Estimation of regression and dynamic dependence paremeters for non‐stationary multinomial time series

In a time‐series regression setup, multinomial responses along with time dependent observable covariates are usually modelled by certain suitable dynamic multinomial logistic probabilities. Frequently, the time‐dependent covariates are treated as a realization of an exogenous random process and one is interested in the estimation of both the regression and the dynamic dependence parameters conditional on this realization of the covariate process. There exists a partial likelihood estimation approach able to deal with the general dependence structures arising from the influence of both past covariates and past multinomial responses on the covariates at a given time by sequentially conditioning on the history of the joint process (response and covariates), but it provides standard errors for the estimators based on the observed information matrix, because such a matrix happens to be the Fisher information matrix obtained by conditioning on the whole history of the joint process. This limitation of the partial likelihood approach holds even if the covariate history is not influeced by lagged response outcomes. In this article, a general formulation of the auto‐covariance structure of a multinomial time series is presented and used to derive an explicit expression for the Fisher information matrix conditional on the covariate history, providing the possibility of computing the variance of the maximum likelihood estimators given a realization of the covariate process for the multinomial‐logistic model. The difference between the standard errors of the parameter estimators under these two conditioning schemes (covariates Vs. joint history) is illustrated through an intensive simulation study based on the premise of an exogenous covariate process.     

Nonlinear stochastic modeling to improve state estimation in process monitoring and control

State estimation from plant measurements plays an important role in advanced monitoring and control technologies, especially for chemical processes with nonlinear dynamics and significant levels of process and sensor noise. Several types of state estimators have been shown to provide high‐quality estimates that are robust to significant process disturbances and model errors. These estimators require a dynamic model of the process, including the statistics of the stochastic disturbances affecting the states and measurements. The goal of this article is to introduce a design method for nonlinear state estimation including the following steps: (i) nonlinear process model selection, (ii) stochastic disturbance model selection, (iii) covariance identification from operating data, and (iv) estimator selection and implementation. Results on the implementation of this design method in nonlinear examples (CSTR and large dimensional polymerization process) show that the linear time‐varying autocovariance least‐squares technique accurately estimates the noise covariances for the examples analyzed, providing a good set of such covariances for the state estimators implemented. On the estimation implementation, a case study of a chemical reactor demonstrates the better capabilities of MHE when compared with the extended Kalman filter. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Scalable inference for space‐time Gaussian Cox processes

The log‐Gaussian Cox process is a flexible and popular stochastic process for modeling point patterns exhibiting spatial and space‐time dependence. Model fitting requires approximation of stochastic integrals which is implemented through discretization over the domain of interest. With fine scale discretization, inference based on Markov chain Monte Carlo is computationally burdensome because of the cost of matrix decompositions and storage, such as the Cholesky, for high dimensional covariance matrices associated with latent Gaussian variables. This article addresses these computational bottlenecks by combining two recent developments: (i) a data augmentation strategy that has been proposed for space‐time Gaussian Cox processes that is based on exact Bayesian inference and does not require fine grid approximations for infinite dimensional integrals, and (ii) a recently developed family of sparsity‐inducing Gaussian processes, called nearest‐neighbor Gaussian processes, to avoid expensive matrix computations. Our inference is delivered within the fully model‐based Bayesian paradigm and does not sacrifice the richness of traditional log‐Gaussian Cox processes. We apply our method to crime event data in San Francisco and investigate the recovery of the intensity surface.     

Estimation in Functional Lagged Regression

The paper introduces a functional time series (lagged) regression model. The impulse‐response coefficients in such a model are operators acting on a separable Hilbert space, which is the function space L² in applications. A spectral approach to the estimation of these coefficients is proposed and asymptotically justified under a general nonparametric condition on the temporal dependence of the input series. Since the data are infinite‐dimensional, the estimation involves a spectral‐domain dimension‐reduction technique. Consistency of the estimators is established under general data‐dependent assumptions on the rate of the dimension‐reduction parameter. Their finite‐sample performance is evaluated by a simulation study that compares two ad hoc approaches to dimension reduction with an alternative, asymptotically justified method.     

A Plug‐in Bandwidth Selection Procedure for Long‐Run Covariance Estimation with Stationary Functional Time Series

In several arenas of application, it is becoming increasingly common to consider time series of curves or functions. Many inferential procedures employed in the analysis of such data involve the long‐run covariance function or operator, which is analogous to the long‐run covariance matrix familiar to finite‐dimensional time‐series analysis and econometrics. This function may be naturally estimated using a smoothed periodogram type estimator evaluated at frequency zero that relies on the choice of a bandwidth parameter. Motivated by a number of prior contributions in the finite‐dimensional setting, in particular Newey and West ( 1994 ), we propose a bandwidth selection method that aims to minimize the estimator's asymptotic mean‐squared normed error (AMSNE) in L²[0,1]². As the AMSNE depends on unknown population quantities including the long‐run covariance function itself, estimates for these are plugged in in an initial step after which the estimated AMSNE can be minimized to produce an empirical optimal bandwidth. We show that the bandwidth produced in this way is asymptotically consistent with the AMSNE optimal bandwidth, with quantifiable rates, under mild stationarity and moment conditions. These results and the efficacy of the proposed methodology are evaluated by means of a comprehensive simulation study, from which we can offer practical advice on how to select the bandwidth parameter in this setting.     

Reactions in solid particles—A reappraisal of models

Models such as the Ishida–Wen, or even more commonly the shrinking core models (Ginstling–Brounshtein being an important example), have long been used in the analyses of reactions in particles, particularly in solid–solid systems. There have been a few analyses of the validity of the assumptions made in these models, but to date, no comparison has been undertaken of these models against a general model to delineate their regions of applicability in the parameter space. In this article, we present a general unsteady‐state model that subsumes the earlier models as special cases. Nondimensionalization leads to the identification of two governing parameters in the model, a diffusion‐reaction parameter, and a relative abundance parameter. By solving the general model and comparing the solutions with those of the approximate models in the parameter space, conditions under which the approximate models apply, and the errors that result from their application in other situations, have been identified. © 2012 American Institute of Chemical Engineers AIChE J, 58: 3161–3166, 2012     

Robust estimation for the covariance matrix of multi‐variate time series

In this article, we study the robust estimation for the covariance matrix of stationary multi‐variate time series. As a robust estimator, we propose to use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998) . Particularly, the MDPDE is designed to perform properly when the time series is Gaussian. As a special case, we consider the robust estimator for the autocovariance function of univariate stationary time series. It is shown that the MDPDE is strongly consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

Parametric and Semi‐Parametric Efficient Tests for Parameter Instability

This article examines asymptotically point optimal tests for parameter instability in realistic circumstances when little information about the unstable parameter process and error distribution is available. We first show that, under a correctly specified error distribution, if the unstable parameter processes converge weakly to a Wiener process, then any asymptotic optimal tests for structural breaks and time‐varying parameters are asymptotically equivalent. Our finding is then extended to a semi‐parametric set‐up in which the error distribution is treated as an unknown infinite‐dimensional nuisance parameter. We find that semi‐parametric tests can be adaptive without further restrictive conditions on the error distribution.     

Quantile Autoregression for Censored Data

Quantile autoregression (QAR) is particularly attractive for censored data. However, unlike the standard regression models, the autoregressive models must take account of censoring on both response and regressors. In this article, we show that the existing censored quantile regression methods produce consistent estimators for QAR models when using only the fully observed regressors. A new algorithm is proposed to provide a censored QAR estimator by adopting imputation methods. The algorithm redistributes probability mass of censored points appropriately and iterates towards self‐consistent solutions. Monte Carlo simulations and empirical applications are conducted to demonstrate merits of the proposed method.     

Asymptotic Distribution of the Bias Corrected Least Squares Estimators in Measurement Error Linear Regression Models Under Long Memory

This article derives the consistency and asymptotic distribution of the bias corrected least squares estimators (LSEs) of the regression parameters in linear regression models when covariates have measurement error (ME) and errors and covariates form mutually independent long memory moving average processes. In the structural ME linear regression model, the nature of the asymptotic distribution of suitably standardized bias corrected LSEs depends on the range of the values of where d _X ,d _u , and d _ε are the LM parameters of the covariate, ME and regression error processes respectively. This limiting distribution is Gaussian when and non‐Gaussian in the case . In the former case some consistent estimators of the asymptotic variances of these estimators and a log(n)‐consistent estimator of an underlying LM parameter are also provided. They are useful in the construction of the large sample confidence intervals for regression parameters. The article also discusses the asymptotic distribution of these estimators in some functional ME linear regression models, where the unobservable covariate is non‐random. In these models, the limiting distribution of the bias corrected LSEs is always a Gaussian distribution determined by the range of the values of d _ε ? d _u .     

A PARAMETER‐DRIVEN LOGIT REGRESSION MODEL FOR BINARY TIME SERIES

We consider a parameter‐driven regression model for binary time series, where serial dependence is introduced by an autocorrelated latent process incorporated into the logit link function. Unlike in the case of parameter‐driven Poisson log‐linear or negative binomial logit regression model studied in the literature for time series of counts, generalized linear model (GLM) estimation of the regression coefficient vector, which suppresses the latent process and maximizes the corresponding pseudo‐likelihood, cannot produce a consistent estimator. As a remedial measure, in this article, we propose a modified GLM estimation procedure and show that the resulting estimator is consistent and asymptotically normal. Moreover, we develop two procedures for estimating the asymptotic covariance matrix of the estimator and establish their consistency property. Simulation studies are conducted to evaluate the finite‐sample performance of the proposed procedures. An empirical example is also presented.     

On limiting spectral distribution of large sample covariance matrices by VARMA(p,q)

We studied the limiting spectral distribution of large‐dimensional sample covariance matrices of a stationary and invertible VARMA(p,q) model. Relationship of the power spectral density and limiting spectral distribution of large population dimensional covariance matrices of ARMA(p,q) is established. The equation about Stieltjes transform of large‐dimensional sample covariance matrices is also derived. As applications, the classical M‐P law, VAR(1) and VMA(1) can be regarded as special examples.     

Bias Correction Estimation for a Continuous‐Time Asset Return Model with Jumps

In this article, local linear estimators are adapted for the unknown infinitesimal coefficients associated with continuous‐time asset return models with jumps, which can correct the bias automatically due to their simple bias representation. The integrated diffusion models with jumps, especially infinite activity jumps, are mainly investigated. In addition, under mild conditions, the weak consistency and asymptotic normality are provided through the conditional Lindeberg theorem as the time span T→∞ and the sample interval Δ _n →0. Furthermore, our method presents advantages in bias correction through simulation whether jumps belong to the finite activity case or infinite activity case. Finally, the estimators are illustrated empirically through the returns of stock index under 5‐minute high sampling frequency for real application.     

A Generalized Portmanteau Test For Independence Of Two Infinite‐Order Vector Autoregressive Series

Abstract. In many situations, we want to verify the existence of a relationship between multivariate time series. Here, we propose a semiparametric approach for testing the independence between two infinite‐order vector autoregressive (VAR(∞)) series, which is an extension of Hong's [Biometrika (1996c) vol. 83, 615–625] univariate results. We first filter each series by a finite‐order autoregression and the test statistic is a standardized version of a weighted sum of quadratic forms in the residual cross‐correlation matrices at all possible lags. The weights depend on a kernel function and on a truncation parameter. Using a result of Lewis and Reinsel [Journal of Multivariate Analysis (1985) Vol. 16, pp. 393–411], the asymptotic distribution of the test statistic is derived under the null hypothesis and its consistency is also established for a fixed alternative of serial cross‐correlation of unknown form. Apart from standardization factors, the multivariate portmanteau statistic proposed by Bouhaddioui and Roy [Statistics and Probability Letters (2006) vol. 76, pp. 58–68] that takes into account a fixed number of lags can be viewed as a special case by using the truncated uniform kernel. However, many kernels lead to a greater power, as shown in an asymptotic power analysis and by a small simulation study in finite samples. A numerical example with real data is also presented.     

A Wavelet Characterization of Continuous‐Time Periodically Correlated Processes with Application to Simulation

We introduce a wavelet characterization of continuous‐time periodically correlated processes based on a linear combination of infinite‐dimensional stationary processes. The finite version of this linear combination converges to the main process. The first‐order and second‐order estimators based on the wavelets are presented. Under a simple and easy algorithm, the periodically correlated process is simulated for a given autocovariance function. The proposed algorithm has two main advantages: first, it is fast, and second, it is distribution free. We indicate through four examples that the simulated data are periodically correlated with the desired period.     

Robust estimation for copula Parameter in SCOMDY models

In this study, we study the robust estimation for the copula parameter in semiparametric copula‐based multivariate dynamic (SCOMDY) models proposed by Chen and Fan (2006). To this end, instead of the pseudo maximum likelihood estimator in Chen and Fan (2006), we use a minimum density power divergence estimator (MDPDE) proposed by Basu et al. (1998). It is shown that the MDPDE is consistent and asymptotically normal under regularity conditions. We compare the performance between the two estimators when outliers exist through a simulation study.     

Note on the Asymptotic Efficiency of Sample Covariances in Gaussian Vector Stationary Processes

In this note certain results obtained by Porat ( J. Time Ser. Anal. 8 (1987), 205–20) and Kakizawa and Taniguchi ( J. Time Ser. Anal. 15 (1994), 303–11) concerning the asymptotic efficiency of sample autocovariances of a zero-mean Gaussian stationary process are extended to the case of m -vector processes. It is shown that, for Gaussian vector AR( p ) processes, the sample autocovariance matrix at lag k is asymptotically efficient if 0 ≤ k ≤ p . Further, none of the sample autocovariance matrices is asymptotically efficient for Gaussian vector MA( q ) processes.     

On the attainable region for process networks

In this work the attainable region (AR) concept for process networks with outlet flow rate specifications is introduced for the first time. For process unit models to which the infinite dimensional State‐space conceptual framework is applicable, it is shown that identification of AR boundary membership is equivalent to feasibility assessment of an infinite linear program (ILP). A number of important AR properties are then theoretically established, including AR convexity, and representation of the AR in a concentration state space of reduced dimension. Finite dimensional approximations of the aforementioned ILP are then employed in creating increasingly accurate approximations of the AR. A case study for the vapor‐liquid equilibrium‐based separation of a ternary azeotropic mixture is used to illustrate the proposed method. The quantified two‐ and four‐Dimensional ARs indicate that acetone mole fractions above 0.79 (acetone/methanol azeotrope) are attainable for the considered outlet flow rate ratios. © 2013 American Institute of Chemical Engineers AIChE J, 60: 193–212, 2014     

BOOTSTRAPPING STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. In this paper we develop an asymptotic theory for application of the bootstrap to stationary stochastic processes of autoregressive moving-average (ARMA) type, with known order ( p, q ). We give a proof of the asymptotic validity of the bootstrap proposal applied to M estimators for the unknown parameter vector of the process. For this purpose we derive an asymptotic expansion for M estimators in ARMA models and construct an estimate for the unknown distribution function of the residuals which in principle are not observable. A small simulation study is also included.     

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.