Howell Tong

Non-linear time series : a dynamical system approach

Preface Acknowlegement Introduction 1. An introduction to dynamical systems 2. Some non-linear time series models 3. Probability structure 4. Statistical aspects 5. Non-linear least-squares prediction based on non-linear models 6. Case studies

An adaptive estimation of dimension reduction space

Summary. Searching for an effective dimension reduction space is an important problem in regression, especially for high dimensional data. We propose an adaptive approach based on semiparametric models, which we call the (conditional) minimum average variance estimation (MAVE) method, within quite a general setting. The MAVE method has the following advantages. Most existing methods must undersmooth the nonparametric link function estimator to achieve a faster rate of consistency for the estimator of the parameters (than for that of the nonparametric function). In contrast, a faster consistency rate can be achieved by the MAVE method even without undersmoothing the nonparametric link function estimator. The MAVE method is applicable to a wide range of models, with fewer restrictions on the distribution of the covariates, to the extent that even time series can be included. Because of the faster rate of consistency for the parameter estimators, it is possible for us to estimate the dimension of the space consistently. The relationship of the MAVE method with other methods is also investigated. In particular, a simple outer product gradient estimator is proposed as an initial estimator. In addition to theoretical results, we demonstrate the efficacy of the MAVE method for high dimensional data sets through simulation. Two real data sets are analysed by using the MAVE approach.

On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations

We have shown that within the setting of a difference equation it is possible to link ergodicity with stability via the physical notion of energy in the form of a Lyapunov function.

A Multiple-Threshold AR(1) Model

We consider the model Z, = +(0, k ) + +(I, k)Z,_, + a,(k) whenever r,_, < Z,_,S r,., 1S k k 1, with r, = -m and r, = m. Here {+(i, k); i =0 , l ; 1 5 k 5 1) is a sequence of real constants, not necessarily equal, and, for 1 5 k 5 I, {a,(k), t 2 1) is a sequence of i.i.d. random variables with mean 0 and with {a,(k), t 2 1) independent of {a,(j), t 2 1) for j # k. Necessary and sufficient conditions on the constants {+(i, k)} are given for the stationarity of the process. Least squares estimators of the model parameters are derived and, under mild regularity conditions, are shown to be strongly consistent and asymptotically normal. NON-LINEAR TIME SERIES; SETAR MODELS; AUTOREGRESSIVE MODELS; MARKOV CHAINS

Some Comments on the Canadian Lynx Data

SUMMARY We re-examine the annual trappings of the Canadian lynx over the years 1821-1934 (inclusive), which have been reported and analysed extensively. For some references see Elton and Nicholson (1942), Rowan (1950), Moran (1952), Hannan (1960), Kashyap (1973) and Bulmer (1974). This paper shows that an autoregressive (AR) model of order eleven provides an acceptable alternative to the more widely adopted class of models with low order AR PlUS one harmonic component.

Asymmetric least squares regression estimation: A nonparametric approach ∗

This paper considers the nonparametric estimation of regression expectiles and percentiles by using an asymmetric least squares (ALS) approach, in which the squared error loss function is given different weights depending on whether thc residual is positive or negative. The kernel method based on locally linear fit is adopted, which also provides an estimator of the derivative of the regression function. Under the assumption that the observations are strictly stationary and ρ-mixing the asymptotic normality for the estimators of conditional expectiles is established by using the convexity lemma. For a large class of regression models, the ALS approach can be adapted to estimate the conditional percentiles directly. Further, we show that these ALS estimators for conditional percentiles are consistent.

Phase- and density-dependent population dynamics in Norwegian lemmings: interaction between deterministic and stochastic processes

We analysed two 26–year long (1970 to 1995) time–series on annual population growth rates of Norwegian lemmings (Lemmus lemmus) from Finse, south Norway, using a threshold autoregressive (TAR) approach. We demonstrate that the population dynamics is both phase– and density–dependent. The phase–dependence accounts for the observed nonlinearity. We used the deduced stochastic model structure as a basis for evaluating the dynamic properties of this system. The dynamics is characterized either by limit cycles or chaos (the latter with a strong semi–periodic component). Stochasticity is seen to play an important role in the determination of the periodicity. The ecological implications of these statistical and mathematical results are discussed.

On The Estimation of Pr {Y ⩽ X} for Exponential Families

The minimum variance s-unbiased estimator of Pr{Y ⩽ X} is derived for the case in which X and Y are s-independent r.v.s each with a Cdf belonging to the exponential family but otherwise not necessarily of the same type.

On prediction and chaos in stochastic systems

We propose a new measure of sensitivity to initial conditions within a stochastic environment and explore its connection with nonlinear prediction and statistical estimation. We use modern statistical developments to construct and illustrate pointwise predictors and predictive intervals/distributions.

Testing for a linear MA model against threshold MA models

This paper investigates the (conditional) quasi-likelihood ratio test for the threshold in MA models. Under the hypothesis of no threshold, it is shown that the test statistic converges weakly to a function of the centred Gaussian process. Under local alternatives, it is shown that this test has nontrivial asymptotic power. The results are based on a new weak convergence of a linear marked empirical process, which is independently of interest. This paper also gives an invertible expansion of the threshold MA models.