MODELLING FOR THE WAVELET COEFFICIENTS OF ARFIMA PROCESSES

We consider a model for the discrete nonboundary wavelet coefficients of autoregressive fractionally integrated moving average (ARFIMA) processes in each scale. Because the utility of the wavelet transform for the long‐range dependent processes, which many authors have explained in semi‐parametrical literature, is approximating the transformed processes to white noise processes in each scale, there have been few studies in a parametric setting. In this article, we propose the model from the forms of the (generalized) spectral density functions (SDFs) of these coefficients. Since the discrete wavelet transform has the property of downsampling, we cannot directly represent these (generalized) SDFs. To overcome this problem, we define the discrete non‐decimated nonboundary wavelet coefficients and compute their (generalized) SDFs. Using these functions and restricting the wavelet filters to the Daubechies wavelets and least asymmetric filters, we make the (generalized) SDFs of the discrete nonboundary wavelet coefficients of ARFIMA processes in each scale clear. Additionally, we propose a model for the discrete nonboundary scaling coefficients in each scale.     

ON THE INVERTIBILITY OF MULTIVARIATE LINEAR PROCESSES

Abstract. It is shown that a multivariate linear stationary process whose coefficients are absolutely summable is invertible if and only if its spectral density is regular everywhere. This general characterization of invertibility is applied later to the case of a linear process having an autoregressive moving-average (ARMA) representation. Under the usual assumptions, it is deduced that a process Y described by an ARMA(φ, TH) model is invertible if and only if the polynomial detTH( z ) has no roots on the unit circle. Given an invertible process Y which has an ARMA representation, it is finally shown that the process YT , where YT , =ε _i =0^l S _i Y _t-i , is invertible if and only if the matrix S ( z ) =ε _{i =0}^l S _i z ⁱ is of full rank for all z of modulus 1. It follows, in particular, that any subprocess of an invertible ARMA process is also invertible.     

ASYMPTOTIC BEHAVIOUR OF DISCRETE LINEAR PROCESSES

Abstract. We consider the linear process Y _n (ω) =Σ A _k (ω) · X _n-k (ω) on a probability space (Ω, P ) and ask for sufficient conditions in order to get a limit theorem for ( Y _k ) if the corresponding limit theorem for ( X _k ) is true.     

ON LINEAR PROCESSES WITH GIVEN MOMENTS

Abstract. Let X _t = c ₀ Y _t + c ₁ Y _{t -1}+… be a linear process with known coefficients c _k , where Y _t is a strict white noise. Let m ₁, …, m _2r be given numbers. A method is presented to determine whether there exists a distribution of Y _t such that EX ^k _t = m _k for k = 1, …, 2 r . In the positive case, such a distribution of Y _t is described. Some explicit formulas for AR(1) and AR(2) models are derived. The results can be used for simulating a process with given moments of its stationary distribution. The procedure also enables proof that some stationary distributions cannot belong to the given linear process.     

HIGHER-ORDER ASYMPTOTIC PROPERTIES OF A WEIGHTED ESTIMATOR FOR GAUSSIAN ARMA PROCESSES

Abstract. Let { X _t } be a Gaussian ARMA process with spectral density f _θ(Λ), where θ is an unknown parameter. To estimate θ, we propose an estimator θC_w of the Bayes type. Since our standpoint in this paper is different from Bayes's original approach, we call it a weighted estimator. We then investigate various higher-order asymptotic properties of θC_w. It is shown that θC_w is second-order asymptotically efficient in the class of second-order median unbiased estimators. Furthermore, if we confine our discussions to an appropriate class D of estimators, we can show that θC_w is third-order asymptotically efficient in D . We also investigate the Edgeworth expansion of a transformation of θC_w. We can then give the transformation of θC_w which makes the second-order part of the Edgeworth expansion vanish. Finally we consider the problem of testing a simple hypothesis H:θ=θ_o against the alternative A:θ#θ_o. For this problem we propose a class of tests δ_A which are based on the weighted estimator. We derive the X ² type asymptotic expansion of the distribution of S (ζδ_A) under the sequence of alternatives A n :θ=θ_o+ε n ^1/2, ε > 0. We can then compare the local powers of various tests on the basis of their asymptotic expansions.     

A LINEAR ESTIMATION PROCEDURE FOR THE PARAMETERS OF AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. A linear estimation procedure for the parameters of autoregressive moving-average processes is proposed. The basic idea is to write the spectrum for the moving-average part as a linear function of a properly selected set of parameters and to use Chiu's weighted least-squares procedure to reduce the problem to a weighted linear least-squares problem. The proposed procedure finds estimates by solving systems of linear equations and does not need optimization programs. An one-step estimate is also suggested. It is shown that the estimates are asymptotically equal to the commonly used 'approximate' maximum likelihood estimate described in the paper. For Gaussian processes, the estimates obtained by the proposed procedures are asymptotically efficient.     

LEAST SQUARES ESTIMATION OF A SHIFT IN LINEAR PROCESSES

Abstract. This paper considers a mean shift with an unknown shift point in a linear process and estimates the unknown shift point (change point) by the method of least squares. Pre-shift and post-shift means are estimated concurrently with the change point. The consistency and the rate of convergence for the estimated change point are established. The asymptotic distribution for the change point estimator is obtained when the magnitude of shift is small. It is shown that serial correlation affects the variance of the change point estimator via the sum of the coefficients of the linear process. When the underlying process is autoregressive moving average, a mean shift causes overestimation of its order. A simple procedure is suggested to mitigate the bias in order estimation.     

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.     

ON GENERALIZED FRACTIONAL PROCESSES

Abstract. A class of stationary long-memory processes is proposed which is an extension of the fractional autoregressive moving-average (FARMA) model. The FARMA model is limited by the fact that it does not allow data with persistent cyclic (or seasonal) behavior to be considered. Our extension, which includes the FARMA model as a special case, makes use of the properties of the generating function of the Gegenbauer polynomials, and we refer to these models as Gegenbauer autoregressive moving-average (GARMA) models. While the FARMA model has a peak in the spectrum at f = 0, the GARMA process can model long-term periodic behavior for any frequency 0 f 0.5. Properties of the GARMA process are examined and techniques for generation of realizations, model identification and parameter estimation are proposed. The use of the GARMA model is illustrated through simulated examples as well as with classical sunspot data.     

GENERAL LINEAR PROCESSES:A PROPERTY OF THE EMPIRICAL PROCESS APPLIED TO DENSITY AND MODE ESTIMATION

Abstract. General linear processes do not usually satisfy strong mixing conditions. Therefore, we investigate the empirical process based on samples from such a general linear process by using a truncation argument and derive a local fluctuation inequality. It is well known that such a fluctuation inequality is of basic importance in the study of the empirical process. Here it is applied to obtain a rate of almost sure (a.s.) convergence for certain density estimators in the supremum norm. This extends a local result obtained by Chanda. As a direct corollary a rate of a.s. convergence for a mode estimator is obtained.     

RANDOM AGGREGATION OF UNIVARIATE AND MULTIVARIATE LINEAR PROCESSES

Abstract. The paper is devoted to random aggregation of multivariate autoregressive moving-average (ARMA) processes. We derive second-order characteristics of random aggregate models. We show that random aggregation preserves the ARMA structure. Moreover, we specify a functional relation between the initial model poles and aggregate ones. We then examine the case of univariate ARMA processes. Theorem 4 shows that, if the initial process is ARMA( p, q ), the random aggregate process is an ARMA( p*, q* ) model with p* at most equal to p ; * depends, among other things, on the sampling distribution L . This theorem generalizes the well-known results on the topic of time interval aggregation without overlapping.     

A MIXED INTEGER LINEAR PROGRAMMING-BASED TECHNIQUE FOR THE ESTIMATION OF MULTIPLE GROSS ERRORS IN PROCESS MEASUREMENTS

This paper presents a new strategy for detecting, identifying, and estimating gross errors (measurement biases and leaks) in linear steady state processes. The MILP-based gross error detection and identification model is constructed aiming at identifying the minimum number of gross errors and their sizes. One significant advantage of the method is that the detection, identification, and estimation of gross errors can be performed simultaneously without using any test statistics.     

A MIXED INTEGER LINEAR PROGRAMMING-BASED TECHNIQUE FOR THE ESTIMATION OF MULTIPLE GROSS ERRORS IN PROCESS MEASUREMENTS

This paper presents a new strategy for detecting, identifying, and estimating gross errors (measurement biases and leaks) in linear steady state processes. The MILP-based gross error detection and identification model is constructed aiming at identifying the minimum number of gross errors and their sizes. One significant advantage of the method is that the detection, identification, and estimation of gross errors can be performed simultaneously without using any test statistics.     

线性系统非冗余测量网络的优化设计

研究了线性系统非冗余测量网络设计的问题 ,采用混合整数线性规划方法与图论方法相结合建立了考虑可靠性以及精度的要求下建设费用最小的模型 .此模型适用于采用多块测量仪表测量同一物流的问题 ,求解此模型可以确保得到全局最优解 .对文献中例题的计算结果说明了本文方法的正确性     

ESTIMATION OF THE COEFFICIENTS OF A MULTIVARIATE LINEAR FILTER USING THE INNOVATIONS ALGORITHM

It is shown under mild conditions that the estimators of the coefficient matrices obtained by applying the innovations algorithm to the sample covariances of observations of the multivariate linear time series X _t = ∑^∞ _{j =0}ψ _i Z _t , t = 0, ±1, ±2, . . ., are consistent. The asymptotic distribution of the estimators is found to have a very simple form which generalizes the corresponding univariate result of Brockwell and Davis (Simple consistent estimation of the coefficients of a linear filter. In Stochastic Processes and Their Applications . Amsterdam: North- Holland, pp. 47--59). The asymptotic distribution of the corresponding estimator of the spectral density matrix is also derived. Some simulation results are presented to illustrate the small-sample behaviour of the estimators.     

SUBORDINATION OF STATIONARY PROCESSES

Abstract. Let X ={ X ( t ), t ε T ∁ R} be a ( L ₂ -) stationary process and suppose that N ={ N ( t ), t ≥ 0} is an infinitely divisible process, independent of X. Then ={ ( t ) = X ( N ( t )), t ≥ 0} is again a stationary process. In this paper, we relate the spectral properties of the original process X and the derived or subordinated process .     

ON A CLASS OF NONSTATIONARY PROCESSES

Abstract. In this paper a class of nonstationary processes, referred to as multiplicative stationary processes, is investigated. It is shown that although these processes are not stationary with regard to an additive binary operation, i.e. in the classical sense, they are stationary with respect to a multiplicative binary operation. This property is then exploited in such a way as to guarantee essentially the same structure as is available for stationary processes. In particular, suitable definitions for the autocorrelation, power spectrum and linear processes are given. In addition, the Euler process is introduced as the nonstationary or multiplicative stationary dual of the classical autoregressive processes. Some ergodic theorems are also obtained and numerous examples are given.     

FORECASTING OF MULTIVARIATE PERIODIC AUTOREGRESSIVE MOVING-AVERAGE PROCESSES

Abstract. Minimum mean square error forecasting of multivariate autoregressive moving-average processes with periodically varying parameters and orders is considered. General expressions are obtained for the forecasts, their errors and the covariance matrices of the forecast errors. Recursive evaluations of these quantities are shown to follow from the conditional expectation approach. Prediction ellipsoids and intervals for future values of the process are given. Update equations for the forecasts are obtained. The general results are illustrated and verified for a particular case of the process. A simulated example is given.     

OXIDATIVE CRACKING OF LINEAR HYDROCARBONS AT LOW TEMPERATURES

This work describes the oxidative cracking of n-alkanes with molecular oxygen at low temperatures (below 473 K) in an autoclave reactor. An increase of the oxygen consumption rate with increasing hydrocarbon size was observed. Data for n-hexadecane indicate that oxidative cracking is an autocatalytic reaction. The oxidation rate increased with the progress of the reaction. Low molecular weight compounds were formed as the main products. CG and CG-MS analyses of the liquid products found homologous series of oxygen compounds (acids, ketones, and ethers) and short-chain n-alkanes. Our results strongly suggest that oxidative cracking can be employed for processing heavy materials such as polymers and petroleum residues.     

THE SUM OF FINITE MOVING AVERAGE PROCESSES

Abstract. It is well known that the sum of moving average processes is itself a moving average process. Existing theory does not provide formulae relating the innovations in the sum process to those in the component processes when some zeros of the autocovari-ance function of the sum process are on the unit circle. This gap is filled here by first showing that these zeros are necessarily shared by the component processes. Conditions for the innovations in the sum process to be current are given. Throughout the paper contemporary innovations in the component processes are allowed to be correlated, replacing the usual assumption that they are uncorrelated.