Conditional heteroscedasticity test for Poisson autoregressive model

In this article, the problem of interest is testing the conditional heteroscedasticity of Poisson autoregressive model. We construct a non parametric test statistic based on empirical likelihood method. The asymptotic distribution of the proposed statistic is derived and its finite-sample property is examined through Monte Carlo simulations. The simulation results show that the proposed method is good for practical use.     

Variable selection and estimation for high-dimensional spatial autoregressive models

Spatial regression models are important tools for many scientific disciplines including economics, business, and social science. In this article, we investigate postmodel selection estimators that apply least squares estimation to the model selected by penalized estimation in high-dimensional regression models with spatial autoregressive errors. We show that by separating the model selection and estimation process, the postmodel selection estimator performs at least as well as the simultaneous variable selection and estimation method in terms of the rate of convergence. Moreover, under perfect model selection, the ℓ₂ rate of convergence is the oracle rate of $\sqrt{s/n}$, compared with the convergence rate of $\text{◂√▸}\sqrt{s\log p/n}$ in the general case. Here, n is the sample size and p, s are the model dimension and number of significant covariates, respectively. We further provide the convergence rate of the estimation error in the form of $\sup$ norm, and ideally the rate can reach as fast as $\text{◂√▸}\sqrt{\log s/n}$.     

Minimax A‐ and D‐optimal integer‐valued wavelet designs for estimation

The author discusses integer‐valued designs for wavelet estimation of nonparametric response curves in the possible presence of heteroscedastic noise using a modified wavelet version of the Gasser‐Müller kernel estimator or weighted least squares estimation. The designs are constructed using a minimax treatment and the simulated annealing algorithm. The author presents designs for three case studies in experiments for investigating Gompertz's theory on mortality rates, nitrite utilization in bush beans and the impact of crash helmets in motorcycle accidents.     

A diagnostic statistic for functional-coefficient autoregressive models

In this paper, functional coefficient autoregressive (FAR) models proposed by Chen and Tsay (1993) are considered. We propose a diagnostic statistic for FAR models constructed by comparing between parametric and nonparametric estimators of the functional form of the FAR models. We show asymptotic properties of our statistic mathematically and it can be applied to the estimation of the delay parameter and the specification of the functional form of FAR models.     

Bent‐cable regression with autoregressive noise

Motivated by time series of atmospheric concentrations of certain pollutants the authors develop bent‐cable regression for autocorrelated errors. Bent‐cable regression extends the popular piecewise linear (broken‐stick) model, allowing for a smooth change region of any non‐negative width. Here the authors consider autoregressive noise added to a bent‐cable mean structure, with unknown regression and time series parameters. They develop asymptotic theory for conditional least‐squares estimation in a triangular array framework, wherein each segment of the bent cable contains an increasing number of observations while the autoregressive order remains constant as the sample size grows. They explore the theory in a simulation study, develop implementation details, apply the methodology to the motivating pollutant dataset, and provide a scientific interpretation of the bent‐cable change point not discussed previously. The Canadian Journal of Statistics 38: 386–407; 2010 © 2010 Statistical Society of Canada     

Adaptive order selection for autoregressive models

Autoregressive model is a popular method for analysing the time dependent data, where selection of order parameter is imperative. Two commonly used selection criteria are the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), which are known to suffer the potential problems regarding overfit and underfit, respectively. To our knowledge, there does not exist a criterion in the literature that can satisfactorily perform under various situations. Therefore, in this paper, we focus on forecasting the future values of an observed time series and propose an adaptive idea to combine the advantages of AIC and BIC but to mitigate their weaknesses based on the concept of generalized degrees of freedom. Instead of applying a fixed criterion to select the order parameter, we propose an approximately unbiased estimator of mean squared prediction errors based on a data perturbation technique for fairly comparing between AIC and BIC. Then use the selected criterion to determine the final order parameter. Some numerical experiments are performed to show the superiority of the proposed method and a real data set of the retail price index of China from 1952 to 2008 is also applied for illustration.     

Parameter change test for zero-inflated generalized Poisson autoregressive models

In this paper, we consider the problem of testing for parameter change in zero-inflated generalized Poisson (ZIGP) autoregressive models. We verify that the ZIGP process is stationary and ergodic and that the conditional maximum likelihood estimator (CMLE) is strongly consistent and asymptotically normal. Based on these results, we construct CMLE- and residual-based cumulative sum tests and show that their limiting null distributions are a function of independent Brownian bridges. The simulation results are provided for illustration. A real data analysis is performed on some crime data of Australia.     

Variable selection models based on multiple imputation with an application for predicting median effective dose and maximum effect

The statistical methods for variable selection and prediction could be challenging when missing covariates exist. Although multiple imputation (MI) is a universally accepted technique for solving missing data problem, how to combine the MI results for variable selection is not quite clear, because different imputations may result in different selections. The widely applied variable selection methods include the sparse partial least-squares (SPLS) method and the penalized least-squares method, e.g. the elastic net (ENet) method. In this paper, we propose an MI-based weighted elastic net (MI-WENet) method that is based on stacked MI data and a weighting scheme for each observation in the stacked data set. In the MI-WENet method, MI accounts for sampling and imputation uncertainty for missing values, and the weight accounts for the observed information. Extensive numerical simulations are carried out to compare the proposed MI-WENet method with the other competing alternatives, such as the SPLS and ENet. In addition, we applied the MI-WENet method to examine the predictor variables for the endothelial function that can be characterized by median effective dose (ED50) and maximum effect (Emax) in an ex-vivo phenylephrine-induced extension and acetylcholine-induced relaxation experiment.     

Best subset selection of autoregressive models with exogenous variables and generalized autoregressive conditional heteroscedasticity errors

Summary.  We develop an efficient way to select the best subset autoregressive model with exogenous variables and generalized autoregressive conditional heteroscedasticity errors. One main feature of our method is to select important autoregressive and exogenous variables, and at the same time to estimate the unknown parameters. The method proposed uses the stochastic search idea. By adopting Markov chain Monte Carlo techniques, we can identify the best subset model from a large of number of possible choices. A simulation experiment shows that the method is very effective. Misspecification in the mean equation can also be detected by our model selection method. In the application to the stock-market data of seven countries, the lagged 1 US return is found to have a strong influence on the other stock-market returns.     

Testing for variance changes in autoregressive models with unknown order

The problem of change point in autoregressive process is studied in this article. We propose a Bayesian information criterion-iterated cumulative sums of squares algorithm to detect the variance changes in an autoregressive series with unknown order. Simulation results and two examples are presented, where it is shown to have good performances when the sample size is relatively small.     

A wavelet-based time-varying autoregressive model for non-stationary and irregular time series

In this work we propose an autoregressive model with parameters varying in time applied to irregularly spaced non-stationary time series. We expand all the functional parameters in a wavelet basis and estimate the coefficients by least squares after truncation at a suitable resolution level. We also present some simulations in order to evaluate both the estimation method and the model behavior on finite samples. Applications to silicates and nitrites irregularly observed data are provided as well.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Variable selection in partially linear hazard regression for multivariate failure time data

The aim of this paper is to explore variable selection approaches in the partially linear proportional hazards model for multivariate failure time data. A new penalised pseudo-partial likelihood method is proposed to select important covariates. Under certain regularity conditions, we establish the rate of convergence and asymptotic normality of the resulting estimates. We further show that the proposed procedure can correctly select the true submodel, as if it was known in advance. Both simulated and real data examples are presented to illustrate the proposed methodology.     

Variable selection by ensembles for the Cox model

Using Cox regression as the main platform, we study the ensemble approach for variable selection. We use a popular real-data example as well as simulated data with various censoring levels to illustrate the usefulness of the ensemble approach, and study the nature of these ensembles in terms of their strength and diversity. By relating these characteristics to the ensemble's selection accuracy, we provide useful insights for how to choose among different ensemble strategies, as well as guidelines for thinking about how to design more effective ensembles.     

Bootstrap-based unit root tests for higher order autoregressive models with GARCH(1, 1) errors

ABSTRACT

Bootstrap-based unit root tests are a viable alternative to asymptotic distribution-based procedures and, in some cases, are preferable because of the serious size distortions associated with the latter tests under certain situations. While several bootstrap-based unit root tests exist for autoregressive moving average processes with homoskedastic errors, only one such test is available when the innovations are conditionally heteroskedastic. The details for the exact implementation of this procedure are currently available only for the first order autoregressive processes. Monte-Carlo results are also published only for this limited case. In this paper we demonstrate how this procedure can be extended to higher order autoregressive processes through a transformed series used in augmented Dickey–Fuller unit root tests. We also investigate the finite sample properties for higher order processes through a Monte-Carlo study. Results show that the proposed tests have reasonable power and size properties.     

Variable selection for semiparametric errors-in-variables regression model with longitudinal data

In this paper, we focus on the variable selection for the semiparametric regression model with longitudinal data when some covariates are measured with errors. A new bias-corrected variable selection procedure is proposed based on the combination of the quadratic inference functions and shrinkage estimations. With appropriate selection of the tuning parameters, we establish the consistency and asymptotic normality of the resulting estimators. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedure. We further illustrate the proposed procedure with an application.     

Asymptotic Behavior of Conditional Least Squares Estimators for Unstable Integer‐valued Autoregressive Models of Order 2

In this paper, the asymptotic behavior of the conditional least squares estimators of the autoregressive parameters, of the mean of the innovations, and of the stability parameter for unstable integer‐valued autoregressive processes of order 2 is described. The limit distributions and the scaling factors are different according to the following three cases: (i) decomposable, (ii) indecomposable but not positively regular, and (iii) positively regular models.     

High‐order Corrected Estimator of Asymptotic Variance with Optimal Bandwidth

Estimation of time‐average variance constant (TAVC), which is the asymptotic variance of the sample mean of a dependent process, is of fundamental importance in various fields of statistics. For frequentists, it is crucial for constructing confidence interval of mean and serving as a normalizing constant in various test statistics and so forth. For Bayesians, it is widely used for evaluating effective sample size and conducting convergence diagnosis in Markov chain Monte Carlo method. In this paper, by considering high‐order corrections to the asymptotic biases, we develop a new class of TAVC estimators that enjoys optimal ‐convergence rates under different degrees of the serial dependence of stochastic processes. The high‐order correction procedure is applicable to estimation of the so‐called smoothness parameter, which is essential in determining the optimal bandwidth. Comparisons with existing TAVC estimators are comprehensively investigated. In particular, the proposed optimal high‐order corrected estimator has the best performance in terms of mean squared error.     

Conditional Akaike Information Criteria for a Class of Poisson Mixture Models with Random Effects

Focusing on the model selection problems in the family of Poisson mixture models (including the Poisson mixture regression model with random effects and zero‐inflated Poisson regression model with random effects), the current paper derives two conditional Akaike information criteria. The criteria are the unbiased estimators of the conditional Akaike information based on the conditional log‐likelihood and the conditional Akaike information based on the joint log‐likelihood, respectively. The derivation is free from the specific parametric assumptions about the conditional mean of the true data‐generating model and applies to different types of estimation methods. Additionally, the derivation is not based on the asymptotic argument. Simulations show that the proposed criteria have promising estimation accuracy. In addition, it is found that the criterion based on the conditional log‐likelihood demonstrates good model selection performance under different scenarios. Two sets of real data are used to illustrate the proposed method.     

Variable selection for semiparametric varying coefficient partially linear model based on modal regression with missing data

Abstract

In this article, we focus on the variable selection for semiparametric varying coefficient partially linear model with response missing at random. Variable selection is proposed based on modal regression, where the non parametric functions are approximated by B-spline basis. The proposed procedure uses SCAD penalty to realize variable selection of parametric and nonparametric components simultaneously. Furthermore, we establish the consistency, the sparse property and asymptotic normality of the resulting estimators. The penalty estimation parameters value of the proposed method is calculated by EM algorithm. Simulation studies are carried out to assess the finite sample performance of the proposed variable selection procedure.