STATIONARITY AND CENTRAL LIMIT THEOREM ASSOCIATED WITH BILINEAR TIME SERIES MODELS

Abstract. Consider the general bilinear times series model where {X_t; t= 0, L1, …} is a p-variate process, C (p x (s+ 1)), A (p x p). B _t(p x p) (1 ≤j≤q) are arbitrary matrices of constants, εT=[εt,…εt-q+1] and {εt; t=0, ±1, …} is a strictly stationary ergodic sequence of random variables. We investigate a set of minimal regularity conditions (on C, A, B _j and {ε_t}) under which we can establish the existence and causality of X _t and the asymptotic normality of the sample mean derived from { X _t}.     

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.     

On robust tail index estimation for linear long‐memory processes

We consider robust estimation of the tail index α for linear long‐memory processes with i.i.d. innovations ε_j following a symmetric α‐stable law (1 < α < 2) and coefficients a_j ～ c·j^?β. Estimates based on the left and right tail respectively are obtained together with a combined statistic with improved efficiency, and a test statistic comparing both tails. Asymptotic results are derived. Simulations illustrate the finite sample performance.     

Optimal convergence rates in non‐parametric regression with fractional time series errors

Consider the estimation of g^(ν), the νth derivative of the mean function, in a fixed‐design non‐parametric regression model with stationary time series errors ξ_i. We assume that , ξ_i are obtained by applying an invertible linear filter to iid innovations, and the spectral density of ξ_i has the form as λ → 0 with constants c_f > 0 and α ∈ (?1,1). Under regularity conditions, the optimal convergence rate of is shown to be with r = (1 ? α)(k ? ν)/(2k+1 ? α). This rate is achieved by local polynomial fitting. Moreover, in spite of including long memory and antipersistence, the required conditions on the innovation distribution turn out to be the same as in non‐parametric regression with iid errors.     

High‐frequency sampling of a continuous‐time ARMA process

Continuous‐time autoregressive moving average (CARMA) processes have recently been used widely in the modelling of non‐uniformly spaced data and as a tool for dealing with high‐frequency data of the form ,n = 0, 1, 2,…, where Δ is small and positive. Such data occur in many fields of application, particularly in finance and in the study of turbulence. This article is concerned with the characteristics of the process , when Δ is small and the underlying continuous‐time process is a specified CARMA process.     

Identification of Persistent Cycles in Non‐Gaussian Long‐Memory Time Series

Abstract. Asymptotic distribution is derived for the least squares estimates (LSE) in the unstable AR(p) process driven by a non‐Gaussian long‐memory disturbance. The characteristic polynomial of the autoregressive process is assumed to have pairs of complex roots on the unit circle. In order to describe the limiting distribution of the LSE, two limit theorems involving long‐memory processes are established in this article. The first theorem gives the limiting distribution of the weighted sum, is a non‐Gaussian long‐memory moving‐average process and (c_n,k,1 ≤ k ≤ n) is a given sequence of weights; the second theorem is a functional central limit theorem for the sine and cosine Fourier transforms      

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.     

Predicting elastic moduli of heterogeneous polymer compositions

A new method for predicting elastic moduli M of heterogeneous polymer compositions is proposed. It is based on a phenomenological adjustment between parallel and series models for upper and lower bound moduli M_U and M_L. Thus, where ?_H is the volume fraction of hard phase, ?_S is the volume fraction of soft phase, and n is the only adjustable parameter since the upper and lower bound moduli are given by and where M_H and M_S are the moduli of the pure hard and soft phases, respectively. Predicted values of M are in agreement with measured values in a number of systems which include polyblends and composite materials of fixed morphology. The significance of n is discussed relative to concentrations in the area of a phase transition for the polyblends or relative to phase morphology in the case of fixed morphology compositions. Interestingly, the relationship, by analogy, is in agreement with measured values of polyblend melt viscosities.     

ON M‐Estimation Under Long‐Range Dependence in Volatility

Abstract. We consider M‐estimation of a location parameter for processes with zero autocorrelations but long‐range dependence in volatility. The observed process is the product of i.i.d. Gaussian observations and a long‐memory Gaussian process. For nonlinear estimators, the rate of convergence depends on the type of the ψ‐function. For skew‐symmetric ψ‐functions, a central limit theorem with ‐rate of convergence holds, under suitable regularity assumptions. This is not true in general for M‐estimators where the ψ‐function is not skewsymmetric.     

Testing for parameter constancy in general causal time‐series models

We consider a process belonging to a large class of causal models including AR(∞), ARCH(∞), TARCH(∞),… processes. We assume that the model depends on a parameter and consider the problem of testing for change in the parameter. Two statistics and are constructed using quasi‐likelihood estimator of the parameter. Under the null hypothesis that there is no change, it is shown that each of these two statistics weakly converges to the supremum of the sum of the squares of independent Brownian bridges. Under the alternative of a change in the parameter, we show that the test statistic diverges to infinity. Some simulation results for AR(1), ARCH(1), GARCH(1,1) and TARCH(1) models are reported to show the applicability and the performance of our procedure with comparisons to some other approaches.     

A p‐Order signed integer‐valued autoregressive (SINAR(p)) model

In this article, we propose an extension of integer‐valued autoregressive INAR models. Using a signed version of the thinning operator, we define a larger class of ‐valued processes, called SINAR, which can have positive as well as negative correlations. Using a Markov chain method, conditions for stationarity and the existence of moments are investigated. In particular, it is shown that the autocorrelation function of any real‐valued AR process can be recovered with a SINAR process, which improves INAR modeling.     

Sintering of ZnO: I,Densification and Grain Growth*

Densification and grain growth rates at the intermediate stage of the sintering of pressed zinc oxide pellets were measured in air and oxygen. Application of a lattice diffusion model to the results gave calculated diffusion coefficients in good agreement with directly measured diffusion coefficients for zinc. The change of atmosphere from air to oxygen resulted in lowering of grain size and calculated diffusivity; the diffusion decrease is consistent with lattice diffusivity of zinc occurring by an interstitial mechanism. The results can be expressed by the relations: in air, and in oxygen, from 900° to 100°C.     

A Shrinked Forecast in Stationary Processes Favouring Percentage Error

Abstract. In stationary time‐series forecasting, the commonly used criterion for selecting a proper forecast is the mean square error (MSE), which is minimized by the conditional expectation of future observation given the entire past known as a minimum MSE forecast. In this paper, mean square percentage error (MSPE) instead of is used to forecast autoregressive moving average (ARMA)(p,q) series. The suggested forecast takes the form of or (CV_t+1 is the coefficient of variation for one step ahead) times the minimum MSE forecast, which performs better not only in MSPE, but also in mean absolute percentage error (MAPE) than the ordinary MSE forecast in simulation studies. A real data example also supported this result. We conclude that, if percentage error is a prime concern, this shrinked version of MSE forecast performs better than the ordinary forecast in the stationary ARMA(p,q) model.     

Thermal conductivity of simple gases at normal pressures

Thermal conductivity measurements available in the literature for simple gases at normal pressures (approximately 1 atmosphere) were used to obtain the product k^*λ, where the parameter, λ =M^1/2T_c^1/6/P_c^2/3. Separate relationships between k^*λ and T_R resulted for monatomic, diatomic and triatomic gases. The relationships for monatomic gases can be expressed as follows For the diatomic and triatomic gases, linear relationships resulted, when at the same reduced temperatures, their k^*λ values were plotted against (k^*λ)_m on log-log coordinates. These relationships can be expressed in equation form as follows and Thermal conductivities calculated with these relationships have been compared with experimental values and produce an average deviation of 2.8% for the monatomic gases (219 points), 4.3% for the diatomic gases (282 points) and 4.6% for the triatomic gases (242 points). In this treatment, helium and hydrogen do not follow the general pattern and consequently these substances have been treated separately.     

The kinetics and mechanism of the chloride-chlorate reaction

The kinetics of the reaction have been studied at 25°C. in strong acid solution; the effects of acidity, chloride, chlorate and chlorine are reported. A mechanism is postulated to interpret the peculiar features of this reaction as well as the stoichiometry and some of the kinetics of the parallel reaction The mechanism involves HClO₂ and HOCl as intermediates General rate expressions are derived for the formation of chlorine dioxide and chlorine, and the individual rate constants are calculated. An expression is obtained for the relationship between the ratio of chlorine dioxide to chlorine produced and the ratio of chlorate to chloride.     

The shape of a fluid drop approaching an interface

The shape of a fluid drop approaching an interface does not change appreciably with time and is very close to the equilibrium dimensions, in spite of the large pressure gradient which is present in the draining film. This is because the net vertical force due to the excess pressure in the draining film above that in the drop is identical with that for an equilibrium film being zero for a plane interface and —2Rσ sin² ? for a deformable interface. Employing this result in a force balance around the drop which is independent of the bulk interface shows that the area A of the draining film between a fluid drop of volume V and a deformable fluid-liquid interface is given by where σ is the interfacial tension and Δρ the density difference between the drop and surrounding fluid, 1/b is the curvature at the top of the drop and h is the distance between this point and the edge of the draining film which is inclined to the horizontal at an angle ?. When the interface is a rigid plane the overall curvature 1/R of the draining film and the volume v enclosed by it, together with the angle ? are all zero. The limiting cases of the expression for very small and very large drops agree with those previously established for both deformable and rigid interfaces. An approximate expression which applies when cV^2/3 (where c = Δρg/σ) is between 0.6 and 13.5 and which gives A to within ± e% is where for a rigid plane interface and for a deformable interface When the densities of the drop and bulk heavy fluids are equal, but their respective interfacial tensions σ₁₂ and σ₂₃ with the light fluids are different, the expression becomes which estimates A/V^2/3 to within about ± 25% for σ₁₂/σ₂₃ in the range 0.11 to 9.0 and cV^2/3 (where c = Δρg/σ₁₂) between 0.6 and 13.5.     

Discussion of treatments for the kinetics of radical polymerization with primary radical termination

Without knowledge of the familiar geometric mean, the following equation was derived for the radical polymerization with primary radical termination under the condition that reaction between primary radicals is negligible: where both A and B are constants depending on various rate constants. This equation was mathematically and experimentally dealt with.     

Thermal reactions between MII sulphates and acid orthophosphates in the solid state

Two general types of high temperature reactions between M^II sulphates and acid orthophosphates have been found and a survey of some representative examples has been made. The reactions are typified by two examples using Ca²⁺ salts: and . Similar reactions have been found for other alkaline earth sulphates and phosphates and for corresponding mixed alkaline earth systems. The products have been found to be largely determined by the phase relationships in the respective M^IIO–P₂O₅ systems.     

Sodium metabisulphite initiated aqueous polymerisation of methyl methacrylate

Sodium metabisulphite (Na₂S₂O₅) has been found to initiate the aqueous polymerisation of methyl methacrylate (MMA) at 35°C in phosphate buffer solutions of pH 6.85 and a constant ionic strength (μ) of the media in an inert atmosphere of pure nitrogen. The reaction has a well defined induction period which is a function of the concentrations of the initiator, of the monomer and also of temperature. The polymerisation is signalled by the sudden appearance of a haze at the end of the induction period in a given run, and the polymer separates out as a coarse precipitate during the progress of the polymerisation reactions. When the conversion is over 50 per cent complete, the polymerisation media looks like a thick curd if the monomer concentration is relatively high. The rate of polymerisation is found to decrease with conversion or time in a given run, and the initial rate (v_p), obtained by extrapolating the linear yield/time versus time curves to zero time, keeping the conversion below 10 percent, is found as where (I) = initiator concentration in the range, (0.26 to 3.94) × 10^?3 (mol dm^?3), and (M) = monomer concentration in the range 0.019 to 0.141 (mol dm^?3). At high initiator concentrations, the rate of polymerisation is found to decrease. In a given run, the viscosity average molecular weights (M _v) of the polymers is found to increase quickly with a conversion of up to 25 to 30 percent, and then slowly with the further increase in conversion. (M_v) however is found to decrease with the increase of the initiator concentrations at a given conversion but increases with the increase of the monomer concentrations. Hydroquinone inhibits the polymerisation reactions, whereas air is found to increase the induction period, but later enhances the polymerisation rate in the same run. It has been shown that the bisulphite addition reaction of MMA is not important under the experimental conditions, and the polymerisation occurs by the free radical mechanism. The rate constant (k₂) of the reaction, has been estimated from the analytical data as, k₂ = 14.62 × 10^?2 (dm³ mol^?1 s^?1) at 35°C.