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### 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.

### 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.

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

This talk focuses on the problem of testing the null hypothesis $H_{0\boldsymbol{\beta}}:\boldsymbol{\beta}=\boldsymbol{\beta}_o$ under a semiparametric partly linear regression model, $y_i=\boldsymbol{x}_i' +g(t_i)+\epsilon_i$, $1\leq i\leq n$, 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.

### Estimação de um modelo de equações simultâneas usando o método da regressão quantílica

O método da regressão quantílica é um método de estimação que se baseia numa generalização do conceito de regressão, recorrendo à estimação de quantis da distribuição condicional associada ao modelo. Enquanto que os métodos de estimação usuais em modelos de regressão têm como objectivo estimar o valor médio da distribuição condicional da variável resposta, a regressão quantílica considera a estimação dos quantis da distribuição condicional. Ao ter em conta a estimação de diversos quantis da distribuição, esta técnica permite obter informação mais completa sobre a distribuição condicional no seu todo.

O método da regressão quantílica foi introduzido por Koenker e Bassett (1978) e, dadas as suas potencialidades, tem vindo a ser usado com bons resultados na estimação de parâmetros em diversos modelos estatísticos. Uma das áreas de aplicação em que a regressão quantílica tem despertado maior interesse nos últimos anos, tem sido a área socio-económica. No presente trabalho, vai considerar-se a estimação dos parâmetros de um modelo de equações simultâneas através da regressão quantílica. Os modelos de equações simultâneas são modelos estatísticos fundamentais em econometria; são caracterizados por sistemas de equações que traduzem a dependência de um conjunto de variáveis relativamente a um outro conjunto, admitindo-se a existência de relações de interdependência entre as diversas variáveis. O processo é ilustrado com um modelo proposto por Portela (2001), constituído por duas equações comportamentais e que se destinou ao estudo dos salários e do nível de escolaridade portugueses no período de 1985 a 1997.

### Fluid Buffers - Changing the Behaviour at the Borders

The application of matrix-analytic methods to the resolution of fluid queues has shown a close connection to discrete-state quasi-birth-and-death (QBD) processes. We further explore this similarity and analyze fluid queues with finite and infinite capacities, for which the evolution of the buffer content changes when it is either empty or full. We briefly indicate how the stationary density of the fluid buffer may be computed in an efficient manner.

### Robust Procedures for Semiparametric Partly Linear Autoregression

In many situations, a fully nonparametric autoregressive process, $\{y_t\}$, can neglect a possible linear relation between $y_t$ and any lag $y_{t-k}$ and so, it may be sensible to fit a partly linear autoregressive model.

In the simplest partly linear autoregression model, the stationary process $\{y_t: t\geq 3\}$ satisfies $$$y_t=\beta y_{t-1}+ g(y_{t-2})+\epsilon_t, \label{eq:1:507}$$$ with $\epsilon_t$ i.i.d. independent of $\{y_{t-j}, j\geq 1\}$, $E(\epsilon_t)=0$ finite $E\epsilon_t^2$.

The sensitivity of the least squares estimates to outliers has been extensively described both in the purely parametric and in the nonparametric setting. The sensitivity to outliers of the classical estimates under a partly linear autoregression model ($\ref{eq:1:507}$) is good evidence that robust methods, less sensitive to a single wild spike outlier, would be desirable, since the effect of a single outlier is even worse than in the independent setting.

In this talk, which corresponds to a joint work with Ana Bianco, the problem of obtaining a family of robust estimates for model ($\ref{eq:1:507}$) is addressed introducing a three–step robust procedure whose asymptotic behavior is derived. A robust procedure to choose the smoothing parameter is also discussed. Through a Monte Carlo study, the performance of the proposed estimates is compared with the classical ones. Moreover, a procedure to detect anomalous observations is discussed.

### A semi-Markov approach to the analysis of fluid queues

Fluid queues in a random environment are used to modeltelecommunication networks and are amenable to a variety of approachesfor solution. Recently, it has been shown that they may be solved byfollowing the same renewal-type arguments as for quasi-birth-and-deathprocesses. The advantages are that one may concentrate on thestructural behaviour of the process and rely on fast, numericallystable algorithmic procedures. We shall briefly describe theconceptual approach and discuss the infinite and finite buffer cases.

### Modelos de Contagem e Somas Aleatórias

Dados reais exibem frequentemente padrões de aleatoriedade complexos, que podem ser transmitidos por modelos discretos sofisticados. Muitos desses modelos podem ser agrupados em famílias caracterizadas por relações de recorrência, que permitem desenvolver formas eficientes de cálculo de densidades de somas aleatórias, como a relação de Panjer. Discute-se uma generalização das classes de Panjer, e aplicações ao estudo de somas aleatórias. A aplicação a dados reais justifica reflexão sobre as leis de Zipf-Mandelbrot, discutindo-se ainda uma "lognormal discreta" e em que sentido esta suscita a construção de extensões das leis de Mandelbrot para fenómenos auto-organizativos.

Investigação parcialmente financiada por FCT/POCTI/FEDER (Projecto VEXTRA).

### Nonparametric estimation of the odds-ratios assuming Generalized Additive Models: The multivariate case including interactions

Calculating odds-ratios (OR) and corresponding confidence intervals (CI) for exposures that have been measured using a continuous scale presents important limitations in the traditional practice of analytic epidemiology. Approximations based on linear models require making arbitrary assumptions about the shape of the relation curve, or about its breakpoints. Categorical analyses generally have low statistical efficiency and cut-off points for the categories are in most cases arbitrary and/or opportunistic. Recent publications in epidemiology have shown an interest in the application of a more general flexible regression technique such as the Generalized Additive Models (GAM) (Hastie and Tibshirani, 1990). This modern regression has the advantage of not assuming a parametric relation between exposure and effect, and eliminates the need for the investigator to impose functional assumptions in this respect. The only assumption required is that the effect of the continuous covariate follows an arbitrary continuous smooth function. In this talk we propose the use of GAM to derive the corresponding nonparametric estimates (and CIs) by means of the Local Scoring Algorithm. This procedure permits great flexibility and adequate statistical efficiency. Definition of the nonparametric OR curve is also extended to the case of having second-order interactions between two continuous covariates .In this context, inferential issues are solved by means of bootstrap resampling techniques. Finally, we illustrate these new methods through several real data sets.

### Extremes of Volterra Series Expansions with Heavy-Tailed Innovations

Linear time series models have widely been used in many areas of science. However, it is becoming increasingly clear that linear models do not represent adequately features often exhibited by data sets from environmental and economical processes. One property which is important and needs exploration is the study of sudden bursts of large positive and negative values the sample paths of non-linear processes generally produce.

There is no unifying theory that is applicable to all non-linear systems and consequently the study of such systems has to be restricted to special classes of non-linear models. Volterra series expansions or polynomial systems are one such class. It is well known that Volterra series expansions are known to be the most general non-linear representation for any stationary sequence.

Volterra series expansions have found significant applications in signal processing such as image, video, and speech processing where the non-linearities are mild enough to allow approximations by low order polynomial approximations. In view of these facts, the study of the extremal properties of finite order Volterra series expansions would be highly valuable in better understanding the extremal properties of non-linear processes as well as understanding the order of identification and adequacy of Volterra series, when used as models in signal processing. In fact, such extremal properties may suggest a way of finding finite order Volterra expansions which is consistent with the non-linearities of the observed process.

In this work, we look at the extremal behavior of Volterra series expansions generated by heavy-tailed innovations, via a point process formulation. We also look at some examples of bilinear processes and compare the known results on extremes of such processes with the corresponding adequate Volterra series approximations. A way of determining the proper order of a finite Volterra expansion and hence a finite order polynomial model for the observed non-linear data is also suggested.

### Robust Estimation of the Parameters in the FANOVA Model

We present a robust approach for fitting models with both additive and multiplicative effects of two-way tables. The method is an extension of the median polish technique and uses a robust alternating regression algorithm. The approach is highly robust, and also works well when there are more variables than observations. The model can be simplified to a purely additive or multiplicative model, the latter allows to construct a robust biplot being not predetermined by outliers. The performance of the method will be illustrated by real and artificial examples.

### Uso de Métodos MCMC em Análise Bayesiana de Dados de Sobrevivência

A análise de dados de sobrevivência em geral apresenta dados incompletos e presença de covariáveis. O uso de modelos paramétricos para esses dados pode envolver um grande número de parâmetros. Além disso, os modelos paramétricos usuais podem não ser adequados para muitos conjuntos de dados, pois a função de risco pode ter formas diversas como forma de banheira, multimodalidade, riscos crescentes ou decrescentes, entre outras. Por isso, consideramos modelos mais complexos como modelos de misturas de distribuições paramétricas na presença ou não de covariáveis.

O uso de métodos bayesianos para esses modelos pode ser muito simplificado a partir de métodos de simulação de Monte Carlo em Cadeias de Markov (MCMC), como os algoritmos de Gibbs sampling e Metropolis-Hastings. Também podemos usar critérios bayesianos para discriminar diferentes modelos usados para os dados de sobrevivência, usando estimativas de Monte Carlo a partir das amostras geradas pelo algoritmo Gibbs sampling para a função preditiva. Vários exemplos com dados reais serão considerados para ilustrar a metodologia proposta.

### Taxas de Alarme em Esquemas de Controlo de Qualidade

O desempenho de esquemas de controlo de qualidade é usualmente avaliado à custa de características do run length (RL) - o número de amostras recolhidas até à emissão de um alarme. O average run length (ARL) é de longe a mais popular dessas características e tem sido - extensiva e incorrectamente - utilizado na literatura para descrever o desempenho de um esquema de controlo.

O uso da função taxa de falha de RL foi proposto por Margavio et al. (1995), e quando avaliada em $m=1,2,\dots$, representa a probabilidade de ser emitido alarme pela amostra $m$, sabendo que as $m-1$ amostras anteriores não foram responsáveis pela emissão desse alarme.

Esta função pode ser entendida como uma taxa de alarme, fornece um retrato condicional e mais completo do desempenho de esquemas e será estudada para alguns esquemas de controlo do tipo markoviano.

Daremos destaque à influência das matrizes estocasticamente monótonas no comportamento da taxa de alarme e ilustraremos alguns resultados numéricos e estocásticos que lhe dizem respeito. De notar que tais resultados estocásticos permitem avaliar, por exemplo, o impacto da adopção de head starts no desempenho de esquemas de controlo, de modo qualitativo e mais objectivo.

#### Referências

• Margavio, T.M., Conerly, M.D., Woodall, W.H. and Drake, L.G. (1995). Alarm rates for quality control charts. StatisticsProbability Letters 24, 219-224.

### Modelação Estocástica para Redes sem Fios da Próxima Geração

As redes sem fios da próxima geração são vistas hoje em dia como um dos factores chaves para o desenvolvimento da infra-estrutura de comunicação global emergente. O seu desenho, planeamento e controlo devem ser suportados por modelos de tráfego adequados, nos quais a mobilidade e os novos aspectos de teletráfego serão considerados de uma forma integrada.

### Multiplicative Survival Models based on Counting Processes

Modern survival analysis may be effectively handled within the mathematical framework of counting processes. This theory introduced by Aalen (1972) has been the subject of intense research ever since. In this setting, emphasis is given to construction of the likelihood function due to its importance for both frequentist and Bayesian analysis. Some multiplicative survival models are here presented from a Bayesian perspective, especially frailty models for univariate survival data. A gamma process with independent increments in disjoint intervals is used to model the prior process of the baseline hazard function, and the frailty distribution is assumed to be a gamma distribution. Markov Chain Monte Carlo methods are used to find estimates of several quantities of interest. At last, this approach is illustrated with one example.

### Simultaneous and Multivariate Control Charts for Time Series Data

In this talk we present several new control charts for univariate as well as multivariate time series. All control schemes are EWMA (exponentially weighted moving average) charts.

First, simultaneous control schemes for the mean and the autocovariances of a univariate stationary process are introduced. A multivariate quality characteristic is considered. This quantity is transformed to a one-dimensional variable by using the Mahalanobis distance. The test statistic is obtained by smoothing this variable. Another control chart is based on a multivariate EWMA attempt which is directly applied to our quality characteristic. After that the resulting statistic is transformed to a univariate random variable. Besides modified control charts we consider residual charts, too. In an extensive simulation study all control schemes are compared with each other. The target process is assumed to be an ARMA(1,1) process with normal white noise.

EWMA control charts for multivariate time series were discussed by Kramer and Schmid (1997). Their aim was to find deviations in the mean behaviour. Here we focus on charts detecting changes between the cross-covariances of a multivariate stationary process. The starting point is again a multivariate characteristic. To introduce control charts a similar procedure is chosen as described above. In our comparison study the target process is taken as a 4-variate AR(1) process.

#### References

• Kramer, H. and Schmid, W. (1997). EWMA charts for multivariate time series. Sequential Analysis 16, 131-154.
• Rosolowski, M. and Schmid, W. (2001). EWMA charts for monitoring the mean and the autocovariances of stationary Gaussian process. Submitted for publication.
• Schmid, W. and Sliwa, P. (2001). Monitoring the cross-covariances of a multivariate time series. Technical Report.