Goodness-of-fit tests for binomial AR(1) processes

The binomial AR(1) model describes a nonlinear process with a first-order autoregressive (AR(1)) structure and a binomial marginal distribution. To develop goodness-of-fit tests for the binomial AR(1) model, we investigate the observed marginal distribution of the binomial AR(1) process, and we tackle its autocorrelation structure. Motivated by the family of power-divergence statistics for handling discrete multivariate data, we derive the asymptotic distribution of certain categorized power-divergence statistics for the case of a binomial AR(1) process. Then we consider Bartlett's formula, which is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocorrelations, but which is not applicable when the underlying process is nonlinear. Hence, we derive a novel Bartlett-type formula for the asymptotic distribution of the sample autocorrelations of a binomial AR(1) process, which is then applied to develop tests concerning the autocorrelation structure. Simulation studies are carried out to evaluate the size and power of the proposed tests under diverse alternative process models. Several real examples are used to illustrate our methods and findings.     

First order autoregressive time series with negative binomial and geometric marginals

In this paper we present first order autoregressive (AR(1)) time series with negative binomial and geometric marginals. These processes are the discrete analogues of the gamma and exponential processes introduced by Sim (1990). Many properties of the processes discussed here, such as autocorrelation, regression and joint distributions, are studied.     

Binomial AR(1) processes with innovational outliers

Abstract

Binomial integer-valued AR processes have been well studied in the literature, but there is little progress in modeling bounded integer-valued time series with outliers. In this paper, we first review some basic properties of the binomial integer-valued AR(1) process and then we introduce binomial integer-valued AR(1) processes with two classes of innovational outliers. We focus on the joint conditional least squares (CLS) and the joint conditional maximum likelihood (CML) estimates of models’ parameters and the probability of occurrence of the outlier. Their large-sample properties are illustrated by simulation studies. Artificial and real data examples are used to demonstrate good performances of the proposed models.     

An INAR(1) model based on a mixed dependent and independent counting series

A mixed integer-valued autoregressive model of order one, based on the binomial and the generalized binomial thinning operator is introduced. Geometric marginal distribution is considered. Properties of the model are analysed, unknown parameters are estimated and some numerical results of the estimates are obtained. Finally, model is applied on two real data sets and compared to some relevant models.     

Monitoring correlated processes with binomial marginals

Few approaches for monitoring autocorrelated attribute data have been proposed in the literature. If the marginal process distribution is binomial, then the binomial AR(1) model as a realistic and well-interpretable process model may be adequate. Based on known and newly derived statistical properties of this model, we shall develop approaches to monitor a binomial AR(1) process, and investigate their performance in a simulation study. A case study demonstrates the applicability of the binomial AR(1) model and of the proposed control charts to problems from statistical process control.     

A class of observation-driven random coefficient INAR(1) processes based on negative binomial thinning

Integer-valued time series models and their applications have attracted a lot of attention over the last years. In this paper, we introduce a class of observation-driven random coefficient integer-valued autoregressive processes based on negative binomial thinning, where the autoregressive parameter depends on the observed values of the previous moment. Basic probability and statistics properties of the process are established. The unknown parameters are estimated by the conditional least squares and empirical likelihood methods. Specially, we consider three aspects of the empirical likelihood method: maximum empirical likelihood estimate, confidence region and EL test. The performance of the two estimation methods is compared through simulation studies. Finally, an application to a real data example is provided.     

Binomial AR(1) processes: moments,cumulants, and estimation

The modelling and analysis of count-data time series are areas of emerging interest with various applications in practice. We consider the particular case of the binomial AR(1) model, which is well suited for describing binomial counts with a first-order autoregressive serial dependence structure. We derive explicit expressions for the joint (central) moments and cumulants up to order 4. Then, we apply these results for expressing moments and asymptotic distribution of the squared difference estimator as an alternative to the sample autocovariance. We also analyse the asymptotic distribution of the conditional least-squares estimators of the parameters of the binomial AR(1) model. The finite-sample performance of these estimators is investigated in a simulation study, and we apply them to real data about computerized workstations.     

Slow-explosive AR(1) processes converging to random walk

Abstract

This article investigates slow-explosive AR(1) processes, which converge to a random walk (RW) process with logarithm rates, to fill the gap between nearly non-stationary AR(1) and moderately deviated AR(1) processes, and derives the asymptotics of the least squares estimator using central limit theorems for (reduced) U-statistic. We successfully establish the smooth link between the nearly non-stationary AR(1) and the moderately deviated AR(1) processes. Some novel results are reported, which include the convergence of the least squares estimator to a biased fractional Brownian motion.     

A First-Order Spatial Integer-Valued Autoregressive SINAR(1, 1) Model

Binomial thinning operator has a major role in modeling one-dimensional integer-valued autoregressive time series models. The purpose of this article is to extend the use of such operator to define a new stationary first-order spatial non negative, integer-valued autoregressive SINAR(1, 1) model. We study some properties of this model like the mean, variance and autocorrelation function. Yule-Walker estimator of the model parameters is also obtained. Some numerical results of the model are presented and, moreover, this model is applied to a real data set.     

Zero-Inflated NGINAR(1) process

In this paper, we develop a zero-inflated NGINAR(1) process as an alternative to the NGINAR(1) process (Risti?, Nasti?, and Bakouch 2009 Risti?, M. M., A. S. Nasti?, and H. S. Bakouch. 2009. A new geometric first-order integer-valued autoregressive (NGINAR(1)) process. Journal of Statistical Planning and Inference 139:2218–26.[Crossref], [Web of Science ®] , [Google Scholar]) when the number of zeros in the data is larger than the expected number of zeros by the geometric process. The proposed process has zero-inflated geometric marginals and contains the NGINAR(1) process as a particular case. In addition, various properties of the new process are derived such as conditional distribution and autocorrelation structure. Yule-Walker, probability based Yule-Walker, conditional least squares and conditional maximum likelihood estimators of the model parameters are derived. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Forecasting performances of the model are discussed. Application to a real data set shows the flexibility and potentiality of the new model.     

Fully observed INAR(1) processes

The innovations of an INAR(1) process (integer-valued autoregressive) are usually assumed to be unobservable. There are, however, situations in practice, where also the innovations can be uncovered, i.e. where we are concerned with a fully observed INAR(1) process. We analyze stochastic properties of such a fully observed INAR(1) process and explore the relation between the INAR(1) model and certain metapopulation models. We show how the additional knowledge about the innovations can be used for parameter estimation, for model diagnostics, and for forecasting. Our findings are illustrated with two real-data examples.     

Empirical likelihood estimation for linear regression models with AR(p) error terms with numerical examples

Linear regression models are useful statistical tools to analyze data sets in different fields. There are several methods to estimate the parameters of a linear regression model. These methods usually perform under normally distributed and uncorrelated errors. If error terms are correlated the Conditional Maximum Likelihood (CML) estimation method under normality assumption is often used to estimate the parameters of interest. The CML estimation method is required a distributional assumption on error terms. However, in practice, such distributional assumptions on error terms may not be plausible. In this paper, we propose to estimate the parameters of a linear regression model with autoregressive error term using Empirical Likelihood (EL) method, which is a distribution free estimation method. A small simulation study is provided to evaluate the performance of the proposed estimation method over the CML method. The results of the simulation study show that the proposed estimators based on EL method are remarkably better than the estimators obtained from CML method in terms of mean squared errors (MSE) and bias in almost all the simulation configurations. These findings are also confirmed by the results of the numerical and real data examples.     

Detection and Estimation of Jump Points in Non parametric Regression Function with AR(1) Noise

In this paper, we propose a method based on wavelet analysis to detect and estimate jump points in non parametric regression function. This method is applied to AR(1) noise process under random design. First, the test statistics are constructed on the empirical wavelet coefficients. Then, under the null hypothesis, the critical values of test statistics are obtained. Under the alternative, the consistency of the test is proved. Afterward, the rate of convergence, the estimators of the number, and locations of change points are given theoretically. Finally, the excellent performance of our method is demonstrated through simulations using artificial and real datasets.     

Self-exciting threshold models for time series of counts with a finite range

In this article, an integer-valued self-exciting threshold model with a finite range based on the binomial INARCH(1) model is proposed. Important stochastic properties are derived, and approaches for parameter estimation are discussed. A real-data example about the regional spread of public drunkenness in Pittsburgh demonstrates the applicability of the new model in comparison to existing models. Feasible modifications of the model are presented, which are designed to handle special features such as zero-inflation.     

A generalised NGINAR(1) process with inflated‐parameter geometric counting series

In this paper we propose a new stationary first‐order non‐negative integer valued autoregressive process with geometric marginals based on a generalised version of the negative binomial thinning operator. In this manner we obtain another process that we refer to as a generalised stationary integer‐valued autoregressive process of the first order with geometric marginals. This new process will enable one to tackle the problem of overdispersion inherent in the analysis of integer‐valued time series data, and contains the new geometric process as a particular case. In addition various properties of the new process, such as conditional distribution, autocorrelation structure and innovation structure, are derived. We discuss conditional maximum likelihood estimation of the model parameters. We evaluate the performance of the conditional maximum likelihood estimators by a Monte Carlo study. The proposed process is fitted to time series of number of weekly sales (economics) and weekly number of syphilis cases (medicine) illustrating its capabilities in challenging cases of highly overdispersed count data.     

A comparison of LS/ML and GMM estimation in a simple AR(1) model

This paper compares least squares (LS)/maximum likelihood (ML) and generalised method of moments (GMM) estimation in a simple. Gaussian autoregressive of order one (AR(1)) model. First, we show that the usual LS/ML estimator is a corner solution to a general minimisation problem that involves two moment conditions, while the new GMM we devise is not. Secondly, we examine asymptotic and finite sample properties of the new GMM estimator in comparison to the usual LS/ML estimator in a simple AR(1) model. For both stable and unstable (unit root) specifications, we show asymptotic equivalence of the distributions of the two estimators. However, in finite samples, the new GMM estimator performs better.     

Error autocorrelation revisited: the AR(1) case

The aim of the paper is to consider the implicit restrictions imposed when adopting an AR(1) error term in the context of the linear regression model. It is shown that these restrictions amount to assuming a largely identical temporal structure for all the variables involved in the specification. Implicit in this is the assumption that these variables are mutually Granger non-causal. The main implication of this result is that in most cases when residual autocorrelation is detected boththe OLS and GLS estimators are biased and inconsistent.