# Probability and Statistics Seminar

## Past sessions

18/10/2007, 15:00 — Room P3.10, Mathematics Building
Vladas Pipiras, University of North Carolina at Chapel Hill, USA

### Some research problems on long range dependence

Long range dependence is the property of time series data or corresponding models for which observations are strongly correlated across time lags. It has gained particular attention since the mid 90&rsquo;s, primarily due to its use in modeling data traffic over Internet and other networks. In this talk, the speaker will discuss several, mostly mathematical research problems on long range dependence that he is presently involved in.
27/09/2007, 15:00 — Room V1.31, Civil Engineering Building, IST
Manuel Cabral Morais, Departamento de Matemática e CEMAT, Instituto Superior Técnico

### Stochastic Ordering: from the Lorenz curve to Quality Control

The desire to compare what is random is probably as old as probability itself. A landmark in the history of stochastic ordering (SO) is the pioneering work by Lorenz (1905) in the assessment of income inequality in a population of n individuals. Lorenz - feeling that all of the summary measures then under consideration constituted too much condensation of the data - proposed what is now known as the Lorenz curve and suggested the following rule of interpretation: a high level of income inequality is associated to a severely bent curve. Since this tool was developed by Lorenz, SO has gained widespread acceptance and has been applied, namely, in Biology, Queueing Theory, Reliability Theory, Risk Theory, Scheduling, and Statistical Inference. This presentation will focus on a brief overview of SO and on two applications. In Finance, SO is particularly important in demand and shift effect problems in portfolio selection. As for Quality Control, SO provides decisive insights into how control schemes work in practice and allows the performance comparison of competitive schemes.
21/09/2007, 15:00 — Room P6, Mathematics Building

### Ordenação em excedência de nível e aplicações

12/06/2007, 15:00 — Room P3.31, Mathematics Building
Antonis Economou, DM, University of Athens

### Pricing and equilibrium behavior for a Markovian queue with server vacations

The topic of the talk lies in the intersection of queueing and game theory. More specifically we seek for best balking strategies for the customers in a queueing system with server vacations under various levels of information. The model is the single server Markovian queue with setup times. Whenever a customer leaves this system empty, the server departs immediately to attend to secondary jobs. On the contrary, whenever a customer arrives to an empty system, the server is recalled immediately and it takes an exponential setup time to start service again. We assume a natural reward - cost structure for the customers, which incorporates their desire for service as well as their unwillingness to wait. We examine customers behavior under various levels of information regarding the state of the system at arrival instances. More specifically, a customer may know or not know the state of the server and the number of present customers upon his arrival. We derive equilibrium strategies for the customers under the various levels of information and we study the associated social optimization and profit maximization problems. Analytical and numerical comparisons illustrate further the effect of the information level to the pricing and equilibrium behavior of the system.
07/02/2007, 14:30 — Room P3.31, Mathematics Building
Luís Filipe Meira Machado, Dep. Matemática para a Ciência e Tecnologia, Universidade do Minho

### New methods in multi-state models

The multi-state models may be considered as a generalization of the survival process where several events occur successively over time. The classical multi-state model describes the life history of an individual as moving through various states. Thus at any time, say t, the individual will be in one state. For a single individual, these states may describe whether the subject is healthy, diseased, diseased with complication or dead. Estimation of the transition probabilities in a non-homogeneous illness-death model is considered here. Traditionally, inference for such models is based on the Markov assumption. We review some of these results and propose new estimators, based on a less restrictive (non-Markov) approach.
24/01/2007, 14:30 — Room P3.31, Mathematics Building
Eduardo Luis Trincão da Conceição, Dep. Engenharia Química, Univ. Coimbra

### A reconciliação de dados, o erro de medição com viés, o estimador robusto LTD, o engenheiro, e uma ideia dele

Devido ao erro aleatório inerente aos processos de medição, é usual haver inconsistência entre medições e princípios físicos e químicos básicos. A técnica estatística de estimação que permite obter estimativas dos verdadeiros valores das medições ajustados às leis físicas é conhecida como reconciliação de dados. Um problema que afecta as técnicas de reconciliação clássicas é a presença de outliers nos dados. Um outro problema importante é a existência de viés nas medições, o que ocorre com a inevitável descalibração dos instrumentos de medição. O estimador robusto least trimmed differences (LTD) é inerentemente insensível ao viés dos dados. Daqui decorre o interesse na investigação da sua aplicabilidade ao problema de reconciliação de dados. Sendo um estimador em que o grau de aparamento é fixo a priori, isto limita o grau de eficiência que é possível obter. Uma abordagem possível para contornar este problema é a extensão de uma regra adaptativa proposta por Gervini e Yohai em 2002. Na primeira parte, farei a exposição do problema de reconciliação de dados para sistemas dinâmicos não-lineares, e apresentarei um esboço do ataque deste problema com o estimador LTD. Na segunda parte, apresentarei os resultados de uma simulação de Monte Carlo em que uma pequena variação na regra adaptativa de Gervini e Yohai é aplicada ao estimador least trimmed squares (LTS) e a um estimador de norma-${L}_{p}$ aparado. Além disso, estes estimadores adaptativos são comparados com o estimador MM e o estimador TAU.
08/11/2006, 14:30 — Room P3.31, Mathematics Building
Márcia D'Elia Branco, DE/IME, Universidade de São Paulo

### A Skew-Normal Item Response Theory Family

Normal assumptions for the latent variable and symmetric item characteristics curves have been used in the last 50 years in many psychometric methods for item-response theory (IRT) models. However this assumption can be restrictive for modeling human behavior. This paper introduces a new family of asymmetric models for item response theory, namely the skew-normal item response theory (SN-IRT) model. This family extends the symmetric ogive normal model by considering: a) an accumulated skew-normal distribution for the item characteristic curve and b) a skew-normal distribution for the latent variables modeling the individuals? abilities. Hence, the SN-IRT is a more flexible model for fitting data sets with dichotomous responses. Bayesian inference methodology using two data augmentation approaches for implementing the MCMC methodology is developed. Model selection is considered by using several Bayesian criteria. An application is conducted and the proposed penalization parameter is interpreted in the context of a data set related to a mathematical test applied to Peruvian students in rural schools.
02/11/2006, 14:30 — Room P3.31, Mathematics Building
Robert Weiss, DB/UCLA, University of California

### Hierarchical Models for Combining Phylogenetic Analyses Using an Iterative Re-weighting Algorithm

Molecular phylogeny is the art and science of inferring the family tree and underlying evolutionary parameters relating the molecular sequences from different genes, or viruses, or species and so on. Phylogenetic modeling is computationally challenging and most phylogenetic modeling estimates a single tree to a single set of molecular sequences. We develop a Bayesian hierarchical semi-parametric regression model to combine phylogenetic analyses of sets of HIV-1 nucleotide sequences. We describe several reweighting algorithms for combining completed Markov chain Monte Carlo (MCMC) analyses to shrink parameter estimates while adjusting for data set-specific covariates.
Individual phylogenetic analyses are performed independently using the publicly available software MrBayes (Huelsenbeck and Ronquist, 2001) that fits a computationally intensive Bayesian model using Markov chain Monte Carlo (MCMC) simulation. We place a hierarchical regression model across the individual analyses to estimate parameters of interest within and across analyses. We use a Mixture of Dirichlet processes (MDP) prior for the parameters of interest to relax inappropriate parametric assumptions and to insure the prior distribution for the parameters of interest is continuous. Constructing a large complex model involving all the original data is computationally challenging and would require rewriting the existing stand alone software. Instead we utilize existing MCMC samples from the individual analyses using an iteratively reweighted importance resampling algorithm within MCMC iterations.
27/10/2006, 14:00 — Room P4.35, Mathematics Building

### Uma Abordagem Bayesiana do Modelo ARCH com Potência Assimétrica

Os modelos da classe ARCH (auto-regressivo condicionalmente heterocedástico) modelam a heterocedasticidade constatada em séries temporais económicas. Tal estratégia não apenas melhora a eficiência dos estimadores usuais, mas também fornece uma predição da variância de cada termo do erro. Este trabalho desenvolve uma análise bayesiana do modelo ARCH com potência assimétrica. A análise envolve a estimação de parâmetros, a predição da variância condicional e a selecção de modelos. Os procedimentos de inferência bayesiana são implementados usando-se o Algoritmo de Metropolis-Hastings. O método é aplicado a dados simulados e a uma série de retornos do IBOVESPA.
11/10/2006, 14:30 — Room P3.31, Mathematics Building
Ana M. Bianco, IC/FCEN - Universidade de Buenos Aires

### Tests Robustos en el Modelo de Regresión Logística

En este trabajo se propone un test robusto para el parámetro de regresión de un modelo logístico. El test propuesto es un test tipo Wald, basado en una versión pesada del estimador propuesto por Bianco y Yohai (1996) tal como fue implementado por Croux y Haesbroeck (2003). Se estudia la distribución asintótica del estadístico bajo la hipótesis nula y bajo alternativas contiguas. Se realiza un estudio Monte Carlo para investigar la estabilidad del nivel y la potencia del test bajo contaminación y para comparar el comportamiento del test propuesto en el caso de muestras finitas con el test clásico y con otras propuestas robustas. Finalmente, se ilustra la performance del test propuesto sobre un conjunto de datos reales.
21/06/2006, 11:30 — Room P3.31, Mathematics Building
Guy Latouche, Université Libre de Bruxelles

### Structured Markov Chains in Applied Probability and Numerical Analysis

About thirty years ago, Quasi-Birth-and-Death processes and Skip-Free Markov chains came to the attention of applied probabilists. One of their prominent features is that their analysis requires the resolution of nonlinear equations, involving matrix-polynomial or matrix power series. At first, these were tackled 'in-house' and very soon several algorithms appeared which had their justification grounded, to a large extent, in probabilistic thinking. Soon, these equations caught the attention of numerical analysts who brought to bear their own special way of thinking about such problems and, not surprisingly, obtained improved algorithms in terms of convergence speed or numerical accuracy. The interaction between the two lines of approach are very exciting and this is an attempt to illustrate how the one meshes into the other.
21/06/2006, 10:00 — Room P3.31, Mathematics Building
Wolfgang Schmid, Europe University, Frankfurt (Oder)

### Comparison of Different Estimation Techniques for Portfolio Selection

The main obstacle in the application of the mean-variance portfolio selection is the fact that the moments of the asset returns are unknown. In practice the optimal portfolio weights are estimated by replacing these moments with the classical unbiased sample estimators. We provide a comparison of the exact and the asymptotic distributions of these estimated portfolio weights as well as a sensitivity analysis to shifts in the moments of the asset returns. Furthermore the paper compares the classical estimators of the moments of the asset returns with the recently proposed shrinkage estimators within the framework of portfolio selection. It is shown how the uncertainty about the portfolio weights can be introduced into the performance measurement of trading strategies. The methodology explains the bad out-of-sample performance of the classical Markowitz procedures.
08/06/2006, 15:00 — Room P3.31, Mathematics Building

### Bootstrap Robusto com base na Função de Influência

01/06/2006, 15:00 — Room P3.31, Mathematics Building
Isabel Rodrigues, DMIST

### Testes Robustos para o Modelo das Componentes Principais Comuns

Em alguns métodos estatísticos, como é por exemplo o caso da análise discriminante, poderá ser importante a comparação da estrutura de covariâncias de duas ou mais populações. Muitas vezes a suposição de igualdade das matrizes de covariâncias é claramente inadequada e a estimação das matrizes em separado não respeita o princípio da parcimónia. Com alternativa, alguns autores, com foi o caso de Flury (1988), estudaram modelos com estruturas de covariâncias comuns. Um deste modelos é conhecido como Componentes Principais Comuns (Flury, 1984), por ser uma generalização das Componentes Principais para k grupos e assume que as k matrizes de covariâncias têm valores próprios diferentes mas vectores próprios idênticos. Mais restrito é o modelo proporcional onde se admite que as matrizes de covariâncias diferem apenas de uma constante. Em Flury (1988) foram deduzidos e estudados os estimadores de máxima verosimilhança dos parâmetros destes modelos e construídos testes de razão de verosimilhanças para validar relações entre a estrutura de covariâncias das populações. Contudo, tanto a estimação clássica dos parâmetros com os testes de razão de verosimilhança são em muitas situações sensíveis a observações discordantes. Alternativas robustas de estimação via “plug-in” (PI) e “projection-pursuit” (PP) foram estudadas por Boente, Pires e Rodrigues (2002, 2005a, 2005b). Neste trabalho são propostos alguns procedimentos robustos para testar relações entre as estruturas das covariâncias de k populações.
15/02/2006, 14:30 — Room P3.31, Mathematics Building
Graciela Boente, Universidad de Buenos Aires

### Regiões de Tolerância Multivariadas Robustas

24/01/2006, 14:30 — Room P3.31, Mathematics Building
Jorge Alberto Achcar, Universidade Federal de São Carlos

### Estimators of Sensitivity and Specificity in the Presence of Verification Bias: A Bayesian Approach

Verification bias can occur if some of the patients with test results are not selected to receive the gold standard procedure. Unverified cases frequently are not suggestive to be positives. Consequently, the set of verified cases overestimates the number of true positives and underestimates the number of true negatives. The sensitivity and specificity estimates based only on the patients with verified disease are often biased. In this work, we derive unbiased estimators for sensitivity and specificity using a Bayesian approach. Marginal posterior densities of all parameters are estimated using the Gibbs sampler algorithm. An application to the study of accuracy of Hybrid Capture II in the diagnosis of cervical intraepithelial neoplasia grade 2 and 3 illustrates the proposed methodology.
11/10/2005, 14:30 — Room P4.35, Mathematics Building
Rudolf Dutter, Vienna University of Technology

### Development of a Data Analysis System in R, with Graphical Interface

The computer program system R already offers extremely many powerful data analysis tools. The development of a general graphical user interface is still at the beginning (Fox, 2004). We discuss the historical entrance of an older data analysis system (DAS) with many "new" and powerful features in an R-package. This is designed for graphically oriented analysis with special emphasis on geochemical data. Practical examples from the Kola project are illustrated.
04/10/2005, 11:30 — Room P3.10, Mathematics Building
Antonis Economou, University of Athens

### Exact computations and approximations for the stationary distributions of Markov chains in random environments and applications in queueing and population growth models

We consider a general model for a continuous time Markov chain in random environment. We study certain form of interaction between the process of interest and the environmental process, under which the stationary joint distribution is tractable. More specifically we obtain necessary and sufficient conditions for a generalized product-form stationary distribution. When these conditions fail we propose an alternative technique that transform the original system of the balance equations to an equivalent system. Applications in queueing and population growth models illustrate the scope and the efficiency of the methods.
13/09/2005, 16:00 — Room P4.35, Mathematics Building
Yarema Okhrin, Department of Statistics, University of Frankfurt (Oder), Germany

### Distributional properties and estimation of optimal portfolio

The Markowitz theory of portfolio selection is a classical part of asset allocation. Under the assumption of Gaussian asset returns and investor’s preferences given by the quadratic utility function, we can present the optimal portfolio weights as a function of the first two moments of asset returns. The true moments are unknown to the investor and should be estimated from a sample. Because of this practical applications often suffer from very large or negative portfolio weights. The aim of this project is to assess the distributional properties of estimated portfolio weights and to develop improved estimation procedures. Okhrin and Schmid (2005a) consider the maximum-likelihood estimation of the moments of asset returns. They provide expression for the mean and variance of the estimated portfolio weights of four different types. It appears that the estimated weights are heavily biased in small samples and have very large variance. This explains the empirical evidence from practical applications. It is also shown that the estimated global minimum variance portfolio weights follow multivariate t-distribution, what is of special interest in testing problems. For the portfolio weights that maximize the Sharpe ratio it appears that the moments of order equal or greater than one do not exist. This questions the usefulness of such estimator and makes the results untractable. A classical approach to decrease the volatility of an estimator is shrinkage technique. Using the result of Stein, Jorion (1986) first applied the shrinkage estimation of the expected asset returns to portfolio selection. Recently Ledoit and Wolf (2003, 2004) constructed a shrinkage estimator of the covariance matrix, which is robust against the singularity of sample covariance matrix. Okhrin and Schmid (2005b) applied the shrinkage methodology directly to the optimal portfolio weights by shrinking the classical portfolio weights to the weights obtained from a linear factor model. The optimal shrinkage intensity is derived to minimize the mean-square error. It appears, that the shrinkage estimator is also very successful in the reduction of the variance of portfolio return. Additionally, a new estimator is constructed by using predictive moments from a Bayesian framework with zero-mean prior distribution for the slopes of the factor model.
30/05/2005, 11:30 — Room P3.10, Mathematics Building

### Traffic Modeling of Voice Over IP

Voice over IP (VoIP) uses TCP/IP as the transport network for voice conversations, taking advantage of its statistical multiplexion and allowing the usage of 'free' transport networks as the Internet. The main shortcoming of VoIP are: possibility of data losses and increase of transport delay. These impairments can severely affect the perceived conversation's quality and must be carefully avoided by the appropriate use of queueing analysis, connection admission control algorithms and dimensioning methods. In all these cases, an accurate VoIP traffic modeling is needed. This seminar is aimed to introduce the modeling of VoIP traffic, taking into account the effect of the modern codec used in VoIP: the Voice Activity Detection and the Confort Noise Generation. The one voice source case is first introduced. After that, the multiplexion of several sources is addressed, and the main well known models (Fluid Model and Markov modulated Poisson process) are adapted to the VoIP scenario. Finally a comparative study of the adequacy of the existing models concludes the seminar.

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