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.     

Ultrasonic air-coupled testing of square-shape CFRP composite rods by means of guided waves

In aerospace industry, one of the most important parts of gliders and motor-gliders is a lightweight longeron reinforcement made of carbon fibre reinforced plastics (CFRP) rods, known as Graphlite SM315 composite. During manufacturing, the rods as constructional elements are glued together in epoxy-filled matrix in order to build the arbitrary spar profile. The defects present in single rods such as breakage of fibres, multiple delaminations due to the lack of bonding between fibres and reduction in density affect essentially the strength and the fail-safety of the overall construction.The aim of the present work is to investigate the effects (transmission, reflection, scattering and mode conversion) of guided waves propagation along a square-shape CFRP rod in the case of contactless excitation/reception and interaction with region of multiple delaminations applying the numerical model and performing experiments.The square-shape CFRP composite rods possessing internal artificial delamination type defects have been investigated by numerical modelling and experiments employing the developed air-coupled technique (pitch-catch set-up) for cases of conventional transmission and advanced back-scattering configurations. Numerical predictions of guided wave interaction with a multiple delamination type defect in a CFRP composite rod have been made and the interaction mechanism explained. It is possible to conclude, that the actual sizes of the internal defects have been clearly detected using reception of the back-scattered waves over the edges of the defective regions only.     

RATE OPTIMAL SEMIPARAMETRIC ESTIMATION OF THE MEMORY PARAMETER OF THE GAUSSIAN TIME SERIES WITH LONG-RANGE DEPENDENCE

There exist several estimators of the memory parameter in long- memory time series models with the spectrum specified only locally near zero frequency. In this paper we give an asymptotic lower bound for the minimax risk of any estimator of the memory parameter as a function of the degree of local smoothness of the spectral density at zero. The lower bound allows one to evaluate and compare different estimators by their asymptotic behaviour, and to claim the rate optimality for any estimator attaining the bound. A log-periodogram regression estimator, analysed by Robinson (Log-periodogram regression of time series with long range dependence. Ann. Stat. 23 (1995), 1048--72), is then shown to attain the lower bound, and is thus rate optimal.     

On the Optimal Segment Length for Parameter Estimates for Locally Stationary Time Series

We discuss the behaviour of parameter estimates when stationary time series models are fitted locally to non-stationary processes which have an evolutionary spectral representation. A particular example is the estimation for an autoregressive process with time-varying coefficients by local Yule–Walker estimates. The bias and the mean squared error for the parameter estimates are calculated and the optimal length of the data segment is determined.     

Application of the ultrasonic pulse-echo technique for quality control of the multi-layered plastic materials

The main aim of the ultrasonic pulse-echo technique application for quality control is to characterize the internal structure of the object under investigation. The advantage of such technique is a possibility to perform non-contact and one-side access measurements, and to investigate the internal structure of multi-layered materials as well. The presented novel application of the ultrasonic pulse-echo technique for characterization of the multi-layered plastic materials covers the complete attenuation measurement in the frequency domain and is based on ill-posed Tikhonov regularization task for each layer separately. The law of the frequency-dependent attenuation has been estimated from the inverse transfer function approximation in the frequency domain. Phase velocity dispersion curves have been estimated in two ways: from the experimental signal phase spectra and from the causal Kramers–Kronig relations. The developed method enables to predict waveforms of the reflected signals from the interfaces of the individual layers in on-line mode.According to this approach, the step-by-step iterative analysis has been performed for each layer using the information about the previous layers. During each step, the acoustic properties of an individual layer, such as density, absorption, ultrasound velocity and phase velocity dispersion, have been recovered using numerical optimization. Optimization has been performed comparing the real ultrasonic signal, reflected by multi-layered object, with the simulated response of the model. The comparison of the predicted waveforms with the experimental ones has shown a good correspondence.     

STUDENTIZING WEIGHTED SUMS OF LINEAR PROCESSES

This article presents a general method for studentizing weighted sums of a linear process where weights are arrays of known real numbers and innovations form a martingale difference sequence. Asymptotical normality for such sums was established in Abadir et al. (2013). This article centres on the estimation of the standard deviation, to make the normal approximation operational. The proposed studentization is easy to apply and robust against unknown types of dependence (short range and long range) in the observations. It does not require the estimation of the parameters controlling the dependence structure. A finite‐sample Monte Carlo simulation study shows the applicability of the proposed methodology for moderate sample sizes. Assumptions for studentization are satisfied by the Nadaraya–Watson kernel type weights used for inference in non‐parametric regression settings.     

Uniform Limit Theory for Stationary Autoregression

Abstract. First order autoregression is shown to satisfy a limit theory which is uniform over stationary values of the autoregressive coefficient ρ = ρ_n ∈ [0, 1) provided (1 ? ρ_n)n → ∞. This extends existing Gaussian limit theory by allowing for values of stationary ρ that include neighbourhoods of unity provided they are wider than O(n^?1), even by a slowly varying factor. Rates of convergence depend on ρ and are at least but less than n. Only second moments are assumed, as in the case of stationary autoregression with fixed ρ.     

Simulation of the normal impact of randomly shaped quasi-spherical particles

The paper reports the modelling of randomly shaped particles. An emphasis is placed on the illustration of random properties of particles, using simulations with a controlled probability distribution for the depth of the surface profile. The randomly-shaped quasi-spherical particles were described by applying a multi-sphere approximation and a statistical evaluation technique, and the surface of the particles was approximated using randomly located overlapping subspheres. The concept of statistically similar particles, i.e., particles characterised by having a similar probability distribution for the depth of the surface profile, was employed for these purposes, and an original method involving the application of a stochastic optimisation was developed. The optimization method was demonstrated by generating statistically similar particles. The contact behaviour was investigated by simulating a random particle impact against a wall, using the discrete element method. It was observed that statistically similar particles did not show statistically similar contact characteristics. The results of this study suggested that the refinement of the multi-sphere model (achieved by increasing the number of subspheres) was non-unique, not only in a deterministic context but also in statistical context, and that this subject requires further investigation.