Asymptotic behaviour of estimators of the parameters of nearly unstable INAR(1) models

A sequence of first-order integer-valued autoregressive type (INAR(1)) processes is investigated, where the autoregressive type coefficients converge to 1. It is shown that the limiting distribution of the joint conditional least squares estimators for this coefficient and for the mean of the innovation is normal. Consequences for sequences of Galton{Watson branching processes with unobservable immigration, where the mean of the offspring distribution converges to 1 (which is the critical value), are discussed

Wavelet-Based Detection of Outliers in Poisson INAR(1) Time Series

The presence of outliers or discrepant observations has a negative impact in time series modelling. This paper considers the problem of detecting outliers, additive or innovational, single, multiple or in patches, in count time series modelled by first-order Poisson integer-valued autoregressive, PoINAR(1), models. To address this problem, two wavelet-based approaches that allow the identification of the time points of outlier occurrence are proposed. The effectiveness of the proposed methods is illustrated with synthetic as well as with an observed dataset

Some autoregressive moving average processes with generalized Poisson marginal distributions

Generalized Poisson process, regression, time reversibility, quasi-binomial distribution, quasi-multinomial distribution, vector AR(1) process,

Ordering probability distributions by tail behavior

We introduce, what we call, "the exponent function of a probability distribution" which can be used for ordering probability distributions according to their tail behavior. Some properties of this function as well as the related "failure rate quantile function" are discussed. An ordering among probability distributions based on the right tail exponent of Parzen (1979) is introduced and called "Parzen ordering". Some properties of this ordering along with examples are given.convex ordering dispersive ordering tail classification order statistic Parzen ordering

GQL Versus Conditional GQL Inferences for Non-Stationary Time Series of Counts with Overdispersion

This article proposes an autoregressive model for time series of counts with non-stationary means, variances and covariances as functions of certain time-dependant covariates. For the estimation of the regression, overdispersion and correlation index parameters, a conditional generalized quasilikelihood (CGQL) approach is developed under the assumption that the count responses marginally satisfy the first two moments of a negative binomial distribution. Thus this CGQL approach avoids the use of the likelihood or so-called partial likelihood of the data which are known to be extremely complicated in the present non-stationary time series set-up. It is shown through an extensive simulation study that the proposed CGQL approach performs very well in estimating the parameters of the model. This is also shown that the CGQL approach performs better than an existing GQL approach, especially for the estimation of the overdispersion parameter of the model. Copyright 2008 The Authors

Asymptotic inference for nearly unstable INAR(1) models

A sequence of first-order integer-valued autoregressive (INAR(1)) processes is investigated, where the autoregressive-type coefficient converges to 1. It is shown that the limiting distribution of the conditional least squares estimator for this coefficient is normal and the rate of convergence is n(3/2). Nearly critical Galton-Watson processes with unobservable immigration are also discussed