Asymptotic normality of error density estimator in stationary and explosive autoregressive models

In this paper,we consider the limit distribution of the error density function estima-tor in the rst-order autoregressive models with negatively associated and positively associated random errors.Under mild regularity assumptions,some asymptotic normality results of the residual density estimator are obtained when the autoregressive models are stationary process and explosive process.In order to illustrate these results,some simulations such as con dence intervals and mean integrated square errors are provided in this paper.It shows that the residual density estimator can replace the density\estimator"which contains errors.

Trend Autoregressive Model Exact Run Length Evaluation on a Two-Sided Extended EWMA Chart

The Extended Exponentially Weighted Moving Average(extended EWMA)control chart is one of the control charts and can be used to quickly detect a small shift.The performance of control charts can be evaluated with the average run length(ARL).Due to the deriving explicit formulas for the ARL on a two-sided extended EWMA control chart for trend autoregressive or trend AR(p)model has not been reported previously.The aim of this study is to derive the explicit formulas for the ARL on a two-sided extended EWMA con-trol chart for the trend AR(p)model as well as the trend AR(1)and trend AR(2)models with exponential white noise.The analytical solution accuracy was obtained with the extended EWMA control chart and was compared to the numer-ical integral equation(NIE)method.The results show that the ARL obtained by the explicit formula and the NIE method is hardly different,but the explicit for-mula can help decrease the computational(CPU)time.Furthermore,this is also expanded to comparative performance with the Exponentially Weighted Moving Average(EWMA)control chart.The performance of the extended EWMA control chart is better than the EWMA control chart for all situations,both the trend AR(1)and trend AR(2)models.Finally,the analytical solution of ARL is applied to real-world data in the healthfield,such as COVID-19 data in the United Kingdom and Sweden,to demonstrate the efficacy of the proposed method.

JUMP DETECTION BY WAVELET IN NONLINEAR AUTOREGRESSIVE MODELS

Wavelets are applied to detection of the jump points of a regression function in nonlinear autoregressive model x(t) = T(x(t-1)) + epsilon t. By checking the empirical wavelet coefficients of the data,which have significantly large absolute values across fine scale levels, the number of the jump points and locations where the jumps occur are estimated. The jump heights are also estimated. All estimators are shown to be consistent. Wavelet method ia also applied to the threshold AR(1) model(TAR(1)). The simple estimators of the thresholds are given,which are shown to be consistent.

Empirical likelihood for first-order mixed integer-valued autoregressive model

In this paper, we not only construct the confidence region for parameters in a mixed integer-valued autoregressive process using the empirical likelihood method, but also establish the empirical log-likelihood ratio statistic and obtain its limiting distribution. And then, via simulation studies we give coverage probabilities for the parameters of interest. The results show that the empirical likelihood method performs very well.

The m-delay Autoregressive Model with Application

The classical autoregressive(AR)model has been widely applied to predict future data usingmpast observations over five decades.As the classical AR model required m unknown parameters,this paper implements the AR model by reducing m parameters to two parameters to obtain a new model with an optimal delay called as the m-delay AR model.We derive the m-delay AR formula for approximating two unknown parameters based on the least squares method and develop an algorithm to determine optimal delay based on a brute-force technique.The performance of them-delay AR model was tested by comparing with the classical AR model.The results,obtained from Monte Carlo simulation using the monthly mean minimum temperature in PerthWestern Australia from the Bureau of Meteorology,are no significant difference compared to those obtained from the classical AR model.This confirms that the m-delay AR model is an effective model for time series analysis.

Bootstrap Approaches to Autoregressive Model on Exchange Rates Currency

The use of historical data is important in making the predictions, for instance in the exchange rate. However, in the construction of a model, extreme data or dirtiness of data is inevitable. In this study, AR model is used with the exchange rate historical data (January 2007 until December 2007) for USD/MYR and is divided into 1-, 3- and 6-horizontal months respectively. Since the presence of extreme data will affect the accuracy of the results obtained in a prediction. Therefore, to obtain a more accurate prediction results, the bootstrap approach was implemented by hybrid with AR model coins as the Bootstrap Autoregressive model (BAR). The effectiveness of the proposed model is investigated by comparing the existing and the proposed model through the statistical performance methods which are RMSE, MAE and MAD. The comparison involves 1%, 5% and 10% for each horizontal month. The results showed that the BAR model performed better than the AR model in terms of sensitivity to extreme data, the accuracy of forecasting models, efficiency and predictability of the model prediction. In conclusion, bootstrap method can alleviate the sensitivity of the model to the extreme data, thereby improving the accuracy of forecasting model which also have high prediction efficiency and that can increase the predictability of the model.

AUTOREGRESSIVE MODEL AND POWER SPECTRUM CHARATERISTICS OF CURRENT SIGNAL IN HIGH FREQUENCY GROUP PULSE MICRO-ELECTROCHEMICAL MACHINING

The identification of the inter-electrode gap size in the high frequency group pulse micro-electrochemical machining （HGPECM） is mainly discussed. The auto-regressive（AR） model of group pulse current flowing across the cathode and the anode are created under different situations with different processing parameters and inter-electrode gap size. The AR model based on the current signals indicates that the order of the AR model is obviously different relating to the different processing conditions and the inter-electrode gap size; Moreover, it is different about the stability of the dynamic system, i.e. the white noise response of the Green＇s function of the dynamic system is diverse. In addition, power spectrum method is used in the analysis of the dynamic time series about the current signals with different inter-electrode gap size, the results show that there exists a strongest power spectrum peak, characteristic power spectrum（CPS）, to the current signals related to the different inter-electrode gap size in the range of 0～5 kHz. Therefore, the CPS of current signals can implement the identification of the inter-electrode gap.

Asymptotic Inference in the Random Coefficient Autoregressive Model with Time-functional Variance Noises

This paper considers the random coefficient autoregressive model with time-functional variance noises,hereafter the RCA-TFV model.We first establish the consistency and asymptotic normality of the conditional least squares estimator for the constant coefficient.The semiparametric least squares estimator for the variance of the random coefficient and the nonparametric estimator for the variance function are constructed,and their asymptotic results are reported.A simulation study is presented along with an analysis of real data to assess the performance of our method in finite samples.

Statistical Inference of Partially Linear Spatial Autoregressive Model Under Constraint Conditions

In many application fields of regression analysis,prior information about how explanatory variables affect response variable of interest is often available and can be formulated as constraints on regression coefficients.In this paper,the authors consider statistical inference of partially linear spatial autoregressive model under constraint conditions.By combining series approximation method,twostage least squares method and Lagrange multiplier method,the authors obtain constrained estimators of the parameters and function in the partially linear spatial autoregressive model and investigate their asymptotic properties.Furthermore,the authors propose a testing method to check whether the parameters in the parametric component of the partially linear spatial autoregressive model satisfy linear constraint conditions,and derive asymptotic distributions of the resulting test statistic under both null and alternative hypotheses.Simulation results show that the proposed constrained estimators have better finite sample performance than the unconstrained estimators and the proposed testing method performs well in finite samples.Furthermore,a real example is provided to illustrate the application of the proposed estimation and testing methods.

Polynomial network autoregressive models with divergent orders

We propose a novel polynomial network autoregressive model by incorporating higher-order connected relationships to simultaneously model the effects of both direct and indirect connections. A quasimaximum likelihood estimation method is proposed to estimate the unknown influence parameters, and we demonstrate its consistency and asymptotic normality without imposing any distribution assumption. Moreover,an extended Bayesian information criterion is set for order selection with a divergent upper order. The application of the proposed polynomial network autoregressive model is demonstrated through both the simulation and the real data analysis.

Constructing Confidence Regions for Autoregressive-Model Parameters

We discuss formulas and techniques for finding maximum-likelihood estimators of parameters of autoregressive (with particular emphasis on Markov and Yule) models, computing their asymptotic variance-covariance matrix and displaying the resulting confidence regions;Monte Carlo simulation is then used to establish the accuracy of the corresponding level of confidence. The results indicate that a direct application of the Central Limit Theorem yields errors too large to be acceptable;instead, we recommend using a technique based directly on the natural logarithm of the likelihood function, verifying its substantially higher accuracy. Our study is then extended to the case of estimating only a subset of a model’s parameters, when the remaining ones (called nuisance) are of no interest to us.

Bayesian Estimation and Model Selection for the Spatiotemporal Autoregressive Model with Autoregressive Conditional Heteroscedasticity Errors

The spatial and spatiotemporal autoregressive conditional heteroscedasticity(STARCH) models receive increasing attention. In this paper, we introduce a spatiotemporal autoregressive(STAR) model with STARCH errors, which can capture the spatiotemporal dependence in mean and variance simultaneously. The Bayesian estimation and model selection are considered for our model. By Monte Carlo simulations, it is shown that the Bayesian estimator performs better than the corresponding maximum-likelihood estimator, and the Bayesian model selection can select out the true model in most times. Finally, two empirical examples are given to illustrate the superiority of our models in fitting those data.

A Score Type Test for General Autoregressive Models in Time Series

This paper is devoted to the goodness-of-fit test for the general autoregressive models in time series. By averaging for the weighted residuals, we construct a score type test which is asymptotically standard chi-squared under the null and has some desirable power properties under the alternatives. Specifically, the test is sensitive to alternatives and can detect the alternatives approaching, along a direction, the null at a rate that is arbitrarily close to n-1/2. Furthermore, when the alternatives are not directional, we construct asymptotically distribution-free maximin tests for a large class of alternatives. The performance of the tests is evaluated through simulation studies.

JPEG stream soft-decoding technique based on autoregressive modeling

This paper introduces a new model-based soft decoding techniqt, e to restore the widely used joint photographic expert group （JPEG） streams. The image is modeled as a two dimensional （2D） piecewise stationary autoregressive process, and the decoding task is formulated as a constrained optimization problem. All the constraints are given by the quantization intervals which available at the decoder freely. The autoregressive model serves as an important regularization term of the objective function of the optimization, and the model parameters are solved on the decoded image locally using a weighted total least square method. In addition, a novel bilateral dualside weighting scheme is proposed to minimize the influence of the blocking artifact on the accuracy of parameter estimation. Extensive experimental results suggest that the proposed algorithm systematically improves the quality of JPEG images and also outperforms existing JPEG postprocessing algorithms in a wide bit-rate range both in terms of peak signal-to-noise ratio （PSNR） and subjective quality

Inference for Spatial Autoregressive Models with Infinite Variance Noises

A self-weighted quantile procedure is proposed to study the inference for a spatial unilateral autoregressive model with independent and identically distributed innovations belonging to the domain of attraction of a stable law with index of stabilityα,α∈(0,2).It is shown that when the model is stationary,the self-weighted quantile estimate of the parameter has a closed form and converges to a normal limiting distribution,which avoids the difficulty of Roknossadati and Zarepour(2010)in deriving their limiting distribution for an M-estimate.On the contrary,we show that when the model is not stationary,the proposed estimates have the same limiting distributions as those of Roknossadati and Zarepour.Furthermore,a Wald test statistic is proposed to consider the test for a linear restriction on the parameter,and it is shown that under a local alternative,the Wald statistic has a non-central chisquared distribution.Simulations and a real data example are also reported to assess the performance of the proposed method.

Empirical Likelihood-Based Subset Selection for Partially Linear Autoregressive Models

Based on the empirical likelihood method, the subset selection and hypothesis test for parameters in a partially linear autoregressive model are investigated. We show that the empirical log-likelihood ratio at the true parameters converges to the standard chi-square distribution. We then present the definitions of the empirical likelihood-based Bayes information criteria （EBIC） and Akaike information criteria （EAIC）. The results show that EBIC is consistent at selecting subset variables while EAIC is not. Simulation studies demonstrate that the proposed empirical likelihood confidence regions have better coverage probabilities than the least square method, while EBIC has a higher chance to select the true model than EAIC.

Empirical likelihood inference in autoregressive models with time-varying variances

This paper develops the empirical likelihood(EL)inference procedure for parameters in autore-gressive models with the error variances scaled by an unknown nonparametric time-varying function.Compared with existing methods based on non-parametric and semi-parametric esti-mation,the proposed test statistic avoids estimating the variance function,while maintaining the asymptotic chi-square distribution under the null.Simulation studies demonstrate that the proposed EL procedure(a)is more stable,i.e.,depending less on the change points in the error variances,and(b)gets closer to the desired confidence level,than the traditional test statistic.

MODELING AND PREDICTION CONCERNING TIME SERIES OF FLOOD/DROUGHT RUNS USING THE SELF-EXCITING THRESHOLD AUTOREGRESSIVE MODEL

When linear regressive models such as AR or ARMA model are used for fitting and predicting climatic time series,results are often not sufficiently good because nonlinear variations in the time series.In this paper, a nonlinear self-exciting threshold autoregressive(SETAR)model is applied to modeling and predicting the time series of flood/drought runs in Beijing,which were derived from the graded historical flood/drought records in the last 511 years(1470—1980).The results show that the modeling and predicting with the SETAR model are much better than that of the AR model.The latter can predict the flood/drought runs with a length only less than two years,while the formal can predict more than three-year length runs.This may be due to the fact that the SETAR model can renew the model according to the run-turning points in the process of predic- tion,though the time series is nonstationary.

THE PROBABILISTIC PROPERTIES OF THE NONLINEAR AUTOREGRESSIVE MODEL WITH CONDITIONAL HETEROSKEDASTICITY

In this paper we examine the geometric ergodicities under fairly wide conditions for the following nonlinear autoregressive model

PARTICLE FILTERING BASED AUTOREGRESSIVE CHANNEL PREDICTION MODEL

A particle filtering based AutoRegressive (AR) channel prediction model is presented for cognitive radio systems. Firstly, this paper introduces the particle filtering and the system model. Secondly, the AR model of order p is used to approximate the flat Rayleigh fading channels; its stability is discussed, and an algorithm for solving the AR model parameters is also given. Finally, an AR channel prediction model based on particle filtering and second-order AR model is presented. Simulation results show that the performance of the proposed AR channel prediction model based on particle filtering is better than that of Kalman filtering.