Manuel L. Esquível

Machine Learning Vasicek Model Calibration with Gaussian Processes

In this article, we calibrate the Vasicek interest rate model under the risk neutral measure by learning the model parameters using Gaussian processes for machine learning regression. The calibration is done by maximizing the likelihood of zero coupon bond log prices, using mean and covariance functions computed analytically, as well as likelihood derivatives with respect to the parameters. The maximization method used is the conjugate gradients. The only prices needed for calibration are zero coupon bond prices and the parameters are directly obtained in the arbitrage free risk neutral measure.

A conditional gaussian martingale algorithm for global optimization

A new stochastic algorithm for determination of a global minimum of a real valued continuous function defined on K, a compact set of ℝn, having an unique global minimizer in K is introduced and studied, a context discussion is presented and implementations are used to compare the performance of the algorithm with other algorithms. The algorithm may be thought to belong to the random search class but although we use Gaussian distributions, the mean is changed at each step to be the intermediate minimum found at the preceding step and the standard deviations, on the diagonal of the covariance matrix, are halved from one step to the next. The convergence proof is simple relying on the fact that the sequence of intermediate random minima is an uniformly integrable conditional Gaussian martingale.

Advances in Regression, Survival Analysis, Extreme Values, Markov Processes and Other Statistical Applications

Part I Invited Sessions: Youden Square with Split Units (Stanislaw Franciszek Mejza and Shinji Kuriki).- Likelihood and PLS Estimators for Structural Equation Modeling: an Assessment of Sample Size, Skewness and Model Misspecification Effects (Manuel J. Vilares and Pedro S. Coelho).- Part II Communications: A Parametric Cure Model with Covariates (Ana M. Abreu and Cristina S. Rocha).- Survival Analysis Applied to the Study of Time from Diagnosis of HIV-1 Infection to AIDS in Portugal (Marta Alves, Cristina S. Rocha and Maria Teresa Paixao).- A new Independence Test for VaR Violations (P. Araujo Santos and M. I. Fraga Alves).- Discrimination Between Parametric Survival Models for Removal Times of Bird Carcasses in Scavenger Removal Trials at Wind Turbines Sites (Regina Bispo, Joana Bernardino, Tiago A. Marques and Dinis Pestana).- Uniformity (M. F. Brilhante, M. Malva, S. Mendonca, D. Pestana, F. Sequeira and S. Velosa).- Asymptotic Comparison at Optimal Levels of Minimum-Variance Reduced-Bias Tail Index Estimators (Frederico Caeiro and M. Ivette Gomes).- Extremal Behavior of the Generalized Integer-valued Random Coefficient Autoregressive Process (Luisa Canto e Castro, Dulce Gomes and Maria da Graca Temido).- Models of Individual Growth in a Random Environment: Study and Application of First Passage Times (Clara Carlos, Carlos A. Braumann and Patricia A. Filipe).- Generalized Linear Mixed Effects Model in the Analysis of Longitudinal Discrete Data (Eunice Carrasquinha, M. Helena Goncalves and M. Salome Cabral).- Risk Assessment on Campylobacter in Broiler Meat at Slaughter Level in Portugal (Marta Castel-Branco, Marilia Antunes, Patricia Inacio and Miguel Cardo).- Predicting and Treating Missing Data with Boot.EXPOS (Clara Cordeiro and M. Manuela Neves).- Bayesian Genetic Mapping of Binary Trait Loci (Cesar Correia, Nuno Sepulveda and Carlos Daniel Paulino).- Concomitant Latent Class Models Applied to Mathematics Education (Maria Eugenia Ferrao and Jose G. Dias).- Evaluating Discriminant Analysis Results (Ana Sousa Ferreira and Margarida Cardoso).- Distribution of the Number of Losses in Busy-periods of MX /G/1/n Systems (Fatima Ferreira, Antonio Pacheco and Helena Ribeiro).- Misleading Signals in Simultaneous Residual Schemes for the Process Mean and Variance of AR(1) Processes: A Stochastic Ordering Approach (Patricia Ferreira Ramos, Manuel Cabral Morais and Antonio Pacheco).- Conditional EVT for VAR Estimation: Comparison with a new Independence Test (M. I. Fraga Alves and P. Araujo Santos).- Asymptotic Distribution of the Maximum for a Chaotic Economic Model (Ana Cristina Moreira Freitas).- Adaptive Choice of Thresholds and the Bootstrap Methodology: An Empirical Study (M. Ivette Gomes, Fernanda Figueiredo and M. Manuela Neves).- Distributional Properties of Generalized Threshold ARCH Models (E. Goncalves and N. Mendes-Lopes).- Preliminary Results on Confidence Intervals for Open Bonus Malus (Gracinda R. Guerreiro, Joao T. Mexia and Maria F. Miguens).- Study of the Electrocardiographic Fluctuations on Brugada Syndrome Screening (Carla Henriques, Ana Cristina Matos and Luis Ferreira dos Santos).- Circulatory System Diseases and Neoplasms Mortality on Older Portuguese Population: A Spatio-temporal Analysis by Age and Sex (Sandra Lagarto, Carla Nunes, Dulce Gomes and Maria Filomena Mendes).- Absolute Diffusion Process: Sensitivity Measures (Manuela Larguinho, Jose Carlos Dias and Carlos A. Braumann).- Scaling Exponents in Heart Rate Variability (Argentina Leite, Maria Eduarda Silva and Ana Paula Rocha).- Prediction of Dementia Patients: A Comparative Approach Using Parametric vs. non Parametric Classifiers (Joao Maroco, Dina Silva, Manuela Guerreiro, Alexandre de Mendonca and Isabel Santana).- Pareto Scale Mixtures (Miguel Martins Felgueiras).- Fitting Johnsons SB distribution to Forest Tree Diameter (Ayana Mateus and Margarida Tome).- On a Continuous Time Stock Price Model With two Mean Reverting Regimes (Pedro P. Mota).- Generalized F Tests in Models With Random Perturbations: The Truncated Normal Case (Celia Nunes, Dario Ferreira, Sandra Ferreira and Joao T. Mexia).- Generalized Linear Models, Generalized Additive Models and Neural Networks: Comparative Study in Medical Applications (Ana Luisa Papoila, Cristina Rocha, Carlos Geraldes and Patricia Xufre).- Joint Regression Analysis and Incorporation of Environmental Variables in Stochastic Frontier Production Function: An Application to Experimental Data of Winter Rye (Dulce Gamito Pereira and Ana Sampaio).- On the Maximum and Minimum of a Stationary Random Field (Luisa Pereira).- Publication Bias and Meta-analytic Syntheses (D. Pestana, M. L. Rocha, R. Vasconcelos and S. Velosa).- Self-perception of Health Status and Socio-economic Differences in the Use of Health Services (Alexandra Pinto, Victor Lobo, Fernando Bacao and Helena Bacelar-Nicolau).- Comparison of Modal Variables Using Multivariate Analysis (Isabel Pinto Doria, Aurea Sousa, Helena Bacelar-Nicolau and Georges Le Calve).- Disentangling the Relationship Between Entrepreneurship and Job Creation by Poisson Mixture Regressions (Leandro P. Pontes and Jose G. Dias).-Simulation Study of the Calibration Technique in the Extremal Index Estimation (D. Prata Gomes, Joao T. Mexia and M. Manuela Neves).- Semi-parametric Building of the Optimal Screening Region in Supervised Classification (Sandra Ramos, Maria Antonia Amaral Turkman and Marilia Antunes).- An Application of Statistical Methods of Indirect Estimation and Projection of Internal Migration Flows Within the Portuguese Mainland (Maria FilomenaMendes, Antonio Caleiro, Sandra Lagarto and Filipe Ribeiro).- Robust Clustering Method for the Detection of Outliers: Using AIC to Select the Number of Clusters (Carla M. Santos-Pereira and Ana M. Pires).- HLA Allele and Haplotype Frequencies of the Portuguese Bone Marrow Donors Registry (Ricardo Sao Joao, Ana Luisa Papoila, Dario Ligeiro and Helder Trindade).- Independent Component Analysis for Extended Time Series in Climate Data (Fernando Sebastiao and Irene Oliveira).- Life Satisfaction: a MIMIC Approach With a Discrete Latent Variable (Patricia Serra, Jose G. Dias and Maria de Fatima Salgueiro).- An Application of MRMC ROC Curves on Radiology (Carina Silva-Fortes, Maria Antonia Amaral Turkman, Luis Lanca, Ricardo Silva and Goncalo Marques).- Some Remarks About Gibbs Variable Selection Performance (Julia Teles and Maria Antonia Amaral Turkman).- Dependence of Multivariate Extremes (C. Viseu, L. Pereira, A.P. Martins and H. Ferreira).

Probability Generating Functions for Discrete Real-Valued Random Variables

The probability generating function is a powerful technique for studying the law of finite sums of independent discrete random variables taking integer positive values. For real-valued discrete random variables, the well-known elementary theory of Dirichlet series and the symbolic computation packages available nowadays, such as Mathematica 5, allow us to extend this technique to general discrete random variables. Being so, the purpose of this work is twofold. First, we show that discrete random variables taking real values, nonnecessarily integer or rational, may be studied with probability generating functions. Second, we intend to draw attention to some practical ways of performing the necessary calculations.

On a continuous time stock price model with regime switching, delay, and threshold

Motivated by the need to describe bear-bull market regime switching in stock prices, we introduce and study a stochastic process in continuous time with two regimes, threshold and delay, given by a stochastic differential equation. When the difference between the regimes is simply given by a different set of real valued parameters for the drift and diffusion coefficients, with changes between regimes depending only on these parameters, we show that if the delay is known there are consistent estimators for the threshold as long we know how to classify a given observation of the process as belonging to one of the two regimes. When the drift and diffusion coefficients are of geometric Brownian motion type we obtain a model with parameters that can be estimated in a satisfactory way, a model that allows differentiating regimes in some of the NYSE 21 stocks analyzed and also, that gives very satisfactory results when compared to the usual Black–Scholes model for pricing call options.

On Some Auto-Induced Regime Switching Double-Threshold Glued Diffusions

Regime switching processes are usually defined with an external random source driving the regime changes. In this article, we define and study a regime switching diffusion considering two thresholds, and regime switching occurring, by a change in the diffusion drift and volatility, whenever the trajectory touches the upper threshold after having crossed, or touched, the lower threshold or touches the lower threshold after having crossed, or touched, the upper threshold. We develop an estimation procedure for the thresholds and for the regime parameters of the diffusions. We show that a generalized Black–Scholes model with the regime switching diffusion as the law of the risky asset is arbitrage free and complete under an additional hypothesis on the diffusion coefficients of the two regime diffusions.

On a spread model for portfolio credit risk modeling

We propose a model for determining the spread of a credit portfolio based on the actuarial principle. In this model the spread is a function of the recovery rate and of the probability of default. In an application to data, from a consumer credit portfolio of a Cape Verde bank, we estimate the recovery rate by a beta regression and the probability of default by a logistic regression, both regressions using as independent variables sociodemographic information and consumer credit contract variables in the data set. We show that the data support the possibility of defining a spread for each client - the borrower - that is coherent with the portfolio spread given by the model.

On the Evolution and Asymptotic Analysis of Open Markov Populations: Application to Consumption Credit

In this paper, we study, by means of randomized sampling, the long-run stability of some open Markov population fed with time-dependent Poisson inputs. We show that state probabilities within transient states converge—even when the overall expected population dimension increases without bound—under general conditions on the transition matrix and input intensities. Following the convergence results, we obtain ML estimators for a particular sequence of input intensities, where the sequence of new arrivals is modeled by a sigmoidal function. These estimators allow for the forecast, by confidence intervals, of the evolution of the relative population structure in the transient states. Applying these results to the study of a consumption credit portfolio, we estimate the implicit default rate.

Pseudo Maximum Likelihood and Moments Estimators for Some Ergodic Diffusions

When \(\left (X_t\right )_{t\geq 0}\) is an ergodic process, the density function of Xt converges to some invariant density as t →∞. We will compute and study some asymptotic properties of pseudo moments estimators obtained from this invariant density, for a specific class of ergodic processes. In this class of processes we can find the Cox-Ingersoll & Ross or Dixit & Pindyck processes, among others. A comparative study of the proposed estimators with the usual estimators obtained from discrete approximations of the likelihood function will be carried out.

On some statistical models with a random number of observations

We extend some classical statistical inference to the case of a random number of observations with a stabilized distribution: namely, in the normal model, inference for the mean with known and unknown variance and inference for the variance. We describe some useful models for the number of observations obtained by truncation or translation of usual models given by integer-valued random variables: Poisson, binomial, geometric, and negative binomial. We present an efficient random search algorithm for the computation of the quantiles of the relevant statistics, we describe an interval estimation procedure for the extended model, and we propose a parametric bootstrap simulation study to validate the proposed procedure.