A half-graph depth for functional data

A recent and highly attractive area of research in statistics is the analysis of functional data. In this paper a new definition of depth for functional observations is introduced based on the notion of "half-graph" of a curve. It has computational advantages with respect to other concepts of depth previously proposed. The half-graph depth provides a natural criterion to measure the centrality of a function within a sample of curves. Based on this depth a sample of curves can be ordered from the center outward and L-statistics are defined. The properties of the half-graph depth, such as the consistency and uniform convergence, are established. A simulation study shows the robustness of this new definition of depth when the curves are contaminated. Finally real data examples are analyzed

Bootstrap tests for unit roots based on LAD estimation.

In this paper we propose a new bootstrap test for unit roots in first-order autoregressive models based on least absolute deviation (LAD) estimators. It is well known that the behaviour of this estimator when the distribution is heavy tailed is very good compared with least-squares estimation. The innovations distribution dependence of the LAD asymptotic law is overcome using bootstrap, which automatically approaches the target distribution. Our strategy avoids the usual problem of estimating the variance matrix and the density at zero, and makes also unnecessary the construction of distribution free statistics through linear combinations with the least-squares estimator. We provide the bootstrap functional limit theory necessary to prove the asymptotic validity of the procedure. Moreover, a large simulation study shows that our test has very good power behaviour compared with others proposed in the literature.Publicad

A half-region depth for functional data.

A new definition of depth for functional observations is introduced based on the notion of “half-region” determined by a curve. The half-regiondepth provides a simple and natural criterion to measure the centrality of a function within a sample of curves. It has computational advantages relative to other concepts of depth previously proposed in the literature which makes it applicable to the analysis of high-dimensional data. Based on this deptha sample of curves can be ordered from the center-outward and order statistics can be defined. The properties of the half-regiondepth, such as consistency and uniform convergence, are established. A simulation study shows the robustness of this new definition of depth when the curves are contaminated. Finally, real data examples are analyzed.This research was partially supported by Spanish Ministry of Education and Science grants BEC 2002-03769, SEJ2005-06454, SEJ2007-67734, SEJ2905 and ECO2008-05080.Publicad

Effects of parameter estimation on prediction densities: a bootstrap approach.

We use abootstrap procedure to study the impact of parameterestimation on predictiondensities, focusing on seasonal ARIMA processes with possibly non normal innovations. We compare predictiondensities obtained using the Box and Jenkins approach with bootstrapdensities which may be constructed either taking into account parameterestimation variability or using parameter estimates as if they were known parameters. By means of Monte Carlo experiments, we show that the average coverage of the intervals is closer to the nominal value when intervals are constructed incorporating parameter uncertainty. The effects of parameterestimation are particularly important for small sample sizes and when the error distribution is not Gaussian. We also analyze the effect of the estimation method on the shape of predictiondensities comparing predictiondensities constructed when the parameters are estimated by Ordinary Least Squares (OLS) and by Least Absolute Deviations (LAD). We show how, when the error distribution is not Gaussian, the average coverage and length of intervals based on LAD estimates are closer to nominal values than those based on OLS estimates. Finally, the performance of the bootstrap intervals is illustrated with two empirical examples.Acknowledgements The authors are very grateful to the referees for comments that helped to improve the paper. Also, we are grateful to Lory A. Thombs for her comments, to Eva Senra for providing the Italian IPI data and to Regina Kaiser for helping us to clean the series of outliers. Financial support was provided by the European Union project ERBCHRXCT 940514 and by projects CICYT PB95-0299, DGICYT PB96-0111 from the Spanish Government and Cátedra de Calidad BBV.Publicad

Stability under contamination of robust regression estimators based on differences of residuals

A reasonable approach to robust regression estimation is minimizing a robust scale estimator of the pairwise differences of residuals. We introduce a large class of estimators based on this strategy extending ideas of Yohai and Zamar (1993) and Croux, Rousseeuw and Hossjer (1994). The asymptotic robustness properties of the estimators in this class are addressed using the maxbias curve. We provide a general principle to compute this curve and present explicit formulae for several particular cases including generalized versions of S-, R- and '!-estimators. Finally, the most stable estimator in the class, that is, the estimator with the minimum maxbias curve, is shown to be the set of coefficients that minimizes an appropriate quantile of the distribution of the absolute pairwise differences of residuals

Bootstrap prediction intervals for power-transformed time series

In this paper, we propose a bootstrap procedure to construct prediction intervals for future values of a variable after a linear ARIMA model has been fitted to its power transformation. The procedure is easy to implement and provides a useful tool in empirical applications given that it is often the case that, for example, the log-transformation is modeled when the variable of interest for prediction is the original one. The advantages over existing methods for computing prediction intervals of power-transformed time series are that the proposed bootstrap intervals incorporate the variability due to parameter estimation and do not rely on distributional assumptions neither on the original variable nor on the transformed one. We derive the asymptotic distribution and show the good behavior of the bootstrap approach versus alternative procedures by means of Monte Carlo experiments. Finally, the procedure is illustrated by analyzing three real-time series data sets.The authors are very grateful to Lutz Kilian for providing the RXR and NBRX series and to the associate editor Antonio García Ferrer and two anonymous referees for valuable comments that have been very helpful to improve the paper. Financial support was provided by projects DGES PB96-0111 and BEC 2002-03720 from the Spanish Government and Catedra de Calidad from BBV.Publicad

Bootstrap prediction for returns and volatilities in GARCH models.

A new bootstrap procedure to obtain prediction densities of returns and volatilities of GARCH processes is proposed. Financial market participants have shown an increasing interest in prediction intervals as measures of uncertainty. Furthermore, accurate predictions of volatilities are critical for many financial models. The advantages of the proposed method are that it allows incorporation of parameter uncertainty and does not rely on distributional assumptions. The finite sample properties are analyzed by an extensive Monte Carlo simulation. Finally, the technique is applied to the Madrid Stock Market index, IBEX-35.Acknowledgements: We are very grateful for their helpful comments by three anonymous referees, the editor Stephen Pollock and seminar participants at the Universities of Valladolid, New South Wales and Canterbury and the June 2001 Time Series Workshop of Arrabida, the September 2001 International Conference on Modelling Volatility (Perth) and the June 2002 International Symposium on Forecasting (Dublin). We are also grateful to Gregorio Serna for providing the data set analyzed in this paper and to Dolores Redondas for helping us with the figures. Financial support was provided by projects DGES PB96-0111 and BEC2002-03720 from the Spanish Government and Cátedra de Calidad from BBVAPublicad

Forecasting time series with sieve bootstrap.

In this paper we propose bootstrap methods for constructing nonparametric prediction intervals for a general class of linear processes. Our approach uses the AR(∞)-sieve bootstrap procedure based on residual resampling from an autoregressive approximation to the given process. We present a Monte Carlo study comparing the finite sample properties of the sieve bootstrap with those of alternative methods. Finally, we illustrate the performance of the proposed method with a real data example.We would like to thank Mike Wiper, two referees and the coordinating editor for carefully reading that greatly improved the paper. This research was partially supported by the Dirección General de Educación Superior project DGES PB96-0111 and Cátedra de Calidad BBVA.Publicad

On sieve bootstrap prediction intervals.

In this paper we consider a sieve bootstrap method for constructing nonparametric prediction intervals for a general class of linear processes. We show that the sieve bootstrap provides consistent estimators of the conditional distribution of future values given the observed data.We would like to thank Mike Wiper for his careful reading which greatly improved the paper. This research was partially supported by the CYCIT project BEC 2000-0167 and by the Cátedra de Calidad BBVA.Publicad