Aggregation of predictors for nonstationary sub-linear processes and online adaptive forecasting of time varying autoregressive processes

In this work, we study the problem of aggregating a finite number of predictors for nonstationary sub-linear processes. We provide oracle inequalities relying essentially on three ingredients: (1) a uniform bound of the $\ell^1$ norm of the time varying sub-linear coefficients, (2) a Lipschitz assumption on the predictors and (3) moment conditions on the noise appearing in the linear representation. Two kinds of aggregations are considered giving rise to different moment conditions on the noise and more or less sharp oracle inequalities. We apply this approach for deriving an adaptive predictor for locally stationary time varying autoregressive (TVAR) processes. It is obtained by aggregating a finite number of well chosen predictors, each of them enjoying an optimal minimax convergence rate under specific smoothness conditions on the TVAR coefficients. We show that the obtained aggregated predictor achieves a minimax rate while adapting to the unknown smoothness. To prove this result, a lower bound is established for the minimax rate of the prediction risk for the TVAR process. Numerical experiments complete this study. An important feature of this approach is that the aggregated predictor can be computed recursively and is thus applicable in an online prediction context.Comment: Published at http://dx.doi.org/10.1214/15-AOS1345 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A TWO-STEP ESTIMATOR FOR A SPATIAL LAG MODEL OF COUNTS: THEORY, SMALL SAMPLE PERFORMANCE AND AN APPLICATION

Several spatial econometric approaches are available to model spatially correlated disturbances in count models, but there are at present no structurally consistent count models incorporating spatial lag autocorrelation. A two-step, limited information maximum likelihood estimator is proposed to fill this gap. The estimator is developed assuming a Poisson distribution, but can be extended to other count distributions. The small sample properties of the estimator are evaluated with Monte Carlo experiments. Simulation results suggest that the spatial lag count estimator achieves gains in terms of bias over the aspatial version as spatial lag autocorrelation and sample size increase. An empirical example deals with the location choice of single-unit start-up firms in the manufacturing industry in the US between 2000 and 2004. The empirical results suggest that in the dynamic process of firm formation, counties dominated by firms exhibiting (internal) increasing returns to scale are at a relative disadvantage even if localization economies are presentcount model, location choice, manufacturing, Poisson, spatial econometrics

Modelling adaptive complex behaviour with an application to the stock markets dynamics

In this paper we review a simple agent-based model of adaptive complex behaviour that shows how the interaction of different agent's profit-oriented decisions leads to a wide spectra of organizational possibilities. We comment on some potential applications of this model to the social and life sciences, and later focus on the modelling of the stock market dynamics. We show how some ~f the features of stock price series, and in particular extreme events such as speculative bubbles and crashes, can be obtained when certain conditions are satisfied by most of the investors' preferences

Non-Gaussian Geostatistical Modeling using (skew) t Processes

We propose a new model for regression and dependence analysis when addressing spatial data with possibly heavy tails and an asymmetric marginal distribution. We first propose a stationary process with $t$ marginals obtained through scale mixing of a Gaussian process with an inverse square root process with Gamma marginals. We then generalize this construction by considering a skew-Gaussian process, thus obtaining a process with skew-t marginal distributions. For the proposed (skew) $t$ process we study the second-order and geometrical properties and in the $t$ case, we provide analytic expressions for the bivariate distribution. In an extensive simulation study, we investigate the use of the weighted pairwise likelihood as a method of estimation for the $t$ process. Moreover we compare the performance of the optimal linear predictor of the $t$ process versus the optimal Gaussian predictor. Finally, the effectiveness of our methodology is illustrated by analyzing a georeferenced dataset on maximum temperatures in Australi

A new statistic and practical guidelines for nonparametric Granger causality testing

Upon illustrating how smoothing may cause over-rejection in nonparametric tests for Granger non-causality, we propose a new test statistic for which problems of this type can be avoided. We develop asymptotic theory for the new test statistic, and perform a simulation study to investigate the properties of the new test in comparison with its natural counterpart, the Hiemstra-Jones test. Our simulation results indicate that, if the bandwidth tends to zero at the appropriate rate as the sample size increases, the size of the new test remains close to nominal, while the power remains large. Transforming the time series to uniform marginals improves the behavior of both tests. In applications to Standard and Poor's index volumes and returns, the Hiemstra-Jones test suggests that volume Granger-causes returns. However, the evidence for this gets weaker if we carefully apply the recommendations suggested by our simulation study.

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page