Contrary to most of the papers in the literature of investment under uncertainty we study models that not only capture the timing, but also the size of the investment. We consider a monopoly setting as well as a duopoly setting and compare the results with the standard models in which the firms do not have the capacity choice. Our main results are the following. First, for low uncertainty values the follower chooses a higher capacity than the leader and for high uncertainty values the leader chooses a higher capacity. Second, compared to the model without capacity choice, the monopolist and the follower invest later in a higher capacity for higher values of uncertainty. However, the leader will invest earlier in a higher capacity for higher values of uncertainty. The reverse results apply for lower values of uncertainty.
In many situations, the random variables take values in a Riemannian manifold $(M, g)$ instead of $\mathbb{R}^d$, and this structure needs to be taken into account when we generate estimation procedures. For the nonparametric regression model, we study two families of robust estimators for the regression function when the explanatory variables take values in a Riemannian manifold.
In this talk, we will give a brief introduction of the geometric objects needed to define the nonparametric estimators adapted to a manifold. We discuss the classical proposals and we introduce two families of robust estimators for the regression function. We show the asymptotic properties obtained for both proposal. Finally, through a simulation study, we compare the behavior of the robust estimators against the alternative classic. This is a joint work with Guillermo Henry.
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Room P2, Mathematics Building, IST
Paulo Rodrigues, Banco de Portugal and Universidade Nova de Lisboa
In this paper we discuss new tests for predictability which are inspired in the work of Vogelsang (1998) on testing for trend. The proposed tests, use by design the same critical values irrespectively of whether the predictor is or and are therefore capable of detecting a more general set of alternatives, which are presently by available procedures (exceptions being the tests of Deo and Chen, 2008 and Maynard and Shimotsu, 2009). Numerical evidence suggests that our proposed procedures have good finite sample performance which coupled with the simplicity of application makes them appealing approaches for empirical research and useful alternatives to available procedures.
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Amphitheatre Pa2, Mathematics Building
Sujit Samanta, Universidade Técnica de Lisboa - Instituto Superior Técnico e CEMAT
This paper considers a single-server queueing model with finite and infinite buffers in which customers arrive according to a discrete-time renewal process. The customers are served one at a time under discrete-time Markovian service process (D-MSP). This service process is similar to the discrete-time Markovian arrival process (D-MAP), where arrivals are replaced with service completions. Using the imbedded Markov chain technique and the matrix-geometric method, we obtain the system-length distribution at a prearrival epoch. We also provide the steady-state system-length distribution at an arbitrary epoch by using the supplementary variable technique and the classical argument based on renewal-theory. The analysis of actual waiting-time (in the queue) distribution (measured in slots) has also been investigated. Further, we derive the coefficient of correlation of the lagged interdeparture intervals. Moreover, computational experiences with a variety of numerical results in the form of tables and graphs are discussed.
During the last decades there has been considerable interest in integer-valued time series models and a large volume of work is now available in specialized monographs. Motivation to study discrete data models comes from the need to account for the discrete nature of certain data sets, often counts of events, objects or individuals. Examples of applications can be found in the analysis of time series of count data in many areas. Among the most successful integer-valued time series models proposed in the literature are the INteger-valued AutoRegressive model of order 1 (INAR(1)). In this talk the statistical and probabilistic properties of the INAR(1) models are reviewed.
At the end of the XIXth Century, the classical definition of Probability and its extension to the continuous case were too restrictive and some geometrical applications, based in ingenious interpretations of Bernoulli-Laplace principle of insufficient reason, led to several paradoxes. David Hilbert, in his celebrated address at the International Congress of Mathematicians of 1900, included the axiomatization of Probability in his list of 23 important unsolved problems. Only in 1933 did Kolmogorov lay down a rigorous setup for Probability, inspired by Fréchet’s idea of using Measure Theory. But before this some other efforts to build up a proper axiomatization of Probability deserve to be more widely credited. Among those, the construction of Diogo Pacheco d’Amorim, in his 1914 doctoral thesis, is one of the most interesting. His discussion of a standard model, based on the idea of random choice instead of the concept of probability itself, seems limited, but his final discussion on how to use the law of large numbers and the central limit theorem to have an objective appraisal of whether sampling made by others, or even by a mechanical device, is indistinguishable from a random choice made by one-self, is impressive, since it anticipates the ideas of Monte Carlo by almost 30 years.
We adapt the method developed in Paige, Trindade, and Fernando (2009) in order to make approximate inference on optimal smoothing parameters for penalized spline, and partially linear models. The method is akin to a parametric bootstrap where Monte Carlo simulation is replaced by saddlepoint approximation, and is applicable whenever the underlying estimator can be expressed as the root of an estimating equation that is a quadratic form in normal random variables. This is the case under a variety of common optimality criteria such as ML, REML, GCV, and AIC. We apply the method to some well-known datasets in the literature, and find that under the ML and REML criteria it delivers a performance that is nearly exact, with computational speeds that are at least an order of magnitude faster than exact methods. Perhaps most importantly, the proposed method also offers a computationally feasible alternative where no known exact methods exist, e.g. GCV and AIC.
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Room P4.35, Mathematics Building
Ana Pires, Universidade Técnica de Lisboa - Instituto Superior Técnico and CEMAT
Gregor Mendel (1822-1884) is almost unanimously recognized as the founder of modern genetics. However, long ago, a shadow of doubt was cast on his integrity by another eminent scientist, the statistician and geneticist, Sir Ronald Fisher (1890-1962), who questioned the honesty of the data that form the core of Mendel's work. This issue, nowadays called "the Mendel-Fisher controversy", can be traced back to 1911, when Fisher first presented his doubts about Mendel's results, though he only published a paper with his analysis of Mendel's data in 1936.
A large number of papers have been published about this controversy culminating with the publication in 2008 of a book (Franklin et al., "Ending the Mendel-Fisher controversy"), aiming at ending the issue, definitely rehabilitating Mendel's image. However, quoting from Franklin et al., "the issue of the `too good to be true' aspect of Mendel's data found by Fisher still stands".
We have submitted Mendel's data and Fisher's statistical analysis to extensive computations and simulations attempting to discover an hidden explanation or hint that could help finding an answer to the questions: is Fisher right or wrong, and if Fisher is right is there any reasonable explanation for the "too good to be true", other than deliberate fraud? In this talk some results of this investigation and the conclusions obtained will be presented.
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Room P3.10, Mathematics Building
Graciela Boente, Universidad de Buenos Aires and CONICET, Argentina
he generalized linear model GLM (McCullagh and Nelder, 1989) is a popular technique for modelling a wide variety of data and assumes that the observations are independent such that the conditional distribution of y|x belongs to the canonical exponential family. In this situation, the mean $E(y|x)$ is modelled linearly through a known link function. Robust procedures for generalized linear models have been considered among others by Stefanski et al. (1986), Künsch et al. (1989), Bianco and Yohai (1996), Cantoni and Ronchetti (2001), Croux and Haesbroeck (2002) and Bianco et al. (2005). Recently, robust tests for the regression parameter under a logistic model were considered by Bianco and Martínez (2009).
In practice, some response variables may be missing, by design (as in two-stage studies) or by happenstance. As it is well known, the methods described above are designed for complete data sets and problems arise when missing responses may be present, while covariates are completely observed. Even if there are many situations in which both the response and the explanatory variables are missing, we will focus our attention only when missing data occur only in the responses. Actually, missingness of responses is very common in opinion polls, market research surveys, mail enquiries, social-economic investigations, medical studies and other scientific experiments, where the explanatory variables can be controlled. This pattern is common, for example, in the scheme of double sampling proposed by Neyman (1938). Hence, we will be interested on robust inference when the response variable may have missing observations but the covariate x is totally observed.
In the regression setting with missing data, a common method is to impute the incomplete observations and then proceed to carry out the estimation of the conditional or unconditional mean of the response variable with the completed sample. The methods considered include linear regression (Yates, 1933), kernel smoothing (Cheng, 1994; Chu and Cheng, 1995) nearest neighbor imputation (Chen and Shao, 2000), semiparametric estimation (Wang et al., 2004, Wang and Sun, 2007), nonparametric multiple imputation (Aerts et al. , 2002, González-Manteiga and Pérez-Gonzalez, 2004), empirical likelihood over the imputed values (Wang and Rao, 2002), among others. All these proposals are very sensitive to anomalous observations since they are based on least squares approaches.
In this talk, we introduce a robust procedure to estimate the regression parameter under a GLM model, which includes, when there are no missing data, the family of estimators previously studied. It is shown that the robust estimates of are root-$n$ consistent and asymptotically normally distributed. A robust procedure to test simple hypothesis on the regression parameter is also considered. The finite sample properties of the proposed procedure are investigated through a Monte Carlo study where the robust test is also compared with nonrobust alternatives.
We will discuss the role of Mathematics in Industry and in innovation processes. The focus will be European and we will look at good examples provided e.g. by the experiences of the network European Consortium for Mathematics in Industry (ECMI). I will also present the ongoing ESF Forward Look: "Mathematics and Industry" (see http://www.ceremade.dauphine.fr/FLMI/FLMI-frames-index.html) and discuss possible future developments on a European scale.
