Cressie–Read Power-Divergence Statistics for Non-Gaussian Vector Stationary Processes

Abstract.  For a class of vector-valued non-Gaussian stationary processes, we develop the Cressie–Read power-divergence (CR) statistic approach which has been proposed for the i.i.d. case. The CR statistic includes empirical likelihood as a special case. Therefore, by adopting this CR statistic approach, the theory of estimation and testing based on empirical likelihood is greatly extended. We use an extended Whittle likelihood as score function and derive the asymptotic distribution of the CR statistic. We apply this result to estimation of autocorrelation and the AR coefficient, and get narrower confidence intervals than those obtained by existing methods. We also consider the power properties of the test based on asymptotic theory. Under a sequence of contiguous local alternatives, we derive the asymptotic distribution of the CR statistic. The problem of testing autocorrelation is discussed and we introduce some interesting properties of the local power.     

On the asymptotic distributions of mean,autocovariance, autocorrelation,crossgovariancb and impulse response estimators of a stationary multidimensional random field

Asymptotic properties of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators of a stationary M-dimensionai (M-D) random field are studied. It is shown that only unbiased-type estimators of autocovariances, autocorrelations, crosscovariances and impulse responses have the asymptotic distributions when M≧ 2. Moreover, the asymptotic distributions of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators are presented.     

Statistical Monitoring of Autocorrelated Simple Linear Profiles Based on Principal Components Analysis

In this article, a transformation method using the principal component analysis approach is first applied to remove the existing autocorrelation within each profile in Phase I monitoring of autocorrelated simple linear profiles. This easy-to-use approach is independent of the autocorrelation coefficient. Moreover, since it is a model-free method, it can be used for Phase I monitoring procedures. Then, five control schemes are proposed to monitor the parameters of the profile with uncorrelated error terms. The performances of the proposed control charts are evaluated and are compared through simulation experiments based on different values of autocorrelation coefficient as well as different shift scenarios in the parameters of the profile in terms of probability of receiving an out-of-control signal.     

局部空间自相关指标对比研究

本文研究了最常用的局部空间自相关指标;局部Moran指数和局部G系数,基于模拟的空间区域;设计了一些有代表性的空间聚集方案进行计算,比较了两种指标的探测结果,得出局部G系数要优于局部Moran指数的初步结论。     

A combined mixed-s-skip sampling strategy to reduce the effect of autocorrelation on the X̄ scheme with and without measurement errors

In order to reduce the effect of autocorrelation on the X¯ monitoring scheme, a new sampling strategy is proposed to form rational subgroup samples of size n. It requires sampling to be done such that: (i) observations from two consecutive samples are merged, and (ii) some consecutive observations are skipped before sampling. This technique which is a generalized version of the mixed samples strategy is shown to yield a better reduction of the negative effect of autocorrelation when monitoring the mean of processes with and without measurement errors. For processes subjected to a combined effect of autocorrelation and measurement errors, the proposed sampling technique, together with multiple measurement strategy, yields an uniformly better zero-state run-length performance than its two main existing competitors for any autocorrelation level. However, in steady-state mode, it yields the best performance only when the monitoring process is subject to a high level of autocorrelation, for any given level of measurement errors. A real life example is used to illustrate the implementation of the proposed sampling strategy.KEYWORDS: Autocorrelation, measurement errors, mixed samples strategy, multiple measurements, skipping sampling strategy, steady-state, zero-state     

On the estimation of serial correlation in Markov-dependent production processes

In this paper, we present a study about the estimation of the serial correlation for Markov chain models which is used often in the quality control of autocorrelated processes. Two estimators, non-parametric and multinomial, for the correlation coefficient are discussed. They are compared with the maximum likelihood estimator [U.N. Bhat and R. Lal, Attribute control charts for Markov dependent production process, IIE Trans. 22 (2) (1990), pp. 181–188.] by using some theoretical facts and the Monte Carlo simulation under several scenarios that consider large and small correlations as well a range of fractions (p) of non-conforming items. The theoretical results show that for any value of p≠0.5 and processes with autocorrelation higher than 0.5, the multinomial is more precise than maximum likelihood. However, the maximum likelihood is better when the autocorrelation is smaller than 0.5. The estimators are similar for p=0.5. Considering the average of all simulated scenarios, the multinomial estimator presented lower mean error values and higher precision, being, therefore, an alternative to estimate the serial correlation. The performance of the non-parametric estimator was reasonable only for correlation higher than 0.5, with some improvement for p=0.5.     

Inference for Local Autocorrelations in Locally Stationary Models

For nonstationary processes, the time-varying correlation structure provides useful insights into the underlying model dynamics. We study estimation and inferences for local autocorrelation process in locally stationary time series. Our constructed simultaneous confidence band can be used to address important hypothesis testing problems, such as whether the local autocorrelation process is indeed time-varying and whether the local autocorrelation is zero. In particular, our result provides an important generalization of the R function acf() to locally stationary Gaussian processes. Simulation studies and two empirical applications are developed. For the global temperature series, we find that the local autocorrelations are time-varying and have a “V” shape during 1910–1960. For the S&P 500 index, we conclude that the returns satisfy the efficient-market hypothesis whereas the magnitudes of returns show significant local autocorrelations.     

Small sample efficiency gains from a first observation correction for hatanakafs estimator of the lagged dependent variable-serial correlation regression model

Evidence presented by Fomby and Guilkey (1983) suggests that Hatanaka's estimator of the coefficients in the lagged dependent variable-serial correlation regression model performs poorly, not because of poor selection of the estimate of the autocorrelation coefficient, but because of the lack of a first observation correction. This study conducts a Monte Carlo investigationof the small sample efficiency gains obtainable from a first observation correction suggested by Harvey (1981). Results presented here indicate that substantial gains result from the first observation correction. However, in comparing Hatanaka's procedure with first observation correction to maximum likelihood search, it appears that ignoring the determinantal term of the full likelihood function causes some loss of small sample efficiency. Thus, when computer costsand programming constraints are not binding, maximum likelihood search is to be recommended. In contrast, users who have access to only rudimentary least squares programs would be well served when using Hatanaka's two-step procedure with first     

A NOTE ON ESTIMATING LINEAR TREND IN A REGRESSION MODEL WITH SERIALLY CORRELATED ERROR COMPONENTS

ABSTRACT

We consider a linear trend regression model when the disturbances follow a serially correlated one-way error component model. In this model, we investigate the performance of the Ordinary Least Squares Esitmator (OLSE), First Difference Estimator (FDE), Generalized Least Squares Estimator (GLSE) and the Cochrane-Orcutt-Transformation Estimator (COTE) of slope coefficient in terms of efficiency. The main findings are as follows: (1) when the autocorrelation is close to unity, then the FDE is approximately the GLSE; (2) the OLSE is better than the COTE; and (3) when the value of the autocorrelation is kept constant and T → ∞, the OLSE, COTE and GLSE are asymptotically equivalent whereas the FDE is worse than the other estimators in terms of efficiency.     

Evaluation of Linear Trend Tests Using Resampling Techniques

A number of parametric and non-parametric linear trend tests for time series are evaluated in terms of test size and power, using also resampling techniques to form the empirical distribution of the test statistics under the null hypothesis of no linear trend. For resampling, both bootstrap and surrogate data are considered. Monte Carlo simulations were done for several types of residuals (uncorrelated and correlated with normal and nonnormal distributions) and a range of small magnitudes of the trend coefficient. In particular for AR(1) and ARMA(1, 1) residual processes, we investigate the discrimination of strong autocorrelation from linear trend with respect to the sample size. The correct test size is obtained for larger data sizes as autocorrelation increases and only when a randomization test that accounts for autocorrelation is used. The overall results show that the type I and II errors of the trend tests are reduced with the use of resampled data. Following the guidelines suggested by the simulation results, we could find significant linear trend in the data of land air temperature and sea surface temperature.     

Diagnostics for elliptical linear mixed models with first-order autoregressive errors

For longitudinal time series data, linear mixed models that contain both random effects across individuals and first-order autoregressive errors within individuals may be appropriate. Some statistical diagnostics based on the models under a proposed elliptical error structure are developed in this work. It is well known that the class of elliptical distributions offers a more flexible framework for modelling since it contains both light- and heavy-tailed distributions. Iterative procedures for the maximum-likelihood estimates of the model parameters are presented. Score tests for the presence of autocorrelation and the homogeneity of autocorrelation coefficients among individuals are constructed. The properties of test statistics are investigated through Monte Carlo simulations. The local influence method for the models is also given. The analysed results of a real data set illustrate the values of the models and diagnostic statistics.     

Perturbation diagnostics of autocorrelation coefficients in non linear mixed-effects models with AR(1) errors based on M-estimation

In this work we propose and analyze non linear mixed-effects models for longitudinal data, which are widely used in the fields of economics, biopharmaceuticals, agriculture, and so on. A robust method to obtain maximum likelihood estimates for the parameters is presented, as well as perturbation diagnostics of autocorrelation coefficient in non linear models based on robust estimates and influence curvature. The obtained results are illustrated by plasma concentrations data presented in Davidian and Giltinan, which was analyzed under the non robust situation.     

Interaction Between Autocorrelation and Conditional Heteroscedasticity: A Random-Coefficient Approach

In applied econometrics, we tend to tackle specification problems one at a time rather than considering them jointly. This has serious consequences for statistical inference. One example of this is considering autocorrelation and autoregressive conditional heteroscedasticity (ARCH) separately. In this article we consider a linear regression model with random coefficient autoregressive disturbances that provides a convenient framework to analyze autocorrelation and ARCH simultaneously. Our stationarity conditions and testing results reveal the strong interaction between ARCH and autocorrelation. An empirical example of testing the unbiasedness of experts' expectations of inflation demonstrates that neglecting conditional heteroscedasticity or misspecifying the autocorrelation structure might result in unreliable inference.     

Objective Bayesian hypothesis testing in regression models with first-order autoregressive residuals

This article considers the objective Bayesian testing in the normal regression models with first-order autoregressive residuals. We propose some solutions based on a Bayesian model selection procedure to this problem where no subjective input is considered. We construct the proper priors for testing the autocorrelation coefficient based on measures of divergence between competing models, which is called the divergence-based (DB) priors and then propose the objective Bayesian decision-theoretic rule, which is called the Bayesian reference criterion (BRC). Finally, we derive the intrinsic test statistic for testing the autocorrelation coefficient. The behavior of the Bayes factor-based DB priors is examined by comparing with the BRC in a simulation study and an example.     

THE EFFICIENCY OF THE COCHRANE-ORCUTT PROCEDURE

It is well known that the ordinary least squares (OLS) estimator, though unbiased, is inefficient in the presence of autocorrelated disturbances. Further, it is also widely accepted that the Cochrane-Orcutt (C-O) estimator is more efficient than the OLS estimator. However, Kadiyala (1968) and Maeshiro (1976, 1978) have argued that OLS is more efficient than C-O when the independent variable is trended and the autocorrelation coefficient is positive. We re-examine this issue and show that C-O is more efficient than OLS for the model without an intercept term.     

Effect of autocorrelation when estimating the trend of a time series via penalized least squares with controlled smoothness

This paper studies the effect of autocorrelation on the smoothness of the trend of a univariate time series estimated by means of penalized least squares. An index of smoothness is deduced for the case of a time series represented by a signal-plus-noise model, where the noise follows an autoregressive process of order one. This index is useful for measuring the distortion of the amount of smoothness by incorporating the effect of autocorrelation. Different autocorrelation values are used to appreciate the numerical effect on smoothness for estimated trends of time series with different sample sizes. For comparative purposes, several graphs of two simulated time series are presented, where the estimated trend is compared with and without autocorrelation in the noise. Some findings are as follows, on the one hand, when the autocorrelation is negative (no matter how large) or positive but small, the estimated trend gets very close to the true trend. Even in this case, the estimation is improved by fixing the index of smoothness according to the sample size. On the other hand, when the autocorrelation is positive and large the simulated and estimated trends lie far away from the true trend. This situation is mitigated by fixing an appropriate index of smoothness for the estimated trend in accordance to the sample size at hand. Finally, an empirical example serves to illustrate the use of the smoothness index when estimating the trend of Mexico’s quarterly GDP.     

Optimal designs for multivariate logistic mixed models with longitudinal data

This paper considers the optimal design problem for multivariate mixed-effects logistic models with longitudinal data. A decomposition method of the binary outcome and the penalized quasi-likelihood are used to obtain the information matrix. The D-optimality criterion based on the approximate information matrix is minimized under different cost constraints. The results show that the autocorrelation coefficient plays a significant role in the design. To overcome the dependence of the D-optimal designs on the unknown fixed-effects parameters, the Bayesian D-optimality criterion is proposed. The relative efficiencies of designs reveal that both the cost ratio and autocorrelation coefficient play an important role in the optimal designs.     

A QUASI‐LOCALLY MOST POWERFUL TEST FOR CORRELATION IN THE CONDITIONAL VARIANCE OF POSITIVE DATA

A test is derived for short‐memory correlation in the conditional variance of strictly positive, skewed data. The test is quasi‐locally most powerful (QLMP) under the assumption of conditionally gamma data. Analytical asymptotic relative efficiency calculations show that an alternative test, based on the first‐order autocorrelation coefficient of the squared data, has negligible relative power to detect correlation in the conditional variance. Finite‐sample simulation results confirm the poor performance of the squares‐based test for fixed alternatives, as well as demonstrating the poor performance of the test based on the first‐order autocorrelation coefficient of the raw (levels) data. The robustness of the QLMP test, both to misspecification of the conditional distribution and to misspecification of the dynamics, is also demonstrated using simulation. The test is illustrated using financial trade durations data.     

Multivariate Quality Control Chart for Autocorrelated Processes

Traditional multivariate statistical process control (SPC) techniques are based on the assumption that the successive observation vectors are independent. In recent years, due to automation of measurement and data collection systems, a process can be sampled at higher rates, which ultimately leads to autocorrelation. Consequently, when the autocorrelation is present in the data, it can have a serious impact on the performance of classical control charts. This paper considers the problem of monitoring the mean vector of a process in which observations can be modelled as a first-order vector autoregressive VAR (1) process. We propose a control chart called Z-chart which is based on the single step finite intersection test (Timm, 1996). An important feature of the proposed method is that it not only detects an out of control status but also helps in identifying variable(s) responsible for the out of control situation. The proposed method is illustrated with the help of suitable illustrations.     

Comparison of spline estimator at various levels of autocorrelation in smoothing spline non parametric regression for longitudinal data

The purpose of this research are: (1) to obtain spline function estimation in non parametric regression for longitudinal data with and without considering the autocorrelation between data of observation within subject, (2) to develop the algorithm that generates simulation data with certain autocorrelation level based on size of sample (N) and error variance (EV), and (3) to establish shape of spline estimator in non parametric regression for longitudinal data to simulation with various level of autocorrelation, as well as compare DM and TM approaches in predicting spline estimator in the data simulation with different of autocorrelation observational data on within subject. The results of the application are as follows: (a) implementation of smoothing spline with penalized weighted least square (PWLS) approach with or without consideration of autocorrelation in general (in all sizes and all error variances levels) provides significantly different spline estimator when the autocorrelation level >0.8; (b) based on size comparison, spline estimator in non parametric regression smoothing spline with PLS approach with (DM), or without (DM) consideration of autocorrelation showed significantly different result in level of autocorrelation > 0.8 (in overall size, moderate and large sample size), and > 0.7 (in small sample size); (c) based on level of variance, spline estimator in non parametric regression smoothing spline with PLS approach with (DM), or without (DM) consideration of autocorrelation showed significantly different result in level of autocorrelation > 0.8 (in overall level of variance, moderate and large variance), and > 0.7 (in small variance).