Robust discrimination between long‐range dependence and a change in mean

In this article we introduce a robust to outliers Wilcoxon change‐point testing procedure, for distinguishing between short‐range dependent time series with a change in mean at unknown time and stationary long‐range dependent time series. We establish the asymptotic distribution of the test statistic under the null hypothesis for L₁ near epoch dependent processes and show its consistency under the alternative. The Wilcoxon‐type testing procedure similarly as the CUSUM‐type testing procedure (of Berkes I., Horváth L., Kokoszka P. and Shao Q. 2006. Ann.Statist. 34:1140–1165), requires estimation of the location of a possible change‐point, and then using pre‐ and post‐break subsamples to discriminate between short and long‐range dependence. A simulation study examines the empirical size and power of the Wilcoxon‐type testing procedure in standard cases and with disturbances by outliers. It shows that in standard cases the Wilcoxon‐type testing procedure behaves equally well as the CUSUM‐type testing procedure but outperforms it in presence of outliers. We also apply both testing procedure to hydrologic data.     

Change‐point detection in panel data

We consider N panels and each panel is based on T observations. We are interested to test if the means of the panels remain the same during the observation period against the alternative that the means change at an unknown time. We provide tests which are derived from a likelihood argument and they are based on the adaptation of the CUSUM method to panel data. Asymptotic distributions are derived under the no change null hypothesis and the consistency of the tests are proven under the alternative. The asymptotic results are shown to work in case of small and moderate sample sizes via Monte Carlo simulations.     

Least tail‐trimmed squares for infinite variance autoregressions

We develop a robust least squares estimator for autoregressions with possibly heavy tailed errors. Robustness to heavy tails is ensured by negligibly trimming the squared error according to extreme values of the error and regressors. Tail‐trimming ensures asymptotic normality and super‐‐convergence with a rate comparable to the highest achieved amongst M‐estimators for stationary data. Moreover, tail‐trimming ensures robustness to heavy tails in both small and large samples. By comparison, existing robust estimators are not as robust in small samples, have a slower rate of convergence when the variance is infinite, or are not asymptotically normal. We present a consistent estimator of the covariance matrix and treat classic inference without knowledge of the rate of convergence. A simulation study demonstrates the sharpness and approximate normality of the estimator, and we apply the estimator to financial returns data. Finally, tail‐trimming can be easily extended beyond least squares estimation for a linear stationary AR model. We discuss extensions to quasi‐maximum likelihood for GARCH, weighted least squares for a possibly non‐stationary random coefficient autoregression, and empirical likelihood for robust confidence region estimation, in each case for models with possibly heavy tailed errors.     

Non‐stationary autoregressive processes with infinite variance

Consider an AR(p) process , where {?_t} is a sequence of i.i.d. random variables lying in the domain of attraction of a stable law with index 0<α<2. This time series {Y_t} is said to be a non‐stationary AR(p) process if at least one of its characteristic roots lies on the unit circle. The limit distribution of the least squares estimator (LSE) of for {Y_t} with infinite variance innovation {?_t} is established in this paper. In particular, by virtue of the result of Kurtz and Protter (1991) of stochastic integrals, it is shown that the limit distribution of the LSE is a functional of integrated stable process. Simulations for the estimator of β and its limit distribution are also given.     

Stationary bootstrapping for non‐parametric estimator of nonlinear autoregressive model

We consider stationary bootstrap approximation of the non‐parametric kernel estimator in a general kth‐order nonlinear autoregressive model under the conditions ensuring that the nonlinear autoregressive process is a geometrically Harris ergodic stationary Markov process. We show that the stationary bootstrap procedure properly estimates the distribution of the non‐parametric kernel estimator. A simulation study is provided to illustrate the theory and to construct confidence intervals, which compares the proposed method favorably with some other bootstrap methods.     

Testing for Change in Long‐Memory Stochastic Volatility Time Series

In this article, change‐point problems for long‐memory stochastic volatility (LMSV) models are considered. A general testing problem which includes various alternative hypotheses is discussed. Under the hypothesis of stationarity the limiting behavior of CUSUM‐ and Wilcoxon‐type test statistics is derived. In this context, a limit theorem for the two‐parameter empirical process of LMSV time series is proved. In particular, it is shown that the asymptotic distribution of CUSUM test statistics may not be affected by long memory, unlike Wilcoxon test statistics which are typically influenced by long‐range dependence. To avoid the estimation of nuisance parameters in applications, the usage of self‐normalized test statistics is proposed. The theoretical results are accompanied by an analysis of Standard & Poor's 500 daily closing indices with respect to structural changes and by simulation studies which characterize the finite sample behavior of the considered testing procedures when testing for changes in mean and in variance.     

Likelihood inference for discriminating between long‐memory and change‐point models

We develop a likelihood ratio (LR) test procedure for discriminating between a short‐memory time series with a change‐point (CP) and a long‐memory (LM) time series. Under the null hypothesis, the time series consists of two segments of short‐memory time series with different means and possibly different covariance functions. The location of the shift in the mean is unknown. Under the alternative, the time series has no shift in mean but rather is LM. The LR statistic is defined as the normalized log‐ratio of the Whittle likelihood between the CP model and the LM model, which is asymptotically normally distributed under the null. The LR test provides a parametric alternative to the CUSUM test proposed by Berkes et al. (2006) . Moreover, the LR test is more general than the CUSUM test in the sense that it is applicable to changes in other marginal or dependence features other than a change‐in‐mean. We show its good performance in simulations and apply it to two data examples.     

Testing for parameter stability in nonlinear autoregressive models

In this article we develop testing procedures for the detection of structural changes in nonlinear autoregressive processes. For the detection procedure, we model the regression function by a single layer feedforward neural network. We show that CUSUM‐type tests based on cumulative sums of estimated residuals, that have been intensively studied for linear regression, can be extended to this case. The limit distribution under the null hypothesis is obtained, which is needed to construct asymptotic tests. For a large class of alternatives, it is shown that the tests have asymptotic power one. In this case, we obtain a consistent change‐point estimator which is related to the test statistics. Power and size are further investigated in a small simulation study with a particular emphasis on situations where the model is misspecified, i.e. the data is not generated by a neural network but some other regression function. As illustration, an application on the Nile data set as well as S&P log‐returns is given.     

NON‐STATIONARITY AND QUASI‐MAXIMUM LIKELIHOOD ESTIMATION ON A DOUBLE AUTOREGRESSIVE MODEL

This article first studies the non‐stationarity of the first‐order double AR model, which is defined by the random recurrence equation , where γ₀ > 0, α₀ ≥ 0, and {η_t}is a sequence of i.i.d. symmetric random variables. It is shown that the double AR(1) model is explosive under the condition . Based on this, it is shown that the quasi‐maximum likelihood estimator of (φ₀,α₀) is consistent and asymptotically normal so that the unit root problem does not exist in the double AR(1) model. Simulation studies are carried out to assess the performance of the quasi‐maximum likelihood estimator in finite samples.     

Maximum likelihood estimation for nearly non‐stationary stable autoregressive processes

The maximum likelihood estimate (MLE) of the autoregressive coefficient of a near‐unit root autoregressive process Y_t = ρ_nY_t?1 + ?_t with α‐stable noise {?_t} is studied in this paper. Herein ρ_n = 1 ? γ/n, γ ≥ 0 is a constant, Y₀ is a fixed random variable and ε_t is an α‐stable random variable with characteristic function φ(t,θ) for some parameter θ. It is shown that when 0 < α < 1 or α > 1 and E?₁ = 0, the limit distribution of the MLE of ρ_n and θ are mixtures of a stable process and Gaussian processes. On the other hand, when α > 1 and E?₁ ≠ 0, the limit distribution of the MLE of ρ_n and θ are normal. A Monte Carlo simulation reveals that the MLE performs better than the usual least squares procedures, particularly for the case when the tail index α is less than 1.     

Consistent estimation of the memory parameter for nonlinear time series

Abstract. For linear processes, semiparametric estimation of the memory parameter, based on the log‐periodogram and local Whittle estimators, has been exhaustively examined and their properties well established. However, except for some specific cases, little is known about the estimation of the memory parameter for nonlinear processes. The purpose of this paper is to provide the general conditions under which the local Whittle estimator of the memory parameter of a stationary process is consistent and to examine its rate of convergence. We show that these conditions are satisfied for linear processes and a wide class of nonlinear models, among others, signal plus noise processes, nonlinear transforms of a Gaussian process ξ_t and exponential generalized autoregressive, conditionally heteroscedastic (EGARCH) models. Special cases where the estimator satisfies the central limit theorem are discussed. The finite‐sample performance of the estimator is investigated in a small Monte Carlo study.     

Asymptotic self‐similarity and wavelet estimation for long‐range dependent fractional autoregressive integrated moving average time series with stable innovations

Abstract. Methods for parameter estimation in the presence of long‐range dependence and heavy tails are scarce. Fractional autoregressive integrated moving average (FARIMA) time series for positive values of the fractional differencing exponent d can be used to model long‐range dependence in the case of heavy‐tailed distributions. In this paper, we focus on the estimation of the Hurst parameter H = d + 1/α for long‐range dependent FARIMA time series with symmetric α‐stable (1 < α < 2) innovations. We establish the consistency and the asymptotic normality of two types of wavelet estimators of the parameter H. We do so by exploiting the fact that the integrated series is asymptotically self‐similar with parameter H. When the parameter α is known, we also obtain consistent and asymptotically normal estimators for the fractional differencing exponent d = H ? 1/α. Our results hold for a larger class of causal linear processes with stable symmetric innovations. As the wavelet‐based estimation method used here is semi‐parametric, it allows for a more robust treatment of long‐range dependent data than parametric methods.     

Oracle M‐Estimation for Time Series Models

We propose a thresholding M‐estimator for multivariate time series. Our proposed estimator has the oracle property that its large‐sample properties are the same as of the classical M‐estimator obtained under the a priori information that the zero parameters were known. We study the consistency of the standard block bootstrap, the centred block bootstrap and the empirical likelihood block bootstrap distributions of the proposed M‐estimator. We develop automatic selection procedures for the thresholding parameter and for the block length of the bootstrap methods. We present the results of a simulation study of the proposed methods for a sparse vector autoregressive VAR(2) time series model. The analysis of two real‐world data sets illustrate applications of the methods in practice.     

DOPO‐Based Phosphorus‐Containing Methacrylic (Co)Polymers: Glass Transition Temperature Investigation

A two‐step synthetic procedure is designed for preparing new flame‐retardant methacrylic monomers containing 9,10‐dihydro‐9‐oxa‐10‐phosphaphenanthrene‐10‐oxide (DOPO) as a substituent side group. DOPO and methacrylate moieties are linked by linear aliphatic hydrocarbon spacers (3 to 11 carbon atoms). Copolymerization with methyl methacrylate is carried out leading to copolymers containing between 2 and 10 wt% phosphorus. All homo‐ and copolymers exhibit a unique glass transition temperature (T_g ). A new group contribution for DOPO‐based substituent is extracted that leads to reasonable estimations of T_g s of other published polymers. The Fox equation provides a good estimation of T_g s for most copolymers and for physical blends of poly(methyl methacrylate) (PMMA) and DOPO. When using monomers having three and four carbon atoms in the hydrocarbon spacer, the T_g of copolymers remains close to that of PMMA over a wide range of composition.     

Testing for Change‐Points in Long‐Range Dependent Time Series by Means of a Self‐Normalized Wilcoxon Test

We propose a testing procedure based on the Wilcoxon two‐sample test statistic in order to test for change‐points in the mean of long‐range dependent data. We show that the corresponding self‐normalized test statistic converges in distribution to a non‐degenerate limit under the hypothesis that no change occurred and that it diverges to infinity under the alternative of a change‐point with constant height. Furthermore, we derive the asymptotic distribution of the self‐normalized Wilcoxon test statistic under local alternatives, that is, under the assumption that the height of the level shift decreases as the sample size increases. Regarding the finite sample performance, simulation results confirm that the self‐normalized Wilcoxon test yields a consistent discrimination between hypothesis and alternative and that its empirical size is already close to the significance level for moderate sample sizes.     

Robust estimators under semi‐parametric partly linear autoregression: Asymptotic behaviour and bandwidth selection

Abstract. In this article, under a semi‐parametric partly linear autoregression model, a family of robust estimators for the autoregression parameter and the autoregression function is studied. The proposed estimators are based on a three‐step procedure, in which robust regression estimators and robust smoothing techniques are combined. Asymptotic results on the autoregression estimators are derived. Besides combining robust procedures with M‐smoothers, predicted values for the series and detection residuals, which allow to detect anomalous data, are introduced. Robust cross‐validation methods to select the smoothing parameter are presented as an alternative to the classical ones, which are sensitive to outlying observations. A Monte Carlo study is conducted to compare the performance of the proposed criteria. Finally, the asymptotic distribution of the autoregression parameter estimator is stated uniformly over the smoothing parameter.     

Thermo‐oxidative decomposition kinetics of elastomeric composites based on styrene‐(ethylene‐butylene)‐styrene triblock copolymer and organomontmorillonite

The elastomeric nanocomposites based on organomontmorillonite (OMMT) and styrene‐(ethylene‐butylene)‐styrene (SEBS) thermoplastic elastomer were prepared by melt processing using maleic anhydride grafted SEBS (SEBS‐g‐MA) as compatibilizer. Thermo‐oxidative decomposition behavior of the neat components and the nanocomposites were investigated using thermogravimertic analysis (TGA) in air atmosphere. The isoconversional method is employed to study the kinetics of thermo‐oxidative degradation. The heating modes and the composition of nanocomposites were found to affect the kinetic parameters (E_a, lnA and n). The E_a and lnA values of SEBS, OMMT, and their composites are much higher under dynamic heating than under isothermal heating. The reaction order (n) of OMMT was lower than those of SEBS and their composites. The obtained TG profiles and calculated kinetic parameters indicated that the incorporation of OMMT into SEBS significantly improved the thermal stability both under dynamic heating and under isothermal heating. The simultaneously obtained DSC data showed that the enthalpy of thermal decomposition decreased with OMMT loading. No significant change in the nonisothermal and isothermal stability of the nanocomposites with addition of SEBS‐g‐MA. © 2011 Wiley Periodicals, Inc. J Appl Polym Sci, 2011     

A Robbins–Monro Algorithm for Non‐Parametric Estimation of NAR Process with Markov Switching: Consistency

We approach the problem of non‐parametric estimation for autoregressive Markov switching processes. In this context, the Nadaraya–Watson‐type regression functions estimator is interpreted as a solution of a local weighted least‐square problem, which does not admit a closed‐form solution in the case of hidden Markov switching. We introduce a non‐parametric recursive algorithm to approximate the estimator. Our algorithm restores the missing data by means of a Monte Carlo step and estimates the regression function via a Robbins–Monro step. We prove that non‐parametric autoregressive models with Markov switching are identifiable when the hidden Markov process has a finite state space. Consistency of the estimator is proved using the strong α‐mixing property of the model. Finally, we present some simulations illustrating the performances of our non‐parametric estimation procedure.     

Mid‐chain grafting in PVB‐graft‐PVB

The polymer commonly referred to as poly(vinyl butyral) is actually the statistical terpolymer poly(vinyl butyral‐co‐vinyl alcohol‐co‐vinyl acetate), the main component of the polymeric interlayer in automotive and architectural safety glass, amongst other uses. Here, the effects of ‘self‐grafting’ or ‘auto‐grafting’ in this polymer are examined using size‐exclusion chromatography (SEC) with triple detection. These and supporting experiments (eg, batch‐mode multi‐angle light scattering) allow study of the effects of induced branching in the resultant PVB‐graft‐PVB molecule, and comparison with the ungrafted (though not linear) base polymer. This was done by application and extension of the classic Zimm–Stockmayer long‐chain branching (LCB) theory, by determination of the fractal dimension (d_f) of the polymers and of the change in d_f as a function of molar mass, as well as by the multiplicity of size parameters that are measured in a multi‐detector SEC experiment. Copyright © 2004 Society of Chemical Industry     

Thermal analysis of poly(3‐hydroxybutyrate‐co‐3‐hydroxyvalerate) irradiated under vacuum

Poly(3‐hydroxybutyrate‐co‐3‐hydroxyvalerate) (PHBV) was irradiated by ⁶⁰Co γ‐rays (doses of 50, 100 and 200 kGy) under vacuum. The thermal analysis of control and irradiated PHBV, under vacuum was carried out by thermogravimetric analysis (TGA) and differential scanning calorimetry (DSC). The tensile properties of control and irradiated PHBV were examined by using an Instron tensile testing machine. In the thermal degradation of control and irradiated PHBV, a one‐step weight loss was observed. The derivative thermogravimetric curves of control and irradiated PHBV confirmed only one weight‐loss step change. The onset degradation temperature (T_o) and the temperature of maximum weight‐loss rate (T_p) of control and irradiated PHBV were in line with the heating rate (°C min^?1). T_o and T_P of PHBV decreased with increasing radiation dose at the same heating rate. The DSC results showed that ⁶⁰Co γ‐radiation significantly affected the thermal properties of PHBV. With increasing radiation dose, the melting temperature (T_m) of PHBV shifted to a lower value, due to the decrease in crystal size. The tensile strength and fracture strain of the irradiated PHBV decreased, hence indicating an increased brittleness. Copyright © 2004 Society of Chemical Industry