Bayesian Unit Root Test in Nonnormal AR(1) Model

In this paper, we approximate the distribution of disturbances by the Edgeworth series distribution and propose a Bayesian analysis in a nonnormal AR(1) model. We derive the posterior distribution of the autocorrelation and the posterior odds ratio for unit roots hypothesis in the AR(1) model when the first four cumulants of the Edgeworth series distribution are finite and the higher order cumulants are negligible. We also apply the posterior analysis to eight real exchange rates and investigate whether these exchange rates behave like a random walk or not.     

TESTING FOR A UNIT ROOT IN AN AR(1) TIME SERIES USING IRREGULARLY OBSERVED DATA

Abstract. For an AR(1) model having a unit root with nonconsecutively observed or missing data we consider the ordinary least squares estimator, the one-step Newton-Raphson estimator and an ordinary least squares type estimator which is a simple approximation of the Newton-Raphson estimator. It is shown that the limiting distributions of these estimators of the unit root are the same as those of the regression estimators as tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Statist. Assoc. 74 (1979), 427–31) for the complete data situation. Simulation results show that our proposed unit root tests perform very well for small samples.     

Likelihood ratio tests for the structural change of an AR(p) model to a Threshold AR(p) model

This article considers the likelihood ratio (LR) test for the structural change of an AR model to a threshold AR model. Under the null hypothesis, it is shown that the LR test converges weakly to the maxima of a two‐parameter vector Gaussian process. Using the approach in Chan and Tong (1990)and Chan (1991), we obtain a parameter‐free limiting distribution when the errors are normal. This distribution is novel and its percentage points are tabulated via a Monte Carlo method. Simulation studies are carried out to assess the performance of the LR test in the finite sample and a real example is given.     

A Median-Unbiased Estimator of the AR(1) Coefficient

A proof is given that the median of the ratios of consecutive observations of a stationary first-order autoregressive process X_t = α X _{t −1} + Y_t with P ( Y_t ≥ 0) = P ( Y_t ≤ 0) = 1/2 and P ( X_t = 0) = 0 is a median-unbiased estimator of α.     

A general form of color difference formula based on color discrimination ellipsoid parameters

A general color difference formula has been derived based on the parameters of color discrimination ellipsoids in the CIE 1976 L*a*b* color space. By using different orders of approximation, the general formula resembles the basic forms of the current formulae. The method described in this article suggests a framework for modifying the CIE 1976 L*a*b* color difference formula.     

THE STABILITY OF THE AR(1) PROCESS WITH AN AR(1) COEFFICIENT

Abstract. In this paper we consider a simple time varying coefficient ARMA process:the AR (1) process with an AR (1) coefficient. A basic requirement of the process is that the output has finite variance, and we derive a condition on the parameters for this to be satisfied. The analysis is complicated by the interaction between the equations for the data and the varying coefficient.     

ASYMPTOTIC INFERENCES FOR AN AR(1) MODEL WITH A CHANGE POINT: STATIONARY AND NEARLY NON‐STATIONARY CASES

This article examines the asymptotic inference for AR(1) models with a possible structural break in the AR parameter β near the unity at an unknown time k₀. Consider the model y_t = β₁y_{t − 1}I{t ≤ k₀} + β₂y_{t − 1}I{t > k₀} + ϵ_t, t = 1,2, … ,T, where I{ ⋅ } denotes the indicator function. We examine two cases: case I | β₁ | < 1,β₂ = β_2T = 1 − c ∕ T; and case II β₁ = β_1T = 1 − c ∕ T, | β₂ | < 1, where c is a fixed constant, and {ϵ_t,t ≥ 1} is a sequence of i.i.d. random variables, which are in the domain of attraction of the normal law with zero means and possibly infinite variances. We derive the limiting distributions of the least squares estimators of β₁ and β₂ and that of the break‐point estimator for shrinking break for the aforementioned cases. Monte Carlo simulations are conducted to demonstrate the finite‐sample properties of the estimators. Our theoretical results are supported by Monte Carlo simulations.     

CONTEMPORANEOUS AGGREGATION OF TRIANGULAR ARRAY OF RANDOM‐COEFFICIENT AR(1) PROCESSES

We discuss contemporaneous aggregation of independent copies of a triangular array of random‐coefficient processes with i.i.d. innovations belonging to the domain of attraction of an infinitely divisible law W. The limiting aggregated process is shown to exist, under general assumptions on W and the mixing distribution, and is represented as a mixed infinitely divisible moving average in (4). Partial sums process of is discussed under the assumption EW² < ∞ and a mixing density regularly varying at the ‘unit root’ x = 1 with exponent β > 0. We show that the previous partial sums process may exhibit four different limit behaviors depending on β and the Lévy triplet of W. Finally, we study the disaggregation problem for in spirit of Leipus et al. (2006) and obtain the weak consistency of the corresponding estimator of ϕ(x) in a suitable L₂ space.     

A NOTE ON THE THRESHOLD AR(1) MODEL WITH CAUCHY INNOVATIONS

Abstract. A threshold autoregressive process of the first order with one threshold r and with Cauchy innovations is investigated in the paper. An explicit formula for the stationary density of such process is derived for the special case that r = 0 and that the autoregressive parameters have the same absolute value.     

Spatial batch optimal design based on self-learning Gaussian process models for LPCVD processes

Low pressure chemical vapor deposition (LPCVD) is one of themost important processes during semiconductor manufacturing. However, the spatial distribution of internal temperature and extremely few samples makes it hard to build a good-quality model of this batch process. Besides, due to the properties of this process, the reliability of the model must be taken into consideration when optimizing the MVs. In this work, an optimal design strategy based on the self-learning Gaussian processmodel (GPM)is proposed to control this kind of spatial batch process. The GPMis utilized as the internalmodel to predict the thicknesses of thin films on all spatial-distributed wafers using the limited data. Unlike the conventional model based design, the uncertainties of predictions provided by GPM are taken into consideration to guide the optimal design of manipulated variables so that the designing can be more prudent. Besides, the GPM is also actively enhanced using as little data as possible based on the predictive uncertainties. The effectiveness of the proposed strategy is successfully demonstrated in an LPCVD process.     

Explosive Random‐Coefficient AR(1) Processes and Related Asymptotics for Least‐Squares Estimation

Abstract. Large sample properties of the least‐squares and weighted least‐squares estimates of the autoregressive parameter of the explosive random‐coefficient AR(1) process are discussed. It is shown that, contrary to the standard AR(1) case, the least‐squares estimator is inconsistent whereas the weighted least‐squares estimator is consistent and asymptotically normal even when the error process is not necessarily Gaussian. Conditional asymptotics on the event that a certain limiting random variable is non‐zero is also discussed.     

Limiting distribution of the score statistic under moderate deviation from a unit root in MA(1)

This article derives an asymptotic distribution of Tanaka's score statistic under moderate deviation from a unit root in a moving average model of order one [MA(1)]. The limiting distribution is classified into three types depending on the order of deviation. In the fastest case, the convergence order of the asymptotic distribution continuously changes from the invertible process to the unit root one. In the slowest case, the limiting distribution coincides with one in the invertible process in the distribution sense. This implies that they share a common asymptotic property. The limiting distribution in the intermediate case has the boundary property between the fastest case and the slowest one.     

络合铁与PDS在HPF脱硫装置上的差异性研究

临涣公司一期化产HPF脱硫装置设有两套并联运行的脱硫系统,其脱硫工艺分别采用络合铁与PDS两种技术,结合脱硫原理,从工艺控制和工业运行效果等方面分析了两种脱硫技术的差异性。结果表明:两者最本质的区别在于络合铁脱硫技术可从源头上抑制S₂O₃^2-和SCN^-副盐的产生,而PDS技术会产生两盐;相比PDS脱硫,络合铁脱硫对催化剂浓度、脱硫温度、再生空气量等工艺控制的要求更高;络合铁脱硫可将硫化氢质量浓度控制在50 mg/m³以下,其脱硫效果明显优于PDS脱硫的150 mg/m³~300 mg/m³;在不外排脱硫液的情况下,络合铁脱硫可使副盐基本不增长,而PDS脱硫必须外排脱硫液去提盐工段提盐。     

Biosorption potential of Neurospora crassa cells for decolorization of Acid Red 57 (AR57) dye

Biosorption of Acid Red 57 (AR57) on to Neurospora crassa was studied with variation of pH, contact time, biosorbent and dye concentrations and temperature to determine equilibrium and kinetic models. The AR57 biosorption was fast and equilibrium was attained within 40 min. Langmuir, Freundlich and Dubinin–Radushkevich (D–R) isotherm models were applied to experimental equilibrium data for AR57 biosorption at various temperatures. The equilibrium data fitted very well to all the equilibrium models in the studied concentration range of AR57. Maximum biosorption capacity (q_max) of AR57 on to N. crassa was 2.16 × 10^?4 mol g^?1 at 20 °C. The kinetics of biosorption of AR57 were analyzed and rate constants were derived. The overall biosorption process was best described by a pseudo‐second‐order kinetic model. The changes in Gibbs free energy, enthalpy and entropy of biosorption were also evaluated for the biosorption of AR57 on to N. crassa. The results indicate that the biosorption was spontaneous and exothermic in nature. Copyright © 2006 Society of Chemical Industry     

Color gamut mapping based on a perceptual image difference measure

Gamut mapping is a color transformation technique to solve a problem caused by mismatch of gamuts among imaging devices. One plausible goal of gamut mapping is to find a reproduction that is perceptually closest to the corresponding original image when an exact color matching is not possible. Several measures to quantify the perceptual difference between images have been proposed and applied to the gamut mapping problem. However most of the measures, such as average color difference, are applied on a pixel‐wise basis and show poor correlation with human visual perception. This article describes a model of the perceptual image difference for a given pair of images, which takes the human's contrast sensitivity into account and applies the model to a gamut mapping for generating a reproduction with minimum perceptual image difference. The model has a multispatial‐frequency channel structure with tunable peak gains for each channel, which are determined by psychophysical experiments, so that the model output fits the observer's sensitivity to the image difference. A gamut‐mapped image with minimum perceptual image difference is obtained by an iterative minimization process. To evaluate the proposed method, subjective evaluation experiments are performed to construct ratio scales that measure perceptual image difference of gamut‐mapped reproductions generated by the proposed and pixel‐wise methods. Results show that the reproductions by the proposed method are perceived as perceptually closest to the original, and the model's estimate of perceptual difference correlates better with the experimentally measured perceived image difference than other pixel‐wise measures. © 1999 John Wiley & Sons, Inc. Col Res Appl, 24, 280–291, 1999     

Tests against stationary and explosive alternatives in vector autoregressive models

Abstract. The article proposes new tests for the number of unit roots in vector autoregressive models based on the eigenvalues of the companion matrix. Both stationary and explosive alternatives are considered. The limiting distributions of test statistics depend only on the number of unit roots. Size and power are investigated, and it is found that the new test against some stationary alternatives compares favourably with the widely used likelihood ratio test for the cointegrating rank. The powers are prominently higher against explosive than against stationary alternatives. Some empirical examples are provided to show how to use the new tests with real data.     

Tests Based on Simplicial Depth for AR(1) Models With Explosion

We propose outlier a robust and distribution‐free test for the explosive AR(1) model with intercept based on simplicial depth. In this model, simplicial depth reduces to counting the cases where three residuals have alternating signs. The asymptotic distribution of the test statistic is given by a specific Gaussian process. Conditions for the consistency are given, and the power of the test at finite samples is compared with alternative tests. The new test outperforms these tests in the case of skewed errors and outliers. Finally, we apply the method to crack growth data and compare the results with an OLS approach.     

基于温差函数的低温多效蒸发海水淡化过程热力学分析

从分析低温多效蒸发海水淡化过程的不可逆性出发,建立了海水加热和蒸发以及冷凝器海水预热过程的火用损失关系;导出了反映低温多效蒸发过程热力学效率的不可逆温差函数,为分析、评价过程的热力学效率提供了新的工具;并在温差函数的基础上研究讨论了蒸发器总效数、冷凝器端差、首效加热蒸汽温度和相对热容率等关键因素对低温多效蒸发海水淡化过程热力学效率的影响,为过程的优化、提高过程的能量利用率指明了方向。     

CONFIDENCE REGIONS FOR PARAMETERS IN THE AR(1) MODEL

Abstract. The construction of approximate joint and marginal confidence regions for parameters in the first-order autoregressive time series model is discussed. These regions are based on the large sample distributions of the likelihood ratio (and approximations to it), of the maximum likelihood estimates and of the score statistics. All these approaches are illustrated using a well-known example from Box and Jenkins (Time Series Analysis:Forecasting and Control, revised edn. San Francisco:Holden Day, 1976) and some simulated series. In addition, a simulation study is provided for comparing the coverage properties of the various procedures.