ESTIMATION OF THE FRACTIONAL DIFFERENCE PARAMETER IN THE ARIMA(p, d, q) MODEL USING THE SMOOTHED PERIODOGRAM

Abstract. In recent work on time series analysis considerable interest has been focused on series having the property of long memory. Long memory is a characteristic of time series in which the dependence between distant observations is not negligible. The model that has been most frequently studied, which in some situations shows properties of long memory, is based on the autoregressive integrated moving-average ARIMA( p, d, q ) process. Hosking (Fractional differencing, Biometrika 68 (1) (1981), 165–76) generalized this model by permitting the degree of differencing d to take fractional values. He then demonstrated that for d in the range 0 < d < 0.5 the process is stationary and possesses the long memory property. Our study is based on the ARIMA( p, d, q ) model when d takes any real non-integer value in the interval (-0.5, 0.5). The main aim of our study is to examine methods for estimating the parameters of this model. For estimating d we suggest an estimator based on the smoothed periodogram. Using an empirical approach we compare this estimator with other which are well known in the literature of long memory models, e.g. the raw periodogram regression method and the Hurst coefficient method.     

隔热条对铝合金隔热窗抗风压性能的影响

通过ANSYS软件对铝合金隔热窗进行结构分析,研究了几种不同材料、型号、结构的隔热条在受到不同风压时铝合金隔热窗的应力场分布。研究结果表明,隔热条的材料、型号和结构对于整窗的应力场影响显著。     

AUTOMATIC SEMIPARAMETRIC ESTIMATION OF THE MEMORY PARAMETER OF A LONG-MEMORY TIME SERIES

Abstract. In Geweke and Porter-Hudak's estimator ω d ^(GPH) of the memory parameter of a long-memory process, a critical choice the user must make is the number of frequencies, M , to be used in the regression of the log periodogram on log frequency. This choice is critical in practice because by simply varying M for a given data set it is often possible to obtain a very wide range of values of the estimator. Although Geweke and Porter-Hudak have found that choosing M to be the square root of the sample size gave good results in simulation, they gave no theoretical justification for this choice. Here, we propose automatic criteria for selecting M , and another tuning constant used in related estimates of the memory parameter, based on frequency domain cross-validation. We provide some theoretical and heuristic justification for the proposed criteria. In a simulation study, we compare some of the resulting automatic methods of estimating the memory parameter with existing non-automatic ones.     

ON THE PROBABILITY OF ESTIMATING A DETERMINISTIC COMPONENT IN THE LOCAL LEVEL MODEL

Abstract. A local level model has a deterministic level when the signal-to-noise ratio q is zero. In this paper we investigate the properties of the maximum likelihood estimator of q , paying particular attention to the case where its true value is zero. These properties are shown to be crucially dependent on the initial conditions employed.     

IMPROVED NON-FOURIER FREQUENCY ESTIMATION OF SAMPLED STATIONARY TIME SERIES

A general multi-component cyclic descent least-squares model used in conjunction with a power-of-two fast Fourier transform to perform the harmonic analysis presents numerous advantages over their individual use. Through the appropriate choice of a window taper and a linear filter applied to the time series data, severe leakage is effectively reduced by decreasing the interference between peaks throughout the band. The taper and filter characteristics are dependent on the given application. The resulting assembled frequency components are updated iteratively by means of the least-squares model

A brief portion of this study is devoted to the development of the Fourier transform as a general spectral estimator. Particular attention is given to its limitations. Methods for improving frequency identification in the transform are covered. A brief development of the cyclic descent least-squares model is given as a means of reducing errors inherent in a Fourier analysis. A numerical example is presented depicting the method.     

Plug-in Selection of the Number of Frequencies in Regression Estimates of the Memory Parameter of a Long-memory Time Series

We consider the problem of selecting the number of frequencies, m , in a log-periodogram regression estimator of the memory parameter d of a Gaussian long-memory time series. It is known that under certain conditions the optimal m , minimizing the mean squared error of the corresponding estimator of d , is given by m ^(opt)= Cn ^4/5, where n is the sample size and C is a constant. In practice, C would be unknown since it depends on the properties of the spectral density near zero frequency. In this paper, we propose an estimator of C based again on a log-periodogram regression and derive its consistency. We also derive an asymptotically valid confidence interval for d when the number of frequencies used in the regression is deterministic and proportional to n ^4/5. In this case, squared bias cannot be neglected since it is of the same order as the variance. In a Monte Carlo study, we examine the performance of the plug-in estimator of d , in which m is obtained by using the estimator of C in the formula for m ^(opt) above. We also study the performance of a bias-corrected version of the plug-in estimator of d . Comparisons with the choice m = n ^1/2 frequencies, as originally suggested by Geweke and Porter-Hudak (The estimation and application of long memory time series models. Journal of Time Ser. Anal. 4 (1983), 221–37), are provided.     

Goodness-of-fit Test in Parametric Time Series Models

A goodness-of-fit test is proposed which uses nonparametric curve estimation methods to investigate the fit of parametric models for the spectral density. A test of the null hypothesis that the function has parametric form is considered with a test statistic which compares parametric estimates and nonparametric kernel estimates of the function and its derivatives at a preselected number of points. An important issue for the nonparametric estimator is bandwidth choice, and we propose a data-adaptive method for local bandwidth choice. Under the null hypothesis, asymptotically the test statistic has a χ² distribution. Some practical issues are discussed.     

EMPIRICAL IDENTIFICATION OF MULTIPLE TIME SERIES

Abstract. In the univariate case the problem of empirical identification consists in determining the order parameters p , d and q of ARIMA ( p, d, q ) processes. In this paper we introduce some new techniques for handling the corresponding problem for a multiple time series X ( t ) with the main emphasis on AR and MA models. Types of joint nonstationarity (or rather almost nonstationarity) are defined and a method of analyzing such structures based on the ordered eigenvalues of the function C ( t ) = K ( t ) K ^-1(0) is discussed, where K ( t ) is the covariance function of X ( t ). It is proposed that the further identification procedure should be based on two X ² statistics and on the estimated trace and eigenvalues of C ( t ), the matrix correlation function p ( t ) and the matrix partial correlation function P ( t ). The suitability for identification purposes of each of these functions is examined in terms of such properties as scale-invariance, existence of normalized eigenvalues and standard errors. The methods introduced are illustrated on a 5-dimensional economic time series first studied by Quenouille and on a 4-dimensional smulated MA series.     

Nonlinear spectral density estimation: thresholding the correlogram

Traditional kernel spectral density estimators are linear as a function of the sample autocovariance sequence. The purpose of this article is to propose and analyse two new spectral estimation methods that are based on the sample autocovariances in a nonlinear way. The rate of convergence of the new estimators is quantified, and practical issues such as bandwidth and/or threshold choice are addressed. The new estimators are also compared with traditional ones using flat‐top lag‐windows in a simulation experiment involving sparse time‐series models.     

国内外聚氯乙烯塑料门窗的现状及展望

介绍了未增塑聚氯乙烯塑料门窗的导热系数低、防水、防潮等特点,利用PVC型材作门窗是节约建筑材料的一条重要途径.重点阐述了国内外聚氯乙烯塑料门窗的生产、使用情况及国内产品存在的缺陷,另外还论述了国内塑料门窗的发展趋势,对PVC塑料门窗在国内的发展应用有促进作用.     

ORDER IDENTIFICATION STATISTICS IN STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS:VECTOR AUTOCORRELATIONS AND THE BOOTSTRAP

Abstract. In this paper we consider the vector autocorrelation approach for identifying ARMA ( p, q ) models and use a bootstrap procedure in order to evaluate the distribution of the corresponding sample statistics by means of a resampling scheme for the residuals when p and q are unknown. The asymptotic validity of the bootstrap procedure applied to the vector autocorrelation estimates is established. Some simulations and examples demonstrating the appropriateness of the proposed bootstrap procedure in comparison with large-sample Gaussian approximations are included.     

Transparent Armor Ceramics as Spacecraft Windows

The slow crack growth parameters of several transparent armor ceramics were measured as part of a program to lighten next generation spacecraft windows. Transparent magnesium aluminate (spinel, MgAl₂O₄) and AlON exhibit superior slow crack resistance relative to fused silica, which is the historical material of choice. For spinel, slow crack growth, strength, and fracture toughness are significantly influenced by the grain size, and alumina‐rich phases and porosity at the grain boundaries lead to intergranular fracture in coarse grain spinel. Functions describing the required mass for a desired window life imply that transparent ceramics can lighten window panes from a slow crack growth perspective.     

Breaking of window glass close to fire,II: Circular panes

A heat-conduction equation with linearized radiation cooling boundary conditions is used to calculate the thermal field in a circular window pane heated by thermal radiation (fire) but screened on narrow strips along edges built into the frame. This temperature field is used to calculate a quasi-static thermal stress field in the pane in the first-order planar thermal stresses build up at the edges in a narrow strip of a few times the pane thickness. As the radius of the pane grows the edge stresses quickly approach those of the linear strip pane. The round window does not differ much from the strip window in the geometrical region of practical utility in windows. The results also indicate, without explicit calculations, that in rectangular windows the stresses do not concentrate in the corner regions. As the final result of the theoretical work it seems plausible to use simple, general relationships obtained for the strip window as well for all geometries deviating from linear to evaluate approximately the danger of thermal breakage of a pane.     

Effect of casting solvent on the heterogeneity of films of polymer blends

The miscibility window of the 50/50 w/w poly(styrene-co-4-vinylphenyl)-dimethylsilanol/poly(n-butyl methacrylate) (ST-VPDMS/PBMA) blends prepared from toluene was studied and determined to be in the VPDMS composition range of 4 to 18 mole% in the copolymers. The observed miscibility window was to be compared to the range of 9 to 34 mole% found for the blends prepared from methyl ethyl ketone which was capable of competing for hydrogen bonding. The fact that the observed miscibility windows are influenced by the choice of solvents illustrates that an equilibrium state of polymer mixing is not always attained in solvent casting films and that caution needs to be exercised in studying polymer miscibility when solvent cast films are used.     

塑料门窗抗风压强度计算机辅助设计

介绍了一种新近开发出的用于塑料门窗强度校核的计算机辅助设计软件，该软件是参照最新的国家标准和行业规范，针对塑料门窗的特点开发的，可实现“风荷载计算”，“窗框强度校核”，“玻璃强度校核”和“螺钉强度校核”等功能。     

SOME ASYMPTOTIC PROPERTIES OF THE SAMPLE COVARIANCES OF GAUSSIAN AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. The paper deals with the asymptotic variances of the sample covariances of autoregressive moving average processes. Using state-space representations and some matrix Lyapunov equation theory, closed-form expressions are derived for the asymptotic variances of the sample covariances and for the Cramer-Rao bounds on the process covariances. The main results obtained from these expressions are as follows: For ARMA ( p, q ) processes with p ≥ q , the sample covariance of order n is asymptotically efficient if and only if 0 ≤ n ≤ p – q .
For ARMA ( p, q ) processes with p < q , none of the sample covariances is asymptotically efficient.     

Change-Point Estimation of Fractionally Integrated Processes

In this paper we analyze the least-squares estimator of the change point for fractionally integrated processes with fractionally differencing parameter −0.5 < d < 0.5. When there is a one-time change, we show that the least-squares estimator is consistent and that the rate of convergence depends on d . When there is no change, we find that the least-squares estimator converges in probability to the set {0, 1} for −0.5 < d ≤ 0 but is likely to suggest a spurious change for 0 < d < 0.5. Simulations are also used to illustrate the asymptotic analysis.     

A Test of Linearity for Functional Autoregressive Models

We propose a new test for linearity in time series. We consider an asymptotically stationary functional AR( p ) model on ℜ ^d of the form
X _n = f ( X _{n −1}, ..., X _{n − p} ) + ξ _n ( n ∈ N).
The testing procedure is based on a suitably normalized sum of quadratic deviations between two different estimates of the function f evaluated at q distinct points of ℜ ^dp . The estimators are f^ _n , a recursive version of the non-parametric kernel estimator of f , and  _n , a least squares estimator well suited to the linear case. The main result states that the test statistic has a χ² limit distribution under the null hypothesis. A similar result is derived under the alternative hypothesis for the test statistic corrupted by a non-linear term. Our simulations indicate that our asymptotic results hold for moderate sample sizes when the testing procedure is used carefully     

STOCHASTIC ORDERS OF MAGNITUDE ASSOCIATED WITH TWO-STAGE ESTIMATORS OF FRACTIONAL ARIMA SYSTEMS

Abstract. Two-stage estimators have been proposed in fractional autoregressive integrated moving-average (ARIMA) systems which first estimate the long-run features of the system semi-parametrically and then estimate the short-run features by usual methods in a second stage. Although asymptotic theory is available for the estimates in the first stage of such a procedure, we are aware of no results concerning the estimates in the second stage. In this paper we provide a stochastic order of magnitude associated with an estimator in this class and discuss this result.     

ESTIMATION OF COEFFICIENTS OF TIME SERIES REGRESSION WITH A NONSTATIONARY ERROR PROCESS

Abstract. We treat a problem of estimating unknown coefficients of a time series regression when the variance of the error changes with time, i.e. when a process which the error term obeys is nonstationary. First, we show the weak consistency of the ordinary least squares estimator for the coefficients of a polynomial regression under some assumptions on the covariance structure of the error process. Next, we propose a nonparametric method for estimating the variance of the error process and a weighted least squares estimator of the regression coefficients, which is constructed by using the estimator of the variance. We investigate statistical properties of our proposed estimator in the following way. We consider the prediction of a future value of a linear trend by using our proposed estimator and evaluate its prediction error. By simulation studies, we compare the prediction error of the predictor constructed by using our proposed estimator with the prediction errors obtained for other estimators including the ordinary least squares estimator when the variance of the error process increases with time and the sample sizes are small. As a result, our proposed estimator seems to be reasonable.