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Concept# Théorie de l'estimation

Résumé

En statistique, la théorie de l'estimation s'intéresse à l'estimation de paramètres à partir de données empiriques mesurées ayant une composante aléatoire. Les paramètres décrivent un phénomène physique sous-jacent tel que sa valeur affecte la distribution des données mesurées. Un estimateur essaie d'approcher les paramètres inconnus à partir des mesures. En théorie de l'estimation, deux approches sont généralement considérées:

- l'approche probabiliste (décrite ici) suppose que les données mesurées sont aléatoires avec une distribution de probabilités dépendant des paramètres d'intérêt
- l'approche ensembliste suppose que le vecteur des données mesurées appartient à un ensemble qui dépend du vecteur des paramètres.

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Cours associés (49)

FIN-403: Econometrics

The course covers basic econometric models and methods that are routinely applied to obtain inference results in economic and financial applications.

MICRO-428: Metrology

The course deals with the concept of measuring in different domains, particularly in the electrical, optical, and microscale domains. The course will end with a perspective on quantum measurements, which could trigger the ultimate revolution in metrology.

MATH-442: Statistical theory

The course aims at developing certain key aspects of the theory of statistics, providing a common general framework for statistical methodology. While the main emphasis will be on the mathematical aspects of statistics, an effort will be made to balance rigor and intuition.

Our brain continuously self-organizes to construct and maintain an internal representation of the world based on the information arriving through sensory stimuli. Remarkably, cortical areas related to different sensory modalities appear to share the same functional unit, the neuron, and develop through the same learning mechanism, synaptic plasticity. It motivates the conjecture of a unifying theory to explain cortical representational learning across sensory modalities. In this thesis we present theories and computational models of learning and optimization in neural networks, postulating functional properties of synaptic plasticity that support the apparent universal learning capacity of cortical networks. In the past decades, a variety of theories and models have been proposed to describe receptive field formation in sensory areas. They include normative models such as sparse coding, and bottom-up models such as spike-timing dependent plasticity. We bring together candidate explanations by demonstrating that in fact a single principle is sufficient to explain receptive field development. First, we show that many representative models of sensory development are in fact implementing variations of a common principle: nonlinear Hebbian learning. Second, we reveal that nonlinear Hebbian learning is sufficient for receptive field formation through sensory inputs. A surprising result is that our findings are independent of specific details, and allow for robust predictions of the learned receptive fields. Thus nonlinear Hebbian learning and natural statistics can account for many aspects of receptive field formation across models and sensory modalities. The Hebbian learning theory substantiates that synaptic plasticity can be interpreted as an optimization procedure, implementing stochastic gradient descent. In stochastic gradient descent inputs arrive sequentially, as in sensory streams. However, individual data samples have very little information about the correct learning signal, and it becomes a fundamental problem to know how many samples are required for reliable synaptic changes. Through estimation theory, we develop a novel adaptive learning rate model, that adapts the magnitude of synaptic changes based on the statistics of the learning signal, enabling an optimal use of data samples. Our model has a simple implementation and demonstrates improved learning speed, making this a promising candidate for large artificial neural network applications. The model also makes predictions on how cortical plasticity may modulate synaptic plasticity for optimal learning. The optimal sampling size for reliable learning allows us to estimate optimal learning times for a given model. We apply this theory to derive analytical bounds on times for the optimization of synaptic connections. First, we show this optimization problem to have exponentially many saddle-nodes, which lead to small gradients and slow learning. Second, we show that the number of input synapses to a neuron modulates the magnitude of the initial gradient, determining the duration of learning. Our final result reveals that the learning duration increases supra-linearly with the number of synapses, suggesting an effective limit on synaptic connections and receptive field sizes in developing neural networks.

Time series modeling and analysis is central to most financial and econometric data modeling. With increased globalization in trade, commerce and finance, national variables like gross domestic productivity (GDP) and unemployment rate, market variables like indices and stock prices and global variables like commodity prices are more tightly coupled than ever before. This translates to the use of multivariate or vector time series models and algorithms in analyzing and understanding the relationships that these variables share with each other. Autocorrelation is one of the fundamental aspects of time series modeling. However, traditional linear models, that arise from a strong observed autocorrelation in many financial and econometric time series data, are at times unable to capture the rather nonlinear relationship that characterizes many time series data. This necessitates the study of nonlinear models in analyzing such time series. The class of bilinear models is one of the simplest nonlinear models. These models are able to capture temporary erratic fluctuations that are common in many financial returns series and thus, are of tremendous interest in financial time series analysis. Another aspect of time series analysis is homoscedasticity versus heteroscedasticity. Many time series data, even after differencing, exhibit heteroscedasticity. Thus, it becomes important to incorporate this feature in the associated models. The class of conditional heteroscedastic autoregressive (ARCH) models and its variants form the primary backbone of conditional heteroscedastic time series models. Robustness is a highly underrated feature of most time series applications and models that are presently in use in the industry. With an ever increasing amount of information available for modeling, it is not uncommon for the data to have some aberrations within itself in terms of level shifts and the occasional large fluctuations. Conventional methods like the maximum likelihood and least squares are well known to be highly sensitive to such contaminations. Hence, it becomes important to use robust methods, especially in this age with high amounts of computing power readily available, to take into account such aberrations. While robustness and time series modeling have been vastly researched individually in the past, application of robust methods to estimate time series models is still quite open. The central goal of this thesis is the study of robust parameter estimation of some simple vector and nonlinear time series models. More precisely, we will briefly study some prominent linear and nonlinear models in the time series literature and apply the robust S-estimator in estimating parameters of some simple models like the vector autoregressive (VAR) model, the (0, 0, 1, 1) bilinear model and a simple conditional heteroscedastic bilinear model. In each case, we will look at the important aspect of stationarity of the model and analyze the asymptotic behavior of the S-estimator.

Powerful mathematical tools have been developed for trading in stocks and bonds, but other markets that are equally important for the globalized world have to some extent been neglected. We decided to study the shipping market as an new area of development in mathematical finance. The market in shipping derivatives (FFA and FOSVA) has only been developed after 2000 and now exhibits impressive growth. Financial actors have entered the field, but it is still largely undiscovered by institutional investors. The first part of the work was to identify the characteristics of the market in shipping, i.e. the segmentation and the volatility. Because the shipping business is old-fashioned, even the leading actors on the world stage (ship owners and banks) are using macro-economic models to forecast the rates. If the macro-economic models are logical and make sense, they fail to predict. For example, the factor port congestion has been much cited during the last few years, but it is clearly very difficult to control and is simply an indicator of traffic. From our own experience it appears that most ship owners are in fact market driven and rather bad at anticipating trends. Due to their ability to capture large moves, we chose to consider Lévy processes for the underlying price process. Compared with the macro-economic approach, the main advantage is the uniform and systematic structure this imposed on the models. We get in each case a favorable result for our technology and a gain in forecasting accuracy of around 10% depending on the maturity. The global distribution is more effectively modelled and the tails of the distribution are particularly well represented. This model can be used to forecast the market but also to evaluate the risk, for example, by computing the VaR. An important limitation is the non-robustness in the estimation of the Lévy processes. The use of robust estimators reinforces the information obtained from the observed data. Because maximum likelihood estimation is not easy to compute with complex processes, we only consider some very general robust score functions to manage the technical problems. Two new class of robust estimators are suggested. These are based on the work of F. Hampel ([29]) and P. Huber ([30]) using influence functions. The main idea is to bound the maximum likelihood score function. By doing this a bias is created in the parameters estimation, which can be corrected by using a modification of the following type and as proposed by F. Hampel. The procedure for finding a robust estimating equation is thus decomposed into two consecutive steps : Subtract the bias correction and then Bound the score function. In the case of complex Lévy processes, the bias correction is difficult to compute and generally unknown. We have developed a pragmatic solution by inverting the Hampel's procedure. Bound the score function and then Correct for the bias. The price is a loss of the theoretical properties of our estimators, besides the procedure converges to maximum likelihood estimate. A second solution to for achieving robust estimation is presented. It considers the limiting case when the upper and lower bounds tend to zero and leads to B-robust estimators. Because of the complexity of the Lévy distributions, this leads to identification problems.

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