Juan Vilar

Non-linear time series clustering based on non-parametric forecast densities

The problem of clustering time series is studied for a general class of non-parametric autoregressive models. The dissimilarity between two time series is based on comparing their full forecast densities at a given horizon. In particular, two functional distances are considered: L^1 and L^2. As the forecast densities are unknown, they are approximated using a bootstrap procedure that mimics the underlying generating processes without assuming any parametric model for the true autoregressive structure of the series. The estimated forecast densities are then used to construct the dissimilarity matrix and hence to perform clustering. Asymptotic properties of the proposed method are provided and an extensive simulation study is carried out. The results show the good behavior of the procedure for a wide variety of nonlinear autoregressive models and its robustness to non-Gaussian innovations. Finally, the proposed methodology is applied to a real dataset involving economic time series.

Functional Prediction for the Residual Demand in Electricity Spot Markets

This paper deals with the prediction of residual demand curves in electricity spot markets, as a tool for optimizing bidding strategies in the short-term. Two functional models are formulated and empirically compared with the naïve method, which is the reference model in most of the practical applications found in industry. The first one is a functional nonparametric model that estimates the residual demand as a function of past residual demands, while the second one uses also electricity demand and wind power forecasts as explanatory variables. The proposed models have been tested using real data from the Spanish day-ahead market over a period of two years. The analysis of these results has motivated the development of a new forecasting strategy based on the selective combination of forecasts, taking advantage of the effect of wind fluctuations on the residual demand. This new forecasting approach outperforms the naïve method in all circumstances.

Classifying Time Series Data: A Nonparametric Approach

A general nonparametric approach to identify similarities in a set of simultaneously observed time series is proposed. The trends are estimated via local polynomial regression and classified according to standard clustering procedures. The equality of the trends is checked using several nonparametric test statistics whose asymptotic distributions are approximated by a bootstrap procedure. Once the estimated trends are removed from the model, the residual series are grouped by means of a nonparametric cluster method specifically designed for time series. Such a method is based on a disparity measure between local linear smoothers of the spectra of the series. The performance of the proposed methodology is illustrated by means of its application to a particular financial data example. The dependence of the observations is a crucial factor in this work and is taken into account throughout the study.

Nonparametric analysis of aggregate loss models

This paper describes a nonparametric approach to make inferences for aggregate loss models in the insurance framework. We assume that an insurance company provides a historical sample of claims given by claim occurrence times and claim sizes. Furthermore, information may be incomplete as claims may be censored and/or truncated. In this context, the main goal of this work consists of fitting a probability model for the total amount that will be paid on all claims during a fixed future time period. In order to solve this prediction problem, we propose a new methodology based on nonparametric estimators for the density functions with censored and truncated data, the use of Monte Carlo simulation methods and bootstrap resampling. The developed methodology is useful to compare alternative pricing strategies in different insurance decision problems. The proposed procedure is illustrated with a real dataset provided by the insurance department of an international commercial company.

Finite sample performance of density estimators from unequally spaced data

For broad classes of deterministic and random sampling schemes {[tau]k}, exact mean integrated squared error (MISE) expressions for the kernel estimator of the marginal density of a first-order continuous-time autoregressive process are derived. The obtained expressions show that the effect on MISE due to both the sampling scheme and the sampling rate is significant for finite samples. The results are also extended to a case where the irregular observations are generated from a mixture of first-order continuous-time processes.

Bayesian Analysis of Aggregate Loss Models

This paper describes a Bayesian approach to make inference for aggregate loss models in the insurance framework. A semiparametric model based on Coxian distributions is proposed for the approximation of both the interarrival time between claims and the claim size distributions. A Bayesian density estimation approach for the Coxian distribution is implemented using reversible jump Markov Chain Monte Carlo (MCMC) methods. The family of Coxian distributions is a very flexible mixture model that can capture the special features frequently observed in insurance claims. Furthermore, given the proposed Coxian approximation, it is possible to obtain closed expressions of the Laplace transforms of the total claim count and the total claim amount random variables. These properties allow us to obtain Bayesian estimations of the distributions of the number of claims and the total claim amount in a future time period, their main characteristics and credible intervals. The possibility of applying deductibles and maximum limits is also analyzed. The methodology is illustrated with a real data set provided by the insurance department of an international commercial company.

A Bootstrap Test for the Equality of Nonparametric Regression Curves Under Dependence

A bootstrap procedure for testing the equality of several regression curves under dependence conditions is proposed. The errors are assumed to follow different ARMA structures. A test statistic based on the functional distances between nonparametric estimators of the regression functions is considered. The critical test values are obtained using a resampling method that takes into account the correlation structure. The consistency of the bootstrap procedure is established and its finite sample performance is investigated in a Monte Carlo study. Simulations show that our bootstrap-based test outperforms the asymptotic test. Applications are illustrated with a real data example.

Confidence and prediction intervals in semi-functional partial linear regression

Semi-functional partial linear regression model allows to deal with a nonparametric and a linear component within the functional regression. Naive and wild bootstrap procedures are proposed to approximate the distribution of the estimators for each component in the model, and their asymptotic validities are obtained in the context of dependence data, under α-mixing conditions. Based on that bootstrap procedures, confidence intervals can be obtained for each component in the model, which can be also extended to deal with prediction intervals and prediction densities.