# Probability and Statistics Seminar

## Past sessions

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.
04/02/2005, 14:30 — Room P3.31, Mathematics Building
Frank Critchley, The Open University, UK

### Skewness a la mode?

A new approach to measuring skewness of univariate distributions is developed. A corresponding notion of kurtosis follows naturally. Further developments are briefly indicated.
02/02/2005, 14:00 — Room P3.10, Mathematics Building
Isabel Pereira, Departamento de Matemática, Universidade de Aveiro / U&D Matemática e Aplicações

### Propriedades, Estimação e Predição em Modelos Bilineares com Erros Exponenciais

Em muitas situações reais, além de se estar perante fenómenos com saltos em instantes aleatórios, as observações que constituem a série poderão apresentar um grande enviesamento, serem estritamente positivas com valores muito pequenos, próximos de zero. Um processo que poderá modelar este tipo de situações, e que se irá considerar neste trabalho, é o modelo bilinear $\mathrm{BL}\left(1,0,1,1\right)$ com erros exponenciais. Em particular, obtêm-se as condições sob as quais o modelo é estritamente estacionário e apresentam-se algumas propriedades da distribuição estacionária, em termos dos seus momentos. Sugerem-se duas metodologias para a estimação de parâmetros, no domínio temporal e no domínio da frequência, respectivamente a abordagem bayesiana e o critério de Whittle. Os procedimentos propostos são ilustrados e comparados através de um estudo de simulação. Finalmente, faz-se ainda uma breve análise de predição, usando a metodologia Bayesiana para fazer a previsão da observação futura.
14/01/2005, 14:00 — Room P3.31, Mathematics Building

### Caracterização Estatística de Tráfego Internet

12/01/2005, 14:00 — Room P3.31, Mathematics Building
Graciela Boente, Departamento de Matemática, Universidade de Buenos Aires

### Robust bandwidth selectors in semiparametric partly linear regression models

Consider a semiparametric partly linear model, with response variable $y$ and covariates ${x}_{1},\dots ,{x}_{p}$ and $t$. This model can be a suitable choice when one suspects that the response depends linearly on $x$, but that it is nonlinearly related to $t$. Least square estimators have been studied by several authors. All these estimators, as nonparametric estimators, depend on a smoothing parameter that should be chosen by the practitioner. As it is well known, large bandwidths produce estimators with small variance but high bias, while small values produce more wiggly curves. This trade-off between bias and variance lead to several proposals to select the smoothing parameter, such as cross-validation procedures and plug-in methods. It is well known that, both in linear regression and in nonparametric regression, least squares estimators can be seriously affected by anomalous data. The same statement holds for partly linear models. To avoid that problem, Bianco and Boente (2003) considered a three-step robust estimate for the regression parameter and the regression function. In this talk, we will introduce a robust plug-in selector for the bandwidth, under a partly linear model with fixed design which converges to the optimal one and leads to robust data-driven estimates of the regression function and the regression parameter. Our plug-in proposal is based on nonparametric robust estimates of the $j$-th derivatives, which extends the proposals given when $j=2$. We define an empirical influence measure for data-driven bandwidth selectors and, through it, we study the sensitivity of the plug-in selector. We use a Monte Carlo study to compare the performance of the classical approach and of the resistant selectors under normality and contamination. It appears that the robust selector compares favourably to its competitor, despite the need to select a pilot bandwidth. When combined with the three-step procedure proposed by Bianco and Boente (2003), it leads to robust data-driven estimates both of the regression function and the regression parameter.
22/11/2004, 17:15 — Room P3.10, Mathematics Building
Carlos Alberto B. Pereira, Instituto de Matemática e Estatística, Universidade de São Paulo, Brasil

### Informação Estatística e Análise Pré-posteriori

Esta palestra tem um carácter académico mais do que científico. O nosso objectivo é discutir o conceito de informação estatística como nos foi apresentado pelo Professor Dev Basu. Apresentaremos um exemplo simples de bolas em urnas. Com esse exemplo discutiremos o conceito de informação e mostraremos que podemos perder informação quando realizamos novos experimentos. Usaremos o conceito de Informação de DeGroot e suficiência de Blackwell para escolher experimentos. Por último mostraremos como podemos estabelecer o tamanho de amostra mínimo para atingir objectivos bem definidos.
03/11/2004, 15:00 — Room P3.31, Mathematics Building
Vilius Benetis, Technical University of Denmark

### Capacity Management in Cellular Hierarchical Networks

An efficient utilisation of the radio resources in mobile communications is of a great importance. In general a high degree of sharing is efficient, but requires service protection mechanisms to guarantee the Quality of Service for all customers. We study the effect of cell breathing and overlapping along with hierarchical cell structures. We show that by call packing we obtain a high utilisation. The transformation from cell-based network to direct routing network model is used to carry out calculations. The models in discussion are a generalisation of the Erlang-B formula, including general arrival processes and multi-rate (multi-media) traffic for second and third generation systems.
03/06/2004, 11:00 — Room P10, Mathematics Building
Margarida Rocheta, Liliana Marum e Susana Tereso, IBET - Grupo Pinus

### Um olhar sobre o reino vegetal: a biotecnologia aplicada ao pinheiro bravo

Neste seminário focam-se aspectos gerais sobre o desenvolvimento de uma planta. Em particular é discutida a questão de clonagem de pinheiros feita a partir de árvores melhoradas, de valor florestal comprovado. No seguimento desta questão discute-se como é possível criopreservar embriões de pinheiro por tempo indefinido. Uma vez que o termo chave é transformação genética, neste seminário será discutida o que é, como é efectuada e quais os resultados.
27/04/2004, 14:30 — Room P4.35, Mathematics Building
Marcel F. Neuts, Department of Systems and Industrial Engineering, The University of Arizona

### Exploring Finite Markov Chains by the Systematic Computation of Descriptors

We try to gain insight into the deeper physical behavior of a finite Markov chain by systematically computing quantities related to the visits to a string of nested sets of states. The choice of the successive states added to the nested sets is called an exploratory strategy. The strategy is constructed by focusing of the physical property to be explored. Quantities that serve as criteria in one strategy are reported as descriptors for the other strategies. This is a promising tool for the exploration of finite discrete-time Markov chains. Similar methods can be developed for continuous-time chains and Markov renewal processes, but the required computational methods are substantially different. We believe that this methodology may find applications, among other areas, in genetics and linguistics. The existing Markov chain analysis should be complemented by data analytic procedures applied to real or simulated data bases. The exploration in parallel of the Markov chains and suitable data sets can serve to develop the skills needed to gain reliable insights from the models and from the data sets.

Este Seminário é uma organização conjunta do CEMAT- Grupo 3 e da Sociedade Portuguesa de Estatística

20/04/2004, 11:00 — Room P3.10, Mathematics Building
Nuno Borralho, Director Florestal, Instituto de Investigação da Floresta e Papel, RAIZ

### Determinar o mérito genético de uma árvore

Um dos elementos centrais do sucesso do melhoramento genético é a capacidade de poder prever, com base num modelo genético simples e em métodos estatísticos apropriados, qual o valor genético de um indivíduo. Esta ciência designa-se por Genética Quantitativa. O seu objecto de estudo é poder dizer qual o mérito dos genes que contém (em relação à média da população a que pertence) e que levam a que tenha uma performance melhor. Irei apresentar resumidamente o modelo genético subjacente, os métodos estatísticos mais comuns e os desafios de análise estatística que encontramos na análise de dados reais.
16/04/2004, 13:00 — Room V003, Civil Engineering Building
Marcel F. Neuts, Department of Systems and Industrial Engineering, The University of Arizona

### A random walk on the unit interval

We examine a Markov chain model for a random walk on the unit interval. That model is the subject of several problems, starting with 6.4.34, pp. 321 ff. in the book Marcel F. Neuts, "Algorithmic Probability: A Collection of Problems", Chapman and Hall, New York, New York, 1995. The random walk is studied by analytic, numerical and computer-experimental methods. Each of these approaches complement the others. Together, they offer an example of the diverse, hybrid methods that can be brought to bear on the contemporary problems of applied mathematics.
30/03/2004, 14:30 — Room P3.31, Mathematics Building
Susana S. Neves, ITQB, Instituto de Tecnologia Química e Biológica, Univ. Nova de Lisboa

### Phylogenetics: discovering the tree of life

Brief history of phylogenetic analysis, and some of the related controversies ("cladistics vs. phenetics"). Introduction to terms, principles and methods of phylogenetics, including parsimony, likelihood and distance based approaches. Particular emphasis will be given to the analysis of DNA sequence data. Some of the problems that affect phylogenetic analysis will be discussed, such as homology assessment, horizontal gene transfer (HGT), "gene trees vs. species trees". Information on the most commonly used software will be provided, including a short demonstration on the use of PAUP* (Phylogenetic Analysis Using Parsimony, *and other methods).

The presentation will be in Portuguese (or in English, if requested), with transparencies in English.

16/03/2004, 14:30 — Room P3.31, Mathematics Building
Cláudia Nunes, Departamento de Matemática, IST

Nesta apresentação é feita uma análise de um modelo espaço-temporal, o qual descreve o movimento colectivo de partículas no espaço e no tempo. Considera-se um conjunto E (eventualmente infinito, mas numerável), ao qual chegam partículas de acordo com uma cadeia Markov (K,X). K designa o ambiente aleatório que rege o processo (sendo que K é uma cadeia de Markov homogénea) e X designa o número de partículas que entram no conjunto E em cada instante. Assume-se que o número de partículas geradas em determinado instante depende apenas da transição ocorrida na cadeia K, pelo que o processo bivariado (K,X) é uma cadeia Markov modulada.

Uma vez entradas no conjunto E, as partículas movem-se ao longo dos elementos de E de forma condicionalmente independente (dada a transição no ambiente e o número de partículas geradas), e de acordo com uma cadeia de Markov absorvente em tempo finito.

Para este sistema é feita uma análise do tipo sample path, apresentando-se nomeadamente leis de médias para diversos funcionais de interesse, nomeadamente:

• Taxa de throughput de A para B, onde A e B são subconjuntos (disjuntos) de E;
• Taxa de novas visitas a um dado subconjunto.

Este trabalho é um trabalho conjunto de Nelson Antunes, Cláudia Nunes e António Pacheco.

05/02/2004, 14:30 — Amphitheatre Pa2, Mathematics Building
Daniel Gianola, University of Wisconsin-Madison, USA

### Quantitative genetic models for describing simultaneous and recursive relationships between phenotypes

12/01/2004, 14:30 — Room P4.35, Mathematics Building
Stefano M. Pagnotta, Università degli Studi del Sannio a Benevento, Italy

### Approximation algorithms for the estimate of the MCD and a new proposal

In the multidimensional framework the robust estimation of the location and the covariance matrix is a highly expensive computational task. A popular estimator is the Minimum Covariance Determinant (MCD; Rousseeuw, 1984, 1985). Different authors proposed approximation algorithms for this estimator. Recently Rousseeuw and van Driessen (1999) seem to stop the competition in providing a fast and good approximation to the MCD with their procedure called FAST-MCD. This algorithm works fine when the spatial configuration of data contains either radial outliers or clusters of outliers having dispersion higher than that of the good points. When the cluster of outlying observations has a dispersion lower than that of the good points the FAST-MCD shows some drawbacks. This behavior highlights some remarks about the robustness of the MCD. In the talk we review the MCD estimator and some algorithms for its approximation, we discuss about the source of failure of the estimator, and we present a new procedure.
05/01/2004, 14:30 — Room P4.35, Mathematics Building
Rui Paulo, National Institute of Statistical Sciences and Statistical and Applied Mathematical Sciences Institute, North Carolina, USA

### Default Priors for Gaussian Processes

Motivated by the statistical evaluation of complex computer models, we deal with the issue of objective prior specification for the parameters of Gaussian processes. In particular, we derive the Jeffreys-rule, independence Jeffreys and reference priors for this situation, and prove that the resulting posterior distributions are proper under a quite general set of conditions. Another prior specification strategy, based on maximum likelihood estimates, is also considered, and all priors are then compared on the grounds of the frequentist properties of the ensuing Bayesian procedures. Computational issues are also addressed in the paper, and we illustrate the proposed solutions by means of an example taken from the field of complex computer model validation.
03/12/2003, 10:00 — Room P3.10, Mathematics Building
Graciela Boente, CONICET and Universidad de Buenos Aires

### Robust tests for the regression parameter in semiparametric partly linear models

This talk focuses on the problem of testing the null hypothesis ${H}_{0\beta }:\beta ={\beta }_{o}$ under a semiparametric partly linear regression model, ${y}_{i}={x}_{i}\text{'}\beta +g\left({t}_{i}\right)+{\epsilon }_{i}$, $1\le i\le n$, by using a three-step robust estimate for the regression parameter and the regression function. Two families of tests statistics are considered and their asymptotic distribution are studied under the null hypothesis and under contiguous alternatives. A Monte Carlo study is performed to compare the finite sample behavior of the proposed tests with the classical ones.
20/11/2003, 16:00 — Room P3.31, Mathematics Building
Thelma Sáfadi, Departamento de Ciências Exatas, Universidade Federal de Lavras, Lavras- Minas Gerais- Brasil