Asymptotics for stationary very nearly unit root processes

Abstract. This article considers a mean zero stationary first‐order autoregressive (AR) model. It is shown that the least squares estimator and t statistic have Cauchy and standard normal asymptotic distributions, respectively, when the AR parameter ρ_n is very near to one in the sense that 1 ? ρ_n = o(n^?1).     

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.     

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.     

Limit theory for a general class of GARCH models with just barely infinite variance

In this article, limit theory is established for a general class of generalized autoregressive conditional heteroskedasticity models given by ?_t = σ_tη_t and σ_t = f (σ_t?1, σ_t?2,…, σ_t?p, ?_t?1, ?_t?2,…, ?_t?q), when {?_t} is a process with just barely infinite variance, that is, {?_t} is a process with infinite variance but in the domain of normal attraction. In particular, we show that under some regular conditions, converges weakly to a Gaussian process. Applications of the asymptotic results to statistical inference, such as unit root test and sample autocorrelation, are also investigated. The obtained result fills in a gap between the classical infinite variance and finite variance in the literature. Further, when applying our limiting result to Dickey–Fuller (DF) test in a unit root model with integrated generalized autoregressive conditional heteroskedasticity (IGARCH) errors, it just confirms the simulation result of Kourogenis and Pittis (2008) that the DF statistics with IGARCH errors converges in law to the standard DF distribution.     

The Voidage Function for the Drag Force Ratio in a Liquid‐Fluidized Bed

Di Felice (1994) has shown that the ratio of the drag coefficient, C_D, on a sphere in a liquid‐fluidized bed of uniform spheres to the drag coefficient, C_DS, on the same sphere in isolation and subjected to the same superficial liquid velocity, u, is given by a function ?^?β, where β was expressed as an empirical function of the particle Reynolds number, Re = duρ/µ. Here it is shown that C_D/C_DS is well approximated by ?^?mm, where the Richardson‐Zaki index n is a function of the terminal free‐settling Reynolds number, Re_t = du_tρ/µ, and m is 2 plus the slope of the standard log C_DS vs. log Re plot at plot at Re = Re_t. The present model, using the best experimentally confirmed equation for n and a new simple equation for and a new simple equation for m, is compared with that of Di Felice in their respective abilities to predict liquid‐fluidized bed expansion.     

On the spectral density of a class of chaotic time series

The purpose of this paper is to show explicitly the spectral density function of the stationary stochastic process determined by a certain class of two-dimensional maps F_α defined below (α is a parameter in (0, 1)), the random variable φ(x, y) = x and the invariant probability described below. We first define the transformation T_α: [0, 1]←[0, 1] given by T _α(x) = {x/α if 0 ≤x < α and (α(x?α)/1 ?α) if α≤x≤ 1 where α∈ (0, 1) is a constant. The map T_α describes a model for a particle (or the probability of a certain kind of element in a given population) that moves around, in discrete time, in the interval [0, 1]. The results presented here can be stated either for T_α or for F_α but we prefer the latter. The results for T_α can be obtained from the more general setting described by F_α. The map F_α is defined from K = ([0, 1]× (0, α)) ∨ ([0, α]×[α, 1]) ?;R;² to itself and is given by F_α(x, y) = (T_α(x), G_α(x, y)) for (x, y) ∈K, where G _α(x, y) = {αy if 0 ≤x < α and α + ((1 ?α)/α)y if α≤x < 1. The spectral density function of the stationary process with probability ν (invariant for F_α and absolutely continuous with respect to the Lebesgue measure) Z_t = X_t + ξ_t = φ{F^t_α(X₀, Y₀)} + ξ_t for t∈Z where (X₀, Y₀) ∈R² and ξ_t}_t∈Z is a white noise process, is given explicitly (Theorem 1) by f _Z (λ) = f_X(λ) + (σ²_ξ/2π) = (1/2πvar(X_t))[γ{exp(iλ)}?C(0)] + (σ²/2π) for all λ∈[0, 2π), where var(X_t) = (α²?α + 1)(α²? 5α + 5){12(2 ?α)²}^?1, γ is given by Equation (2.10) of Proposition 5 and C(0) = (1 + α²?α³){3(2 ?α)}^?1. We also estimate the parameter α based on a time series.     

Median‐unbiased Estimation and Exact Inference Methods for First‐order Autoregressive Models with Conditional Heteroscedasticity of Unknown Form

Abstract. Consider the first‐order autoregressive model y_t = φy_t?1 + ?_t, t = 1,…, T, with arbitrary initial non‐zero value y₀. Assuming that the error terms ?_t are independently distributed according to median‐zero distributions [ Zieliński (1999) Journal of Time Series Analysis, Vol. 20, p. 477] shows that the estimator conjectured by Hurwicz (1950) Statistical Inference in Dynamic Economic Models. New York, NY: Wiley – the median of the consecutive ratios y_t/y_t?1– is an exactly median‐unbiased estimator of the autoregressive parameter φ. This paper shows that the Hurwicz estimator remains median‐unbiased under more general distributional assumptions, without assuming statistical independence. In particular, no restrictions are placed on the degree of heterogeneity and dependence of the conditional variance process. A computationally efficient method is also proposed to build exact confidence intervals for the autoregressive parameter which are valid in finite samples for any value of φ on the real line.     

Predictions of nucleation theory applied to ehrenfest thermodynamic transitions: II. The effects of pressure and stress

This paper is a sequel to an earlier one on the applicability of classical nucleation theory to second-order transitions in the Ehrenfest sense (1). In each case the approach was to obtain the critical size r_c and energy barrier ΔG_c for the growth of a nucleus of β-phase in an α-phase matrix by a Maclaurin series expansion of the free-energy-density g = (G_β ? G_α)/v_β as a function of θ (in BC-I) and of ΔP and Δσ in this paper where θ = (T ? T_t) is the degree of undercooling and ΔP and Δσ are analogous terms for the hydrostatic pressure shift and tensile stress shift away from the equilibrium transition. The expansion coefficients were determined by the use of thermodynamic relationships. For second-order transitions, r_c = 4γv_β T_t/ΔC_pθ², r_c = 4γ/Δβ(Δp)², and r_c = 4γ/Y_αY_β(Δσ)², respectively, for the three cases. The terms ΔC_p, Δβ, and ΔY denote the differences in heat capacity, compressibility, and Young's modulus, e.g., ΔY = Y_β ? Y_α. The interfacial energy γ_αβ is denoted by γ. The activation energy barriers for the cases developed in this paper were ΔG_c = (16π/3)γ³/(Δβ)² (Δp)⁴ and ΔG_c = (64π/3)γ³Y_α²Y_β²/(ΔY)²(Δσ)⁴. More complicated expressions are given in the paper for the r_c and ΔG_c for first-order transitions. In the long run, these expressions may prove more useful than the ones for second-order because of the modifications expressions for the kinetics of transformations.     

Impact of the Sampling Rate on the Estimation of the Parameters of Fractional Brownian Motion

Abstract. Fractional Brownian motion is a mean‐zero self‐similar Gaussian process with stationary increments. Its covariance depends on two parameters, the self‐similar parameter H and the variance C. Suppose that one wants to estimate optimally these parameters by using n equally spaced observations. How should these observations be distributed? We show that the spacing of the observations does not affect the estimation of H (this is due to the self‐similarity of the process), but the spacing does affect the estimation of the variance C. For example, if the observations are equally spaced on [0, n] (unit‐spacing), the rate of convergence of the maximum likelihood estimator (MLE) of the variance C is . However, if the observations are equally spaced on [0, 1] (1/n‐spacing), or on [0, n²] (n‐spacing), the rate is slower, . We also determine the optimal choice of the spacing Δ when it is constant, independent of the sample size n. While the rate of convergence of the MLE of C is in this case, irrespective of the value of Δ, the value of the optimal spacing depends on H. It is 1 (unit‐spacing) if H = 1/2 but is very large if H is close to 1.     

Sampling,Embedding and Inference for CARMA Processes

A CARMA(p,q) process Y is a strictly stationary solution Y of the pth‐order formal stochastic differential equation a(D)Y_t = b(D)DL_t, where L is a two‐sided Lévy process, a(z) and b(z) are polynomials of degrees p and q respectively, with p > q, and D denotes differentiation with respect to t. Since estimation of the coefficients of a(z) and b(z) is frequently based on observations of the Δ‐sampled sequence , for some Δ > 0, it is crucial to understand the relation between Y and Y^Δ. If then Y^Δ is an ARMA sequence with coefficients depending on those of Y and the crucial problems for estimation are the determination of the coefficients of Y^Δ from those of Y (the sampling problem) and the determination of the coefficients of Y from those of Y^Δ (the embedding problem). In this article we consider both questions and use the results to determine the asymptotic distribution, as n→∞, with Δ fixed, of , where is the quasi‐maximum‐likelihood estimator of the vector of coefficients of a(z) and b(z), based on n consecutive observations of Y^Δ.     

Scaling of Yield Stress of Polythiophene Suspensions under Electric Field

Summary: Electrorheological properties in steady shear of perchloric acid doped poly(3‐thiopheneacetic acid), PTAA, particles in silicone oil were investigated to determine the effects of field strength, particle concentration, doping degree (conductivity values), operating temperature and nonionic surfactant. The PTAA/silicone oil suspensions show the typical ER response of Bingham flow behavior upon the application of electric field. The yield stress increases with electric field strength, E, and particle volume fraction, ?, according to a scaling law of the form, τ_y ∝ E^α · ?^γ. The scaling exponent α approaches the value of 2, predicted by the polarization model, as the particle volume fraction decreases and when the doping level of the particles decreases. The scaling exponent γ tends to unity, as predicted by the polarization model, when the electric field strength is low. The yield stress under electric field initially increases with temperature up to 25 °C, and then levels off. At electric fields above of 1.5 kV/mm, the yield stress increases significantly by up to 50% on addition of small amounts of a nonionic surfactant.

Effect of switching the applied electric field on the viscosity of a 20 wt.‐% highly HClO₄ doped polythiophene suspensions during stress sweep test.     

Sillimanite‐mullite transformation observed in synchrotron X‐ray diffraction experiments

High‐resolution synchrotron powder X‐ray diffraction (XRD) experiments were conducted to clarify the transformation of sillimanite to mullite (mullitization) and determine the mullitization temperature (T_c). We were able to distinguish sillimanite and mullite in the XRD patterns, despite their very similar crystallographic parameters, and to detect the appearance of small mullite peaks among sillimanite peaks. Analysis of the Johnson‐Mehl‐Avrami (JMA) equation for mullitization ratio (ζ) revealed that at temperatures T≥1240°C the mullitization had the same kinetics. The activation energy E at T≥1240°C obtained from the Arrhenius plot was 679.8 kJ mol^?1. In analysis using a time‐temperature‐transformation diagram for mullitization, a mullitization curve of ζ=1% can be described as where t is time, n is a reaction‐mechanism‐dependent parameter determined as 0.324 by JMA‐analysis, k₀ is the frequency factor, E_A is the activation energy for atomic diffusion, and represents the activation energy for nucleation. The results of fitting the data to this equation were T_c=1199°C, A=3.9×10⁶ kJ mol^?1 K^?2, E_A=605 kJ mol^?1, and k₀=3.65×10¹⁵. We conclude that the boundary between sillimanite and mullite+SiO₂ in the phase diagram is ~1200°C.     

A NOTE ON THE EMBEDDING OF DISCRETE-TIME ARMA PROCESSES

Abstract. Let {X_n, n= 0, 1, 2,…} be a discrete-time ARMA(p, q) process with q < p whose autoregressive polynomial has r (not necessarily distinct) negative real roots. According to a recent result of He and Wang (On embedding a discrete-parameter ARMA model in a continuous-parameter ARMA model. J. Time Ser. Anal. 10 (1989), 315–23) there exists a continuous-time ARMA (p', q') process {Y(t), t≥0} with q' < p'=p+r such that {Y(n), n= 0, 1, 2,…} has the same autocorrelation function as {X_n}. In this paper we show that this result is false by considering the case when {X_n} is a discrete-time AR(2) process whose autoregressive polynomial has distinct complex conjugate roots. We identify the proper subset of such processes which are embeddable in a continuous-time ARMA(2, 1) process. We show that every discrete-time AR(2) process with distinct complex conjugate roots can be embedded in either a continuous-tie ARMA(2, 1) process or a continuous-time ARMA(4, 2) process, or in some cases both. We derive an expression for the spectral density of the process obtained by sampling a general continuous-time ARMA(p, q) process (with distinct autoregressive roots) at arbitrary equally spaced time points. The expression clearly shows that the sampled process is a discrete-time ARMA (p', q') process with q' < p.     

Synthesis and biological activity of phthalimide‐based polymers containing 5‐fluorouracil

The attachment of anticancer agents to polymers is a promising approach towards reducing the toxic side‐effects and retaining the potent antitumour activity of these agents. A new tetrahydrophthalimido monomer containing 5‐fluorouracil (ETPFU) and its homopolymer and copolymers with acrylic acid (AA) and with vinyl acetate (VAc) have been synthesized and spectroscopically characterized. The ETPFU contents in poly(ETPFU‐co‐AA) and poly(ETPFU‐co‐VAc) obtained by elemental analysis were 21 mol% and 20 mol%, respectively. The average molecular weights of the polymers determined by gel permeation chromatography were as follows: M_n = 8900 g mol^?1, M_w = 13 300 g mol^?1, M_w/M_n = 1.5 for poly(ETPFU); M_n = 13 500 g mol^?1, M_w = 16 600 g mol^?1, M_w/M_n = 1.2 for poly(ETPFU‐co‐AA); M_n = 8300 g mol^?1, M_w = 11 600 g mol^?1, M_w/M_n = 1.4 poly(ETPFU‐co‐VAc). The in vitro cytotoxicity of the compounds against FM3A and U937 cancer cell lines increased in the following order: ETPFU > 5‐FU > poly(ETPFU) > poly(ETPFU‐co‐AA) > poly(ETPFU‐co‐VAc). The in vivo antitumour activities of all the polymers in Balb/C mice bearing the sarcoma 180 tumour cell line were greater than those of 5‐FU and monomer at the highest dose (800 mg kg^?1). © 2002 Society of Chemical Industry     

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.     

Space‐charge limited conduction in polyaniline films

Polyaniline emeraldine base (PANI‐EB) powder was synthesized by oxidative polymerization of aniline. The PANI‐EB films were prepared by the solution‐casting technique. The temperature‐dependent dc conductivity measured in the range 173–303 K suggests that the PANI‐EB is a quasi‐one‐dimensional disordered conductor. The current‐voltage characteristics of the PANI‐EB films measured in the range 333–383 K showed the SCLC mechanism. The SCLC parameters such as free carrier density (p₀), trap density (p_t), the ratio between free carrier density to the total carrier density (θ), mobility (µ) and the effective hole mobility (µ_eff) were calculated. The activation energy (E_a = 0.32 eV) and the Fermi level (E_F = 0.42 eV) were estimated. As well as these, the trap parameters such as the trap filled limit voltage (V_TFL), the shallow trap density (N_t), the depth of the dominant trap level (E_t ? E_v), the density of states within the hole mobility edge (N_v) and the characteristic energy (E_c) were also calculated and presented. The exponential type of traps distribution with large number of traps was found to be due to the disorder and moisture in the polymer films. Copyright © 2004 Society of Chemical Industry     

Synthesis and biological activity of medium range molecular weight polymers containing exo‐3,6‐epoxy‐1,2,3,6‐tetrahydrophthalimidocaproic acid

A new monomer, exo‐3,6‐epoxy‐1,2,3,6‐tetrahydrophthalimidocaproic acid (ETCA), was prepared by reaction of maleimidocaproic acid and furan. The homopolymer of ETCA and its copolymers with acrylic acid (AA) or with vinyl acetate (VAc) were obtained by photopolymerizations using 2,2‐dimethoxy‐2‐phenylacetophenone as an initiator at 25 °C. The synthesized ETCA and its polymers were identified by FTIR, ¹H NMR and ¹³C NMR spectroscopies. The apparent average molecular weights and polydispersity indices determined by gel permeation chromatography (GPC) were as follows: M_n = 9600 g mol^?1, M_w = 9800 g mol^?1, M_w/M_n = 1.1 for poly(ETCA); M_n = 14 300 g mol^?1, M_w = 16 200 g mol^?1, M_w/M_n = 1.2 for poly(ETCA‐co‐AA); M_n = 17 900 g mol^?1, M_w = 18 300 g mol^?1, M_w/M_n = 1.1 for poly(ETCA‐co‐VAc). The in vitro cytotoxicity of the synthesized compounds against mouse mammary carcinoma and human histiocytic lymphoma cancer cell lines decreased in the following order: 5‐fluorouracil (5‐FU) ≥ ETCA > polymers. The in vivo antitumour activity of the polymers against Balb/C mice bearing sarcoma 180 tumour cells was greater than that of 5‐FU at all doses tested. © 2001 Society of Chemical Industry     

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.     

Synthesis and antitumour activity of medium molecular weight phthalimide polymers of camptothecin

Attachment of anticancer agents to polymers has been demonstrated to improve their therapeutic profiles. A new monomer containing camptothecin, 5‐norbonene‐endo‐2,3‐dicarboxylimidoundecanoyl‐camptothecin (NDUCPT) and its homopolymer and copolymer with acrylic acid (AA) were synthesized and spectroscopically characterized. The NDUCPT content in poly(NDUCPT‐co‐AA) obtained by elemental analysis was 51%. The average molecular weights of the polymers determined by gel permeation chromatography were as follows: M_n = 12 100, M_w = 23 400 g mol^?1, M_w/M_n = 1.93 for poly(NDUCPT), M_n = 15 400, M_w = 28 300 g mol^?1, M_w/M_n = 1.83 for poly(NDUCPT‐co‐AA). The IC₅₀ value of NDUCPT and its polymers against U937 cancer cells was larger than that of CPT. The in vivo antitumour activity of all polymers in Balb/C mice bearing the sarcoma 180 tumour cell line was greater than that of CPT at a dose of 100 mg kg^?1. Copyright © 2003 Society of Chemical Industry