In this talk we consider investment decisions within an uncertain dynamic and competitive framework. Each investment decision involves to determine the timing and the capacity level. In this way we extend the main bulk of the real options theory where the capacity level is given. We consider a monopoly setting as well as a duopoly setting. Our main results are the following. In the duopoly setting we provide a fully dynamic analysis of entry deterrence/accommodation strategies. Contrary to the seminal industrial organization analyses that are based on static models, we find that entry can only be deterred temporarily. To keep its monopoly position as long as possible the first investor overinvests in capacity. In very uncertain economic environments the first investor eventually ends up being the largest firm in the market. If uncertainty is moderately present, a reduced value of waiting implies that the preemption mechanism forces the first investor to invest so soon that a large capacity cannot be afforded. Then it will end up with a capacity level being lower than the second investor.
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Room P3.10, Mathematics Building
Patrícia Ferreira, CEMAT - Departamento de Matemática - IST
Quando se pretende controlar simultaneamente o valor esperado e a variância de um processo é comum utilizar-se um esquema conjunto. Este tipo de esquema é constituído por duas cartas de controlo que operam em simultâneo, uma que controla o valor esperado e outra que controla a variância do processo. A utilização deste tipo de esquemas pode levar à ocorrência de sinais erróneos, associados, por exemplo, às seguintes situações:
o valor esperado do processo está fora de controlo, no entanto a carta para a variância emite um sinal antes da carta usada para controlar o valor esperado;
a variância do processo está fora de controlo mas a carta para o valor esperado é a primeira a emitir sinal.
Os sinais erróneos são sinais válidos que podem levar o operador de controlo de qualidade a desencadear acções inadequadas para corrigir uma causa inexistente. Posto isto, é importante considerar a frequência com que estes sinais ocorrem como uma medida de desempenho dos esquemas conjuntos. Neste trabalho analisa-se o desempenho de esquemas conjuntos do ponto de vista da probabilidade de ocorrência de um sinal erróneo com especial enfoque em esquemas conjuntos para processos univariados i.i.d. e autocorrelacionados.
Neste seminário será abordada a questão do “Qual é o Maior Salto em Comprimento ao alcance do H(h)omem, dado o actual state of the art”? Para responder a essa pergunta será usado o crème de la crème, i.e., os dados são coligidos a partir dos melhores atletas olímpicos na modalidade, a partir da base de dados do World Athletics Competitions - Long Jump Men Outdoors. Esta abordagem do problema é baseada na Teoria de Valores Extremos e as respectivas técnicas estatísticas. Usar-se-ão apenas os melhores desempenhos das World top lists. A estimativa final do potencial recorde, i.e., o limite superior do acontecimento salto em comprimento, permite inferir acerca da melhor marca individual possível, dadas as condições actuais, quer em termos de conhecimento do fenómeno, quer relativamente às condições e regras de registo na modalidade desportiva. Actualmente o recorde de 8,95m é detido por Mike Powell (USA) em Tokyo, 30/08/1991. Em Valores Extremos insere-se na estimativa do limite superior do suporte para uma distribuição no Max-domínio da Gumbel.
Palavras-chave: Valores Extremos em Desporto, Teoria de Valores Extremos, Estimação do Limite Superior do Suporte no Domínio Gumbel, Abordagem Semi-paramétrica para Estatística de Extremos.
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Room P3.10, Mathematics Building
K F Turkman, CEAUL - DEIO - FCUL - University of Lisbon
The Wold Decomposition theorem says that under fairly general
conditions, a stationary time series has a unique linear
causal representation in terms of uncorrelated random variables.
However, The Wold Decomposition theorem gives us a representation,
not a model for , in the sense that we can only recover
uniquely the moments of up to second order from this
representation, unless the input series is a Gaussian sequence. If
we look for models for , then we should look for such model
within the class of convergent Volterra series expansions. If we
have to go beyond second order properties, and many real data sets
from financial and environmental sciences indicate that we should,
then linear models with iid Gaussian input are a very tiny,
insignificant fraction of possible models for a stationary time
series, corresponding to the first term of the infinite order
Volterra expansion. On the other hand, Volterra series expansions
are not particularly useful as a possible class of models, as
conditions of stationarity and invertibility are hard to check, if
not impossible, therefore they have very limited use as models for
time series, unless the input series is observable. From a
prediction point of view, the Projection Theorem for Hilbert spaces
tells us how to obtain the best linear predictor for
within the linear span of , but when
linear predictors are not sufficiently good, it is not
straightforward to find, if possible at all, the best predictor
within richer subspaces constructed over . It is therefore important to look for classes of
nonlinear models to improve upon the linear predictor, which are
sufficiently general, but at the same time are sufficiently
flexible to work with. There are many ways a time series can be
nonlinear. As a consequence, there are many classes of nonlinear
models to explain such nonlinearities, but whose probabilistic
characteristics are difficult to study, not to mention the
difficulties associated with modeling issues. Likelihood based
inference is particularly a difficult issue as for most nonlinear
processes, we can not even write the likelihood. However, recently
there has been very exciting advances in simulation based
inferential methods such as sequential Markov Chain Monte Carlo,
Particle filters and Approximate Bayesian Computation methods for
generalized state space models which we will mention briefly.
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Room P3.10, Mathematics Building
Russell Alpizar-Jara , Research Center in Mathematics and Applications (CIMA-U.E.) Department of Mathematics, University of Évora
Capture-recapture methods have been widely used in Biological
Sciences to estimate population abundance and related demographic
parameters (births, deaths, immigration, or emigration). More
recently, these models have been used to estimate community
dynamics parameters such as species richness, rates of extinction,
colonization and turnover, and other metrics that require
presence/absence data of species counts. In this presentation, we
will use the latest application to illustrate some of the concepts
and the underlying theory of capture-recapture models. In
particular, we will review basic closed-population,
open-population, and combination of closed and open population
models. We will briefly mention about other applications of these
models to Medical, Social and Computer Sciences.
Keywords: Capture-recapture experiments; multinomial and mixture
distributions; non-parametric and maximum likelihood estimation;
population size estimation.
A forma usual de conceptualizar a representação gráfica duma matriz $X_{n\times p}$ de dados de indivíduos $\times$ variáveis consiste em associar um eixo a cada variável e nesse referencial cartesiano representar cada individuo por um ponto, cujas coordenadas são dadas pela linha de $X$ correspondente ao individuo. A popularidade desta representação no espaço dos individuos ($\mathbb{R}^p$) resulta, em grande medida, do facto de ser visualizável para dados bivariados ou tri-variados. No entanto, para um número maior de variáveis ($p \gt 3$) essa vantagem deixa de existir.
Uma representação alternativa é importante na análise e modelação dos dados. No espaço das variáveis, cada eixo corresponde a um individuo e cada variável é representada por um vector a partir da origem, definido pelas $n$ coordenadas da respectiva coluna matricial. Esta representação das variáveis em $\mathbb{R}^n$ tem a enorme vantagem de casar conceitos estatísticos e conceitos geométricos, permitindo uma melhor compreensão dos primeiros. Tem raízes sólidas na escola francesa de análise de dados, mas o seu potencial nem sempre é explorado.
Nesta comunicação começa-se por relembrar os conceitos geométricos correspondentes a indicadores fundamentais da estatística univariada e bivariada (média, desvio padrão, coeficiente de variação ou coeficiente de correlação) ou multivariada (exemplificando com o caso da análise em componentes principais). Aprofunda-se a discussão no contexto de regressões lineares múltiplas, cujos conceitos fundamentais (coeficiente de determinação, as três somas de quadrados e a sua relação fundamental) têm interpretação geométrica no espaço das variáveis.
Seguidamente, discute-se a utilidade desta representação geométrica no estudo das distâncias de Mahalanobis, que desempenham um papel de primeiro plano na estatística multivariada. Mostra-se como as distâncias (ao quadrado) de Mahalanobis medem a inclinação do subespaço de $\mathbb{R}^n$ gerado pelas colunas da matriz centrada dos dados, o subespaço $\mathcal{C}(X_c)$, em relação ao sistema de eixos. Em particular, mostra-se como as distâncias de Mahalanobis ao centro, \[D^2_{x_i,\overline{x}}=(x_i-\overline{x})^t \S^{-1} (x_i-\overline{x}),\] são apenas função de $n$ e do ângulo $\theta_i$ entre o eixo correspondente ao indivíduo $i$ e $\mathcal{C}(X_c)$, enquanto que a distância (ao quadrado) de Mahalanobis entre dois individuos, \[D^2_{x_i,x_j}=(x_i-x_j)^t \S^{-1} (x_i-x_j),\] é também função apenas de $n$ e do ângulo entre $\mathcal{C}(X_c)$ e a bissectriz gerada por $e_i-e_j$, sendo $e_i$ e $e_j$ os vectores canónicos de $\mathbb{R}^n$ associados aos dois individuos. Algumas recentes majorações e outras propriedades importantes destas distâncias (Gath & Hayes, 2006 e Branco & Pires, 2011) são expressão directa destas relações geométricas. Apesar das distâncias de Mahalanobis dizerem respeito aos individuos, os conceitos geométricos que lhes estão associados no espaço das variáveis podem ser explorados para aprofundar e estender esses resultados.
A change in a production process must be detected quickly so
that a corrective action can be taken. Thus, it comes as no
surprise that the run length (RL) is usually used to describe the
performance of a quality control chart.
This popular performance measure has a phase-type distribution
when dealing with Markov-type charts, namely, cumulative sum
(CUSUM) and exponentially weighted moving average (EWMA) charts, as
opposed to a geometric distribution, when standard Shewhart charts
are in use.
In this talk, we briefly discuss sufficient conditions on the
associated probability transition matrix to deal with run lengths
with aging properties such as new better than used in expectation,
new better than used, and increasing hazard rate.
We also explore the implications of these aging properties of
the run lengths, namely when we decide to confront the in control
and out-of-control variances of the run lengths of matched in
control Shewhart and Markov-type control charts.
Keywords
Phase-type distributions; Run length; Statistical process
control; Stochastic ordering.
Bibiography
Morais, M.C. and Pacheco, A. (2012). A note on the aging
properties of the run length of Markov-type control charts.
Sequential Analysis 31, 88-98.
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Room P3.10, Mathematics Building
Verena Hagspiel , CentER, Department of Econometrics and Operations Research Tilburg University, The Netherlands
Our paper contributes to the literature of technology adoption. In
most of these models it is assumed that after the arrival of a new
technology the probability of the next arrival is constant. We
extend this approach by assuming that after the last technology
jump the probability of a new arrival can change. Right after the
arrival of a new technology the intensity equals a specific value
that switches if no new technology arrival has taken place within a
certain period after the last technology arrival. We look at
different scenarios, dependent on whether the firm is threatened by
a drop in the arrival rate after a certain time period or expects
the rate of new arrivals to rise. We analyze the effect of variance
of time between two consecutive arrivals on the optimal investment
timing and show that larger variance accelerates investment in a
new technology. We find that firms often adopt a new technology a
time lag after its introduction, which is a phenomenon frequently
observed in practice. Regarding a firm's technology releasing
strategy we explain why clear signals set by regular and steady
release of new product generations stimulates customers buying
behavior. Depending on whether the arrival rate is assumed to
change or be constant over time, the optimal technology adoption
timing changes significantly. In a further step we add an
additional source of uncertainty to the problem and assume that the
length of the time period after which the arrival intensity changes
is not known to the firm in advance. Here, we find that increasing
uncertainty accelerates investment, a result that is opposite to
the standard real options theory.
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Room P3.10, Mathematics Building
Bruno de Sousa, Instituto de Higiene e Medicina Tropical, UNL, CMDT
A common feature of the health of men across Europe is their higher
rates of premature mortality and shorter life expectancy than
women. Following the publication of the first State of Men's Health
in Europe we sought to explore possible reasons.
We described trends in life expectancy in the European Union
member States (EU27) between 1999 and 2008 using mortality data
obtained from Eurostat. We then used Pollard's decomposition method
to identify the contribution of deaths from different causes and at
different age groups to differences in life expectancy. We first
examined the change in life expectancy for men and for women
between the beginning and end of this period. Second, we examined
the gap in life expectancy between men and women at the beginning
and end of this period.
Between 1999 and 2008 life expectancy in the EU27 increased by
2.77 years for men and by 2.12 years for women. Most of these
improvements were due to reductions in mortality at ages over 60,
with cardiovascular disease accounting for 1.40 years of the
reduction in men. In 2008 life expectancy of men in the EU27 was
6.04 years lower than that of women. Deaths from all major groups
of causes, and at all ages, contribute to this gap, with external
causes contributing 1.00 year, cardiovascular disease 1.75 years
and neoplasms 1.71 years.
Improvements in the life expectancy of men and women have
mostly occurred at older ages. There has been little improvement in
the high rate of premature death in younger men. This would suggest
a need for interventions to tackle the high death rate in younger
men. The demonstration of variations in premature death and life
expectancy seen in men within the new European Commission report,
highlight the impact of poor socio-economic conditions. The more
pronounced adverse effect on the health of men suggests that men
suffer from 'heavy impact diseases' and these are more quickly
life-limiting with women more likely to survive, but with poorer
health.