Extended binomial AR(1) processes with generalized binomial thinning operator

Abstract

In this article, we introduce an extended binomial AR(1) model based on the generalized binomial thinning operator. This operator relaxes the independence assumption of the binomial thinning operator and contains dependent Bernoulli counting series. The new model contains the binomial AR(1) model as a particular case. Some probabilistic and statistical properties are explored. Estimators of the model parameters are derived by conditional maximum likelihood (CML), conditional least squares (CLS) and weighted conditional least squares (WCLS) methods. Some asymptotic properties and numerical results of the estimators are studied. The good performance of the new model is illustrated, among other competitive models in the literature, by an application to the monthly drunken driving counts.     

First-order random coefficients integer-valued threshold autoregressive processes

In this paper, we introduce a first-order random coefficient integer-valued threshold autoregressive process, which is based on binomial thinning. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares and conditional maximum likelihood estimators are derived for both the cases that the threshold variable is known or not. The asymptotic properties of the estimators are established. Moreover, forecasting problem is addressed. Finally, some numerical results of the estimates and a real data example are presented.     

A new geometric INAR(1) process based on counting series with deflation or inflation of zeros

In this paper, we introduce a new non-negative integer-valued autoregressive time series model based on a new thinning operator, so called generalized zero-modified geometric (GZMG) thinning operator. The first part of the paper is devoted to the distribution, GZMG distribution, which is obtained as the convolution of the zero-modified geometric (ZMG) distributed random variables. Some properties of this distribution are derived. Then, we construct a thinning operator based on the counting processes with ZMG distribution. Finally, an INAR(1) time series model is introduced and its properties including estimation issues are derived and discussed. A small Monte Carlo experiment is conducted to evaluate the performance of maximum likelihood estimators in finite samples. At the end of the paper, we consider an empirical illustration of the introduced INAR(1) model.     

A geometric time series model with a new dependent Bernoulli counting series

ABSTRACT

New generalized binomial thinning operator with dependent counting series is introduced. An integer valued time series model with geometric marginals based on this thinning operator is constructed. Main features of the process are analyzed and determined. Estimation of the parameters are presented and some asymptotic properties of the obtained estimators are discussed. Behavior of the estimators is described through the numerical results. Also, model is applied on the real data set and compared to some relevant INAR(1) 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.     

An extension on INAR models with discrete Laplace marginal distributions

The main theme considered in this article is an integer-valued thinning operator with both positive and negative values, its properties, and a new time series with skew discrete Laplace marginals. Some properties of this model are discussed, as well as estimators of unknown parameters, similarities and differences with some other existing models, applications in real-life situations, and identification and approximation of latent processes affecting the concerning process.     

Estimation in a bivariate integer-valued autoregressive process

ABSTRACT

A bivariate integer-valued autoregressive time series model is presented. The model structure is based on binomial thinning. The unconditional and conditional first and second moments are considered. Correlation structure of marginal processes is shown to be analogous to the ARMA(2, 1) model. Some estimation methods such as the Yule–Walker and conditional least squares are considered and the asymptotic distributions of the obtained estimators are derived. Comparison between bivariate model with binomial thinning and bivariate model with negative binomial thinning is given.     

A new geometric first-order integer-valued autoregressive (NGINAR(1)) process

A new stationary first-order integer-valued autoregressive process with geometric marginal distributions is introduced based on negative binomial thinning. Some properties of the process are established. Estimators of the parameters of the process are obtained using the methods of conditional least squares, Yule–Walker and maximum likelihood. Also, the asymptotic properties of the estimators are derived involving their distributions. Some numerical results of the estimators are presented with a discussion to the obtained results. Real data are used and a possible application is discussed.     

A New Markov Binomial Distribution

In this article, we introduce a two-state homogeneous Markov chain and define a geometric distribution related to this Markov chain. We define also the negative binomial distribution similar to the classical case and call it NB related to interrupted Markov chain. The new binomial distribution is related to the interrupted Markov chain. Some characterization properties of the geometric distributions are given. Recursion formulas and probability mass functions for the NB distribution and the new binomial distribution are derived.     

Marshall–Olkin bivariate Weibull distributions and processes

In this paper we introduce a new probability model known as type 2 Marshall–Olkin bivariate Weibull distribution as an extension of type 1 Marshall–Olkin bivariate Weibull distribution of Marshall–Olkin (J Am Stat Assoc 62:30–44, 1967). Various properties of the new distribution are considered. Bivariate minification processes with the two types of Weibull distributions as marginals are constructed and their properties are considered. It is shown that the processes are strictly stationary. The unknown parameters of the type 1 process are estimated and their properties are discussed. Some numerical results of the estimates are also given.     

Minimax prediction in finite populations

In this paper, minimax predictors are considered for some population quantities of interest, under some specific superpopulation models. Minimax predictors are derived under a binomial superpopulation model. The well known ratio estimator is shown to be minimax under the usual simple regression normal model. Nonparametric minimax predictors are considered under some models. Some of the predictors considered also shown to be admis s ible.     

On the relationship between the matrix operators,vech and vecd

We introduce a matrix operator, which we call “vecd” operator. This operator stacks up “diagonals” of a symmetric matrix. This operator is more convenient for some statistical analyses than the commonly used “vech” operator. We show an explicit relationship between the vecd and vech operators. Using this relationship, various properties of the vecd operator are derived. As applications of the vecd operator, we derive concise and explicit expressions of the Wald and score tests for equal variances of a multivariate normal distribution and for the diagonality of variance coefficient matrices in a multivariate generalized autoregressive conditional heteroscedastic (GARCH) model, respectively.     

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.     

On compounded geometric distributions and their applications

Here, we introduce two-parameter compounded geometric distributions with monotone failure rates. These distributions are derived by compounding geometric distribution and zero-truncated Poisson distribution. Some statistical and reliability properties of the distributions are investigated. Parameters of the proposed distributions are estimated by the maximum likelihood method as well as through the minimum distance method of estimation. Performance of the estimates by both the methods of estimation is compared based on Monte Carlo simulations. An illustration with Air Crash casualties demonstrates that the distributions can be considered as a suitable model under several real situations.     

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.     

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 bivariate first-order signed integer-valued autoregressive process

Bivariate integer-valued time series occur in many areas, such as finance, epidemiology, business etc. In this article, we present bivariate autoregressive integer-valued time-series models, based on the signed thinning operator. Compared to classical bivariate INAR models, the new processes have the advantage to allow for negative values for both the time series and the autocorrelation functions. Strict stationarity and ergodicity of the processes are established. The moments and the autocovariance functions are determined. The conditional least squares estimator of the model parameters is considered and the asymptotic properties of the obtained estimators are derived. An analysis of a real dataset from finance and a simulation study are carried out to assess the performance of the model.     

On Shifted Geometric INAR(1) Models Based on Geometric Counting Series

Two types of shifted geometric integer valued autoregressive models of order one (SGINAR(1)) are proposed. Both are based on the thinning operator generated by counting series of i.i.d. geometric random variables. Their correlation properties are derived and compared. Also, regression and conditional variance are considered. Nonparametric estimators of model parameters are obtained and their asymptotic characterizations are given. Finally, these two models are applied to a real-life data set and they are compared to some referent INAR(1) models.     

Matrix variate Cauchy distribution

In this article we introduce the matrix variate Cauchy distribution. Its density function has been derived using independent random matrices having dependent normal entries. Some properties of this distribution are also studied.     

Extended Poisson INAR(1) processes with equidispersion,underdispersion and overdispersion

Real count data time series often show the phenomenon of the underdispersion and overdispersion. In this paper, we develop two extensions of the first-order integer-valued autoregressive process with Poisson innovations, based on binomial thinning, for modeling integer-valued time series with equidispersion, underdispersion, and overdispersion. The main properties of the models are derived. The methods of conditional maximum likelihood, Yule–Walker, and conditional least squares are used for estimating the parameters, and their asymptotic properties are established. We also use a test based on our processes for checking if the count time series considered is overdispersed or underdispersed. The proposed models are fitted to time series of the weekly number of syphilis cases and monthly counts of family violence illustrating its capabilities in challenging the overdispersed and underdispersed count data.