2.1 --- Data Preparation --- p.26 / Chapter 3.2.2 --- ICA Using JADE --- p.27 / Chapter 3.2.3 --- Factor Statistics --- p.28 / Chapter 3.2.4 --- Factor Selection --- p.29 / Chapter 3.2.5 --- Reconstruction and VaR Computation --- p.30 / Chapter 3.3 --- Result and Comparison --- p.30 / Chapter 3.4 --- Problem of Using Kurtosis and Skewness --- p.40 / Chapter 3.5 --- Summary --- p.43 / Chapter 4 --- Mixture of Gaussians and Value-at-Risk Computation --- p.45 / Chapter 4.1 --- Complexity of VaR Computation --- p.45 / Chapter 4.1.1 --- Factor Selection Criteria and Convolution Complexity --- p.46 / Chapter 4.1.2 --- Sensitivity of VaR Estimation to Gaussian Components --- p.47 / Chapter 4.2 --- Gaussian Mixture Model --- p.52 / Chapter 4.2.1 --- Concept and Justification --- p.52 / Chapter 4.2.2 --- Formulation and Method --- p.53 / Chapter 4.2.3 --- Result and Evaluation of Fitness --- p.55 / Chapter 4.2.4 --- Evaluation of Fitness using Z-Transform --- p.56 / Chapter 4.2.5 --- Evaluation of Fitness using VaR --- p.58 / Chapter 4.3 --- VaR Estimation using Convoluted Mixtures --- p.60 / Chapter 4.3.1 --- Portfolio Returns by Convolution --- p.61 / Chapter 4.3.2 --- VaR Estimation of Portfolio Returns --- p.64 / Chapter 4.3.3 --- Result and Analysis --- p.64 / Chapter 4.4 --- Summary --- p.68 / Chapter 5 --- VaR for Portfolio Optimization and Management --- p.69 / Chapter 5.1 --- Review of Concepts and Methods --- p.69 / Chapter 5.2 --- Portfolio Optimization Using VaR --- p.72 / Chapter 5.3 --- Contribution of the VaR by ICA/GMM --- p.76 / Chapter 5.4 --- Summary --- p.79 / Chapter 6 --- Conclusion --- p.80 / Chapter 6.1 --- Future Work --- p.82 / Chapter A --- Independent Component Analysis --- p.83 / Chapter B --- Gaussian Mixture Model --- p.85 / Bibliography --- p.88

In practice, the application and extension of the ICA model depend on the problem and the data to be investigated. We finally focus on GARCH models in finance, and show that estimation of univariate or multivariate GARCH models is actually a nonlinear ICA problem; maximizing the likelihood is equivalent to minimizing the statistical dependence in standardized residuals. ICA can then be used for factor extraction in multivariate factor GARCH models. We also develop some extensions of ICA for this task. These techniques for extracting factors from multivariate return series are compared both theoretically and experimentally. We find that the one based on conditional decorrelation between factors behaves best. / In this thesis, first we consider the problem of source separation of post-nonlinear (PNL) mixtures, which is an extension of ICA to the nonlinear mixing case. With a large number of parameters, existing methods are computation-demanding and may be prone to local optima. Based on the fact that linear mixtures of independent variables tend to be Gaussian, we develop a simple and efficient method for this problem, namely extended Gaussianization. With Gaussianization as preprocessing, this method approximates each linear mixture of independent sources by the Cornish-Fisher expansion with only two parameters. Inspired by the relationship between the PNL mixing model and the Wiener system, extended Gaussianization is also proposed for blind inversion of Wiener systems. / Independent component analysis (ICA) is a recent and powerful technique for recovering latent independent sources given only their mixtures. The basic ICA model assumes that sources are linearly mixed and mutually independent. / Next, we study the subband decomposition ICA (SDICA) model, which extends the basic ICA model to allow dependence between sources by assuming that only some narrow-band source sub-components are independent. In SDICA, it is difficult to determine the subbands of source independent sub-components. We discuss the feasibility of performing SDICA in an adaptive manner. An adaptive method, called band selective ICA, is then proposed for this task. We also investigate the relationship between overcomplete ICA and SDICA and show that band selective ICA can solve the overcomplete ICA problems with sources having specific frequency localizations. Experimental results on separating images of human faces as well as artificial data are presented to verify the powerfulness of band selective ICA. / Zhang Kun. / "July 2005." / Adviser: Lai-Wan Chan. / Source: Dissertation Abstracts International, Volume: 67-07, Section: B, page: 3925. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 218-234). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstract in English and Chinese. / School code: 1307.

A technique called gaussian temporal factor analysis (gaussian TFA) proposed by Xu in 2000 may be used to test the APT model under the mild assumption that the efficient market hypothesis (EMH) is violated. We are motivated to investigate statistical behaviors of the gaussian TFA model. / According to a recent survey by Cochrane (1999), the multi-factor APT model is gaining popularity and recognition over CAPM by the investment community. While empirical evidence shows that mutual funds can earn average returns not explained by the CAPM by following a variety of investment styles, this anomaly could be captured by APT which includes the single-factor CAPM as a special case. Yet, three aspects of APT still cannot be tested in practice. / First, a systematic testing package is proposed for testing gaussian TFA in six dimensions, including factor number, factor loadings, residuals correlations and autoregressive conditional heteroscedasticity (ARCH) effects, economic significance and factor independence, using financial data in Hong Kong. Particularly, a new hypothesis testing approach is proposed for statistically testing independence. / In the finance literature, an objective way to judge whether an asset pricing model is misspecified is by statistical tests. In the past, both the capital asset pricing model (CAPM) and the arbitrage pricing theory (APT) have been the subjects of extensive tests. / Second, we investigate two extensions of the gaussian TFA model in view of ARCH in driving noise residuals. We test the extended models for ARCH as well as other aspects to ensure model specification adequacy. Furthermore, we find that ARCH effects are not quite significant driving noise residuals of the macroeconomic modulate independent state-space model. This may be due to long-term modelling of the market. / Third, we test gaussian TFA from the practical point of view in financial prediction and portfolio management. For prediction, we introduce the gaussian TFA alternative mixture experts (ME) approach for forecasting. For adaptive portfolio management, we derive the gaussian TFA adaptive algorithm for implementing the Sharpe-ratio based adaptive portfolio management under different scenarios. Empirical results reveal that APT-based portfolio management techniques are in general superior to return-based techniques. / by Kai-Chun Chiu. / "July, 2003." / Adviser: Lei Xu. / Source: Dissertation Abstracts International, Volume: 64-09, Section: B, page: 4451. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2003. / Includes bibliographical references (p. 113-125). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest dissertations and theses, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / School code: 1307.

Os modelos geoestatísticos bivariados denem funções aleatórias para dois processos estocásticos com localizações espaciais conhecidas. Pode-se adotar a suposição da existência de um campo aleatório gaussiano latente para cada variável aleatória. A suposição de gaussianidade do processo latente é conveniente para inferências sobre parâmetros do modelo e para obtenção de predições espaciais, uma vez que a distribuição de probabilidade conjunta para um conjunto de pontos do processo latente é também gaussiana. A matriz de covariância dessa distribuição deve ser positiva denida e possuir a estrutura de variabilidade espacial entre e dentre os atributos. Gelfand et al. (2004) e Diggle e Ribeiro Jr. (2007) propuseram estratégias para estruturar essa matriz, porém não existem muitos relatos sobre o uso e avaliações comparativas entre essas abordagens. Neste trabalho foi conduzido um estudo de simulação de modelos geoestatísticos bivariados em conjunto com estimação por máxima verossimilhança e krigagem ordinária, sob diferentes congurações amostrais de localizações espaciais. Também foram utilizados dados provenientes da análise de solo de uma propriedade agrícola com 51,8ha de área, onde foram amostradas 67 localizações georeferenciadas. Foram utilizados os valores mensurados de pH e da saturação por bases do solo, que foram submetidas à análise descritiva espacial, modelagens geoestatísticas univariadas, bivariadas e predições espaciais. Para vericar vantagens quanto à adoção de modelos univariados ou bivariados, a amostra da saturação por bases, que possui coleta mais dispendiosa, foi dividida em uma subamostra de modelagem e uma subamostra de controle. A primeira foi utilizada para fazer a modelagem geoestatística e a segunda foi utilizada para comparar as precisões das predições espaciais nas localizações omitidas no processo de modelagem. / Bivariate geostatistical models dene random functions for two stochastic processes with known spatial locations. Existence of a Gaussian random elds can be assumed for each latent random variable. This Gaussianity assumption for the latent process is a convenient one for the inferences on the model parameters and for spatial predictions once the joint distribution for a set of points is multivariate normal. The covariance matrix of this distribution should be positivede nite and to have the spatial variability structure between and among the attributes. Gelfand et al. (2004) and Diggle e Ribeiro Jr. (2007) suggested strategies for structuring this matrix, however there are few reports on comparing approaches. This work reports on a simulation study of bivariate models together with maximum likelihood estimators and spatial prediction under dierent sets of sampling locations space. Soil sample data from a eld with 51.8 hectares is also analyzed with the two soil attributes observed at 67 spatial locations. Data on pH and base saturation were submitted to spatial descriptive analysis, univariate and bivariate modeling and spatial prediction. To check for advantages of the adoption of univariate or bivariate models, the sample of the more expensive variable was divided into a modeling and testing subsamples. The rst was used to t geostatistical models, and the second was used to compare the spatial prediction precisions in the locations not used in the modeling process.

We present limit theory for tests of homogeneity for finite mixture models. More specifically, we derive the asymptotic distribution of certain random quantities used for testing that a mixture of two distributions is in fact just a single distribution. Our methods apply to cases where the mixture component distributions come from one of a wide class of one-parameter exponential families, both continous and discrete. We consider two random quantities, one related to testing simple hypotheses, the other composite hypotheses. For simple hypotheses we consider the maximum of the standardised score process, which is itself a test statistic. For composite hypotheses we consider the maximum of the efficient score process, which is itself not a statistic (it depends on the unknown true distribution) but is asymptotically equivalent to certain common test statistics in a certain sense. We show that we can approximate both quantities with the maximum of a certain Gaussian process depending on the sample size and the true distribution of the observations, which when suitably normalised has a limiting distribution of the Gumbel extreme value type. Although the limit theory is not practically useful for computing approximate p-values, we use Monte-Carlo simulations to show that another method suggested by the theory, involving using a Studentised version of the maximum-score statistic and simulating a Gaussian process to compute approximate p-values, is remarkably accurate and uses a fraction of the computing resources that a straight Monte-Carlo approximation would.