A bootstrap detection for operational determinism.

We propose a bootstrap detection for operationally deterministic versus stochastic nonlinear modelling and illustrate the method with both simulated and real data sets.

Statistics and dynamical phenomena

A friend of mine, an expert in statistical genomics, told me the following story: At a dinner party, an attractive lady asked him, "What do you do for a living?" He replied, "I model." As my friend is a handsome man, the lady did not question his statement and continued, "What do you model?" "Genes." She then looked at him up and down and said, "Mh, you must be very much in demand." "Yes, very much so, especially after I helped discover a new culprit gene for a common childhood disease." The lady looked puzzled. In this snapshot, I will give you an insight into Statistics, the field that fascinated my friend (and myself) so much. I will concentrate on phenomena that change over time, in other words, dynamical events

Statistical Tests for Lyapunov Exponents of Deterministic Systems

In order to develop statistical tests for the Lyapunov exponents of deterministic dynamical systems, we develop bootstrap tests based on empirical likelihood for percentiles and expectiles of strictly stationary processes. The percentiles and expectiles are estimated in terms of asymmetric least deviations and asymmetric least squares methods. Asymptotic distributional properties of the estimators are established.Bootstrap, chaos, empirical likelihood, expectile, percentile.

Semiparametric penalty function method in partially linear model selection

Model selection in nonparametric and semiparametric regression is of both theoretical and practical interest. Gao and Tong (2004) proposed a semiparametric leave–more–out cross–validation selection procedure for the choice of both the parametric and nonparametric regressors in a nonlinear time series regression model. As recognized by the authors, the implementation of the proposed procedure requires the availability of relatively large sample sizes. In order to address the model selection problem with small or medium sample sizes, we propose a model selection procedure for practical use. By extending the so–called penalty function method proposed in Zheng and Loh (1995, 1997) through the incorporation of features of the leave-one-out cross-validation approach, we develop a semiparametric, consistent selection procedure suitable for the choice of optimum subsets in a partially linear model. The newly proposed method is implemented using the full set of data, and simulations show that it works well for both small and medium sample sizes.Linear model; model selection; nonparametric method; partially linear model; semiparametric method

Adaptive orthogonal series estimation in additive stochastic regression models

In this paper, we consider additive stochastic nonparametric regression models. By approximating the nonparametric components by a class of orthogonal series and using a generalized cross-validation criterion, an adaptive and simultaneous estimation procedure for the nonparametric components is constructed. We illustrate the adaptive and simultaneous estimation procedure by a number of simulated and real examples.Adaptive estimation; additive model; dependent process; mixing condition; nonlinear time series; nonparametric regression; orthogonal series; strict stationarity; truncation parameter

Statistical tests for Lyapunov exponents of deterministic systems.

In order to develop statistical tests for the Lyapunov exponents of deterministic dynamical systems, we develop bootstrap tests based on empirical likelihood for percentiles and expectiles of strictly stationary processes. The percentiles and expectiles are estimated in terms of asymmetric least deviations and asymmetric least squares methods. Asymptotic distributional properties of the estimators are established.

Nested sub-sample search algorithm for estimation of threshold models

Threshold models have been popular for modelling nonlinear phenomena in diverse areas, in part due to their simple fitting and often clear model interpretation. A commonly used approach to fit a threshold model is the (conditional) least squares method, for which the standard grid search typically requires O(n) operations for a sample of size n; this is substantial for large n, especially in the context of panel time series. This paper proposes a novel method, the nested sub-sample search algorithm, which reduces the number of least squares operations drastically to O(log n) for large sample size. We demonstrate its speed and reliability via Monte Carlo simulation studies with finite samples. Possible extension to maximum likelihood estimation is indicated

On moving-average models with feedback

Moving average models, linear or nonlinear, are characterized by their short memory. This paper shows that, in the presence of feedback in the dynamics, the above characteristic can disappear.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ352 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Statistical tests for Lyapunov exponents of deterministic systems

In order to develop statistical tests for the Lyapunov exponents of deterministic dynamical systems, we develop bootstrap tests based on empirical likelihood for percentiles and expectiles of strictly stationary processes. The percentiles and expectiles are estimated in terms of asymmetric least deviations and asymmetric least squares methods. Asymptotic distributional properties of the estimators are established.Bootstrap; chaos; empirical likelihood; expectile; percentile