Broadband semi‐parametric estimation of long‐memory time series by fractional exponential models

This article proposes broadband semi‐parametric estimation of a long‐memory parameter by fractional exponential (FEXP) models. We construct the truncated Whittle likelihood based on FEXP models in a semi‐parametric setting to estimate the parameter and show that the proposed estimator is more efficient than the FEXP estimator by Moulines and Soulier (1999) in linear processes. A Monte Carlo simulation suggests that the proposed estimation is more preferable than the existing broadband semi‐parametric estimation.     

A Self‐Normalized Semi‐Parametric Test to Detect Changes in the Long Memory Parameter

We consider the problem of testing for change points in the long memory parameter. The test relies on semi‐parametric estimation of the long memory parameter, which does not require the complete parametric specification of the whole spectrum. A self‐normalizer utilizing a sequence of recursive semi‐parametric estimators is used to make the asymptotic distribution of the test statistic free of the nuisance scale parameter. We study the asymptotic behavior of the proposed test for situations when there is at most one change point and also when there are an unknown number of change points. Monte Carlo simulations are carried out to examine the finite‐sample performance of the proposed test.     

Estimation of the long‐memory stochastic volatility model parameters that is robust to level shifts and deterministic trends

I provide conditions under which the trimmed FDQML estimator, advanced by McCloskey (2010) in the context of fully parametric short‐memory models, can be used to estimate the long‐memory stochastic volatility model parameters in the presence of additive low‐frequency contamination in log‐squared returns. The types of low‐frequency contamination covered include level shifts as well as deterministic trends. I establish consistency and asymptotic normality in the presence or absence of such low‐frequency contamination under certain conditions on the growth rate of the trimming parameter. I also provide theoretical guidance on the choice of trimming parameter by heuristically obtaining its asymptotic MSE‐optimal rate under certain types of low‐frequency contamination. A simulation study examines the finite sample properties of the robust estimator, showing substantial gains from its use in the presence of level shifts. The finite sample analysis also explores how different levels of trimming affect the parameter estimates in the presence and absence of low‐frequency contamination and long‐memory.     

On the Spectral Density of the Wavelet Coefficients of Long‐Memory Time Series with Application to the Log‐Regression Estimation of the Memory Parameter

Abstract. In recent years, methods to estimate the memory parameter using wavelet analysis have gained popularity in many areas of science. Despite its widespread use, a rigorous semi‐parametric asymptotic theory, comparable with the one developed for Fourier methods, is still lacking. In this article, we adapt to the wavelet setting, the classical semi‐parametric framework introduced by Robinson and his co‐authors for estimating the memory parameter of a (possibly) non‐stationary process. Our results apply to a class of wavelets with bounded supports, which include but are not limited to Daubechies wavelets. We derive an explicit expression of the spectral density of the wavelet coefficients and show that it can be approximated, at large scales, by the spectral density of the continuous‐time wavelet coefficients of fractional Brownian motion. We derive an explicit bound for the difference between the spectral densities. As an application, we obtain minimax upper bounds for the log‐scale regression estimator of the memory parameter for a Gaussian process and we derive an explicit expression of its asymptotic variance.     

Bootstrapping the Local Periodogram of Locally Stationary Processes

Abstract. Locally stationary processes are non‐stationary stochastic processes the second‐order structure of which varies smoothly over time. In this paper, we develop a method to bootstrap the local periodogram of a locally stationary process. Our method generates pseudo local periodogram ordinates by combining a parametric time and non‐parametric frequency domain bootstrap approach. We first fit locally a time varying autoregressive model so as to capture the essential characteristics of the underlying process. A locally calculated non‐parametric correction in the frequency domain is then used so as to improve upon the locally parametric autoregressive fit. As an application, we investigate theoretically the asymptotic properties of the bootstrap method proposed applied to the class of local spectral means, local ratio statistics and local spectral density estimators. Some simulations demonstrate the ability of our method to give accurate estimates of the quantities of interest in finite sample situations and an application to a real‐life data‐set is presented.     

Semiparametric Estimation in Time‐Series Regression with Long‐Range Dependence

Abstract. We consider semiparametric estimation in time‐series regression in the presence of long‐range dependence in both the errors and the stochastic regressors. A central limit theorem is established for a class of semiparametric frequency domain‐weighted least squares estimates, which includes both narrow‐band ordinary least squares and narrow‐band generalized least squares as special cases. The estimates are semiparametric in the sense that focus is on the neighbourhood of the origin, and only periodogram ordinates in a degenerating band around the origin are used. This setting differs from earlier studies on time‐series regression with long‐range dependence, where a fully parametric approach has been employed. The generalized least squares estimate is infeasible when the degree of long‐range dependence is unknown and must be estimated in an initial step. In that case, we show that a feasible estimate which has the same asymptotic properties as the infeasible estimate, exists. By Monte Carlo simulation, we evaluate the finite‐sample performance of the generalized least squares estimate and the feasible estimate.     

The Periodogram of fractional processes1

Abstract. We analyse asymptotic properties of the discrete Fourier transform and the periodogram of time series obtained through (truncated) linear filtering of stationary processes. The class of filters contains the fractional differencing operator and its coefficients decay at an algebraic rate, implying long‐range‐dependent properties for the filtered processes when the degree of integration α is positive. These include fractional time series which are nonstationary for any value of the memory parameter (α ≠ 0) and possibly nonstationary trending (α ≥ 0.5). We consider both fractional differencing or integration of weakly dependent and long‐memory stationary time series. The results obtained for the moments of the Fourier transform and the periodogram at Fourier frequencies in a degenerating band around the origin are weaker compared with the stationary nontruncated case for α > 0, but sufficient for the analysis of parametric and semiparametric memory estimates. They are applied to the study of the properties of the log‐periodogram regression estimate of the memory parameter α for Gaussian processes, for which asymptotic normality could not be showed using previous results. However, only consistency can be showed for the trending cases, 0.5 ≤ α < 1. Several detrending and initialization mechanisms are studied and only local conditions on spectral densities of stationary input series and transfer functions of filters are assumed.     

Semiparametric Inference in Seasonal and Cyclical Long Memory Processes

Several semiparametric estimates of the memory parameter in standard long memory time series are now available. They consider only local behaviour of the spectrum near zero frequency, about which the spectrum is symmetric. However long-range dependence can appear as a spectral pole at any Nyqvist frequency (reflecting seasonal or cyclical long-memory), where the spectrum need display no such symmetry. We introduce Seasonal/Cyclical Asymmetric Long Memory (SCALM) processes that allow differing rates of increase on either side of such a pole. To estimate the two consequent memory parameters we extend two semiparametric methods that were proposed for the standard case of a spectrum diverging at the origin, namely the log-periodogram and Gaussian or Whittle methods. We also provide three tests of symmetry. Monte Carlo analysis of finite sample behaviour and an empirical application to UK inflation data are included. Our models and methods allow also for the possibility of negative dependence, described by a possibly asymmetric spectral zero.     

Parametric and Semi‐Parametric Efficient Tests for Parameter Instability

This article examines asymptotically point optimal tests for parameter instability in realistic circumstances when little information about the unstable parameter process and error distribution is available. We first show that, under a correctly specified error distribution, if the unstable parameter processes converge weakly to a Wiener process, then any asymptotic optimal tests for structural breaks and time‐varying parameters are asymptotically equivalent. Our finding is then extended to a semi‐parametric set‐up in which the error distribution is treated as an unknown infinite‐dimensional nuisance parameter. We find that semi‐parametric tests can be adaptive without further restrictive conditions on the error distribution.     

A Bayesian Non‐Parametric Dynamic AR Model for Multiple Time Series Analysis

In this article, we propose a Bayesian non‐parametric model for the analysis of multiple time series. We consider an autoregressive structure of order p for each of the series and borrow strength across the series by considering a common error population that is also evolving in time. The error populations (distributions) are assumed non‐parametric whose law is based on a series of dependent Polya trees with zero median. This dependence is of order q and is achieved via a dependent beta process that links the branching probabilities of the trees. We study the prior properties and show how to obtain posterior inference. The model is tested under a simulation study and is illustrated with the analysis of the economic activity index of the 32 states of Mexico.     

QMLE for Quadratic ARCH Model with Long Memory

We discuss parametric quasi‐maximum likelihood estimation for quadratic autoregressive conditional heteroskedasticity (ARCH) process with long memory introduced in Doukhan emphet al. (2016) and Grublyt? and ?karnulis (2016) with conditional variance involving the square of inhomogeneous linear combination of observable sequence with square summable weights. The aforementioned model extends the quadratic ARCH model of Sentana ( 1995 ) and the linear ARCH model of Robinson ( 1991 ) to the case of strictly positive conditional variance. We prove consistency and asymptotic normality of the corresponding quasi‐maximum likelihood estimates, including the estimate of long memory parameter 0 < d < 1/2. A simulation study of empirical mean‐squared error is included.     

Conditional variance estimation in regression models with long memory

In this article we study asymptotic properties of a non‐parametric kernel estimator of the conditional variance in a random design model with parametric mean and heteroscedastic errors, for a class of long‐memory errors and predictors. We establish small and large bandwidths asymptotics, which show a different behaviour compared with that of kernel estimators of the conditional mean. We distinguish between an oracle case (i.e. where the errors are directly observed) and a non‐oracle case (where the errors are replaced with residuals) and show non‐equivalence between the oracle and non‐oracle case. We also discuss a practical problem of bandwidth choice. Theoretical results are justified by simulation studies. We apply our theory to DJA and FTSE indices.     

Semi‐interpenetrating polymer networks based on polyurethane and poly(vinyl pyrrolidone) obtained by photopolymerization: Structure‐property relationships and bacterial adhesion

A range of semi‐interpenetrating polymer networks (semi‐IPNs) based on polyurethane (PU) and poly(vinyl pyrrolidone) (PVP) have been synthesized and characterized with respect to their thermodynamic characteristics, morphology, mechanical properties, surface properties, water sorption and bacterial adhesion. The free energies of mixing of PU and PVP in semi‐IPNs have been determined by the vapor sorption method and were shown to be positive for all compositions. The surface properties of semi‐IPNs were investigated using the dynamic contact angle analysis. It was shown that the advancing contact angle changes from 83.1° to 65.3° with increasing PVP from 7.05% to 57.38%. Scanning electron microscopy demonstrated that the semi‐IPNs are two‐phase systems with incomplete phase separation. The mechanical properties reflect the changes in structure of semi‐IPNs with increasing of amounts of PVP in the system. Incorporation of PVP into the semi‐IPN with PU restricts the ability of PVP to sorb water. As infection is likely to be caused by bacterial adherence to biomedical implants, the bacterial adhesion data suggests that the semi‐IPNs with PVP content below 22.52% may be useful for biomedical material applications. Polym. Eng. Sci. 44:940–947, 2004. © 2004 Society of Plastics Engineers.     

Miscibility and thermal and dynamic mechanical behaviour of semi‐interpenetrating polymer networks based on polyurethane and poly(hydroxyethyl methacrylate)

The thermodynamic miscibility and thermal and dynamic mechanical behaviour of semi‐interpenetrating polymer networks (semi‐IPNs) of crosslinked polyurethane (PU) and linear poly(hydroxyethyl methacrylate) (PHEMA) have been investigated. The free energies of mixing of the semi‐IPN components have been determined by the vapour sorption method and it was established that the parameters are positive and depend on the amount of PHEMA in the semi‐IPN samples. Thermal analyses glass transition temperatures evidenced two in the semi‐IPNs in accordance with the investigation of the thermodynamic miscibility of these systems. Dynamic mechanical analysis revealed a pronounced change in the viscoelastic properties of the PU‐based semi‐IPNs with different amounts of PHEMA in the samples. The semi‐IPNs have two distinct tan δ maxima related to the relaxations of the two polymers in their glass temperature domains. The temperature position of PU relaxation maximum tan δ is invariable but its amplitude decreases in the semi‐IPNs with increasing amount of PHEMA in the systems. The tan δ maximum of PHEMA is shifted to a lower temperature and its amplitude decreases with increasing amount of PU in the semi‐IPNs. The segregation degree of components α was calculated using the viscoelastic properties of semi‐IPNs. It was concluded that the studied semi‐IPNs are two‐phase systems with incomplete phase separation. The different levels of immiscibility lead to the different degree of phase separation in the semi‐IPNs with compositions. Copyright © 2004 Society of Chemical Industry     

Semiinterpenetrating polymer networks based on polyurethane and polyvinylpyrrolidone. I. Thermodynamic state and dynamic mechanical analysis

Semiinterpenetrating polymer networks (semi‐IPNs) based on polyurethane (PU) and polyvinylpyrrolidone (PVP) have been synthesized, and their thermodynamic characteristics, thermal properties, and dynamical mechanical properties have been studied to have an insight in their structure as a function of their composition. First, the free energies of mixing of the two polymers in semi‐IPNs based on crosslinked PU and PVP have been determined by the vapor sorption method. It was established that these constituent polymers are not miscible in the semi‐IPNs. The differential scanning calorimetry results evidence the T_g of polyurethane and two T_g for PVP. The dynamic mechanical behavior of the semi‐IPNs has been investigated and is in accordance with their thermal behavior. It was shown that the semi‐IPNs present three distinct relaxations. If the temperature position of PU maximum tan δ is invariable, on the contrary, the situation for the two maxima observed for PVP is more complex. Only the maximum of the highest temperature relaxation is shifted to lower temperature with changing of the semi‐IPNs composition. It was concluded that investigated semi‐IPNs are two‐phase systems with incomplete phase separation. The phase composition was calculated using viscoelastic properties. © 2001 John Wiley & Sons, Inc. J Appl Polym Sci 80: 852–862, 2001     

Phase separation in the polyurethane/poly(2‐hydroxyethyl methacrylate) semi‐interpenetrating polymer networks synthesized by different ways

The thermal, dynamic mechanical analysis, morphology and mechanical properties of semi‐interpenetrating polymer networks based on crosslinked polyurethane (PU) and poly(2‐hydroxyethyl methacrylate) (PHEMA) synthesized by photopolymerization and by thermopolymerization have been investigated. The thermal analysis has evidenced the two glass temperature transitions in the semi‐IPNs and this is confirmed by the thermodynamic miscibility investigation of the systems. The Dynamic Mechanical Analysis spectra have shown that the phase separation is more significant in the thermopolymerized semi‐IPNs: the tan δ peaks of constituent polymers are more distinct and the minimum between the two peaks is deeper. The calculated segregation degree values of semi‐IPN's components are significantly higher for thermopolymerized semi‐IPNs, thereby the process of phase separation in the thermopolymerized semi‐IPNs is more developed. The structures of two series of samples investigated by SEM are completely different. The mechanical properties reflect these changes in structure of semi‐IPNs with increasing amount of PHEMA and with the changing of the method of synthesis. The results suggest that the studied semi‐IPNs are two‐phase systems with incomplete phase separation. The semi‐IPN samples with early stage of phase separation demonstrate higher mechanical characteristics. POLYM. ENG. SCI., 2008. © 2008 Society of Plastics Engineers     

Regulation of micromorphology and proton conductivity of sulfonated polyimide/crosslinked PNIPAm semi‐interpenetrating networks by hydrogen bonding

A series of novel sulfonated polyimide (SPI)/crosslinked poly(N‐isopropylacrylamide) (cPNIPAm) semi‐interpenetrating polymer networks (semi‐IPNs) were synthesized as the proton exchange membranes for direct methanol fuel cells via in situ polymerization. The micromorphology and properties of the semi‐IPN membranes were characterized. The results indicated that the hydrogen bonds between cPNIPAm and SPI in the semi‐IPN structure were a crucial factor for regulating the micromorphology, proton conductivity and other properties of the semi‐IPN membranes. A more uniform sulfonic ionic cluster distribution was observed in the membrane of SPI‐20‐cPNIPAm with equimolar ratio of sulfonic acid groups and amido bonds, which could provide effective proton transport channels. The SPI‐20‐cPNIPAm exhibited a maximum proton conductivity of 0.331 S cm^?1 at 80 ^oC (relative humidity 100%), an optimal selectivity of 8.01 × 10⁵ S s cm^?3 and an improved fuel cell performance of 72 mW cm^?2 compared with both pristine SPI and other semi‐IPN membranes. The SPI‐20‐cPNIPAm semi‐IPN membranes also retained good mechanical properties and thermal stabilities on the whole. © 2014 Society of Chemical Industry     

Discriminant Analysis of Time Series in the Presence of Within‐Group Spectral Variability

Many studies record replicated time series epochs from different groups with the goal of using frequency domain properties to discriminate between the groups. In many applications, there exists variation in cyclical patterns from time series in the same group. Although a number of frequency domain methods for the discriminant analysis of time series have been explored, there is a dearth of models and methods that account for within‐group spectral variability. This article proposes a model for groups of time series in which transfer functions are modelled as stochastic variables that can account for both between‐group and within‐group differences in spectra that are identified from individual replicates. An ensuing discriminant analysis of stochastic cepstra under this model is developed to obtain parsimonious measures of relative power that optimally separate groups in the presence of within‐group spectral variability. The approach possesses favourable properties in classifying new observations and can be consistently estimated through a simple discriminant analysis of a finite number of estimated cepstral coefficients. Benefits in accounting for within‐group spectral variability are empirically illustrated in a simulation study and through an analysis of gait variability.     

Thermal and pH dual‐responsive hydrogels based on semi‐interpenetrating network of poly(N‐isopropylacrylamide) and collagen nanofibrils

Hydrogels with environment‐sensitive properties have great potential applications in the controlled drug release field. In this paper, hybrid hydrogels with semi‐interpenetrating polymer networks (semi‐IPNs), composed of poly(N‐isopropylacrylamide) (PNIPAM) as the thermo‐sensitive component by in situ polymerization and self‐assembled collagen nanofibrils as the pH‐sensitive framework, were prepared for controlled release of methyl violet as a model drug. From Fourier transform infrared spectroscopy and scanning electron microscopy, it was indicated that the crosslinking of PNIPAM in the presence of collagen nanofibrils led to the formation of semi‐IPNs with homogeneous porous structure, and the semi‐IPNs showed improved thermal stability and elastic properties compared with the native collagen as determined using differential scanning calorimetry and rheologic measurements. Furthermore, the semi‐IPNs possessed swelling behaviors quite different from those of neat collagen or PNIPAM hydrogel under various pH values and temperatures. Correspondingly, as expected, the drug release behavior in vitro for semi‐IPNs performed variously compared with that for single‐component semi‐IPNs, which revealed the tunable performance of semi‐IPNs for release ability. Finally the thermo‐ and pH‐responsive mechanism of the semi‐IPNs was illuminated to provide guidance for the application of the thermo‐ and pH‐sensitive collagen‐based hybrid hydrogels in controlled drug delivery systems. © 2019 Society of Chemical Industry