This paper deals with the problem of model selection for a general class of integer-valued time series.
We propose a penalized criterion based on the Poisson quasi-likelihood of the model.
Under certain regularity conditions, the consistency of the procedure as well as the consistency and the asymptotic normality of the Poisson quasi-likelihood estimator of the selected model are established.
Simulation experiments are conducted for some classical models such as Poisson, binary INGARCH and negative binomial model with nonlinear dynamic. Also, an application to a real dataset is provided.
We consider a nonparametric version of the integer-valued GARCH(1,1) model for time series of counts. The link function in the recursion for the variances is not specified by finite-dimensional parameters, but we impose nonparametric smoothness conditions. We propose a least squares estimator for this function and show that it is consistent with a rate that we conjecture to be nearly optimal.
In this article, we study a robust estimation method for a general class of integer-valued time series models. The conditional distribution of the process belongs to a broad class of distribution and unlike classical autoregressive framework, the conditional mean of the process also depends on some multivariate exogenous covariate. We derive a robust inference procedure based on the minimum density power divergence. Under certain regularity conditions, we establish that the proposed estimator is consistent and asymptotically normal. Simulation experiments are conducted to illustrate the empiri…
We consider the structural change in a class of discrete valued time series that the conditional distribution follows a one-parameter exponential family. We propose a change-point test based on the maximum likelihood estimator of the parameter of the model. Under the null hypothesis (of no change), the test statistics converges to a well known distribution, allowing for the calculation of the critical values of the test. The test statistic diverges to infinity under the alternative, that is, the test asymptotically has power one. Some simulation results and real data applications are reported …
This paper proposes a piecewise autoregression for general integer-valued time series. The conditional mean of the process depends on a parameter which is piecewise constant over time. We derive an inference procedure based on a penalized contrast that is constructed from the Poisson quasi-maximum likelihood of the model. The consistency of the proposed estimator is established. From practical applications, we derive a data-driven procedure based on the slope heuristic to calibrate the penalty term of the contrast; and the implementation is carried out through the dynamic programming algorithm…
Reduce-Rank Matrix Integer-Valued Autoregressive Model
School of Mathematics and Statistics, Shanxi University（山西大学数学与统计学院）；School of Applied Sciences, Taiyuan University of Science and Technology（太原科技大学应用科学学院）
This article introduces a general class of heavy-tailed autoregressions for modeling integer-valued time series with outliers. The proposed specification is based on a heavy-tailed mixture of negative binomial distributions that features an observation-driven dynamic equation for the conditional expectation. The existence of a unique stationary and ergodic solution for the class of autoregressive processes is shown under a general contraction condition. The estimation of the model can be easily performed by Maximum Likelihood given the closed form of the likelihood function. The strong consist…
Outlying observations are commonly encountered in the analysis of time series. In this paper the problem of detecting additive outliers in integer-valued time series is considered. We show how Gibbs sampling can be used to detect outlying observations in INAR(1) processes. The methodology proposed is illustrated using examples as well as an observed data set.
An extension of the RINAR(1) process for modelling discrete-time dependent counting processes is considered. The model RINAR(p) investigated here is a direct and natural extension of the real AR(p) model. Compared to classical INAR(p) models based on the thinning operator, the new models have several advantages: simple innovation structure ; autoregressive coefficients with arbitrary signs ; possible negative values for time series ; possible negative values for the autocorrelation function. The conditions for the stationarity and ergodicity, of the RINAR(p) model, are given. For parameter est…
A new class of integer-valued autoregressive models with dynamic survival probability is proposed. The peculiarity of this class of models lies on the specification of the survival probability through a stochastic recurrence equation. The estimation of the model can be performed by maximum likelihood and the consistency of the estimator is proved in a misspecified model setting. The flexibility of the proposed specification is illustrated in a simulation study. An application to a time series of crime reports is presented. The results show how the dynamic survival probability can enhance both …
This paper develops likelihood-based methods for estimation, inference, model selection, and forecasting of continuous-time integer-valued trawl processes. The full likelihood of integer-valued trawl processes is, in general, highly intractable, motivating the use of composite likelihood methods, where we consider the pairwise likelihood in lieu of the full likelihood. Maximizing the pairwise likelihood of the data yields an estimator of the parameter vector of the model, and we prove consistency and, in the short memory case, asymptotic normality of this estimator. When the underlying trawl p…
In various situations in the insurance industry, in finance, in epidemiology, etc., one needs to represent the joint evolution of the number of occurrences of an event. In this paper, we present a multivariate integer-valued autoregressive (MINAR) model, derive its properties and apply the model to earthquake occurrences across various pairs of tectonic plates. The model is an extension of Pedelis & Karlis (2011) where cross autocorrelation (spatial contagion in a seismic context) is considered. We fit various bivariate count models and find that for many contiguous tectonic plates, spatial co…
We study the problem of intervention effects generating various types of outliers in an integer-valued autoregressive model with Poisson innovations. We concentrate on outliers which enter the dynamics and can be seen as effects of extraordinary events. We consider three different scenarios, namely the detection of an intervention effect of a known type at a known time, the detection of an intervention effect of unknown type at a known time and the detection of an intervention effect when both the type and the time are unknown. We develop F-tests and score tests for the first scenario. For the…
In recent days different types of surveillance data are becoming available for public health reasons. In most cases several variables are monitored and events of different types are reported. As the amount of surveillance data increases, statistical methods that can effectively address multivariate surveillance scenarios are demanded. Even though research activity in this field is increasing rapidly in recent years, only a few approaches have simultaneously addressed the integer-valued property of the data and its correlation (both time correlation and cross correlation) structure. In this pap…
Measles presents a unique and imminent challenge for epidemiologists and public health officials: the disease is highly contagious, yet vaccination rates are declining precipitously in many localities. Consequently, the risk of a measles outbreak continues to rise. To improve preparedness, we study historical measles data both pre- and post-vaccine, and design new methodology to forecast measles counts with uncertainty quantification. We propose to model the disease counts as an integer-valued functional time series: measles counts are a function of time-of-year and time-ordered by year. The c…
INteger Auto-Regressive (INAR) processes are usually defined by specifying the innovations and the operator, which often leads to difficulties in deriving marginal properties of the process. In many practical situations, a major modeling limitation is that it is difficult to justify the choice of the operator. To overcome these drawbacks, we propose a new flexible approach to build an INAR model: we pre-specify the marginal and innovation distributions. Hence, the operator is a consequence of specifying the desired marginal and innovation distributions. Our new INAR model has both marginal and…
This paper considers non-negative integer-valued autoregressive processes where the autoregression parameter is close to unity. We consider the asymptotics of this `near unit root' situation. The local asymptotic structure of the likelihood ratios of the model is obtained, showing that the limit experiment is Poissonian. To illustrate the statistical consequences we discuss efficient estimation of the autoregression parameter and efficient testing for a unit root.
We consider together the retrospective and the sequential change-point detection in a general class of integer-valued time series. The conditional mean of the process depends on a parameter $θ^*$ which may change over time. We propose procedures which are based on the Poisson quasi-maximum likelihood estimator of the parameter, and where the updated estimator is computed without the historical observations in the sequential framework. For both the retrospective and the sequential detection, the test statistics converge to some distributions obtained from the standard Brownian motion under the …
Non-negative integer-valued semi-selfsimilar processes are introduced. Levy processes in this class are characterized. Its relation to an AR(1) scheme is derived.
In this paper, we consider the structural change in a class of discrete valued time series, which the true conditional distribution of the observations is assumed to be unknown. The conditional mean of the process depends on a parameter $θ^*$ which may change over time. We provide sufficient conditions for the consistency and the asymptotic normality of the Poisson quasi-maximum likelihood estimator (QMLE) of the model. We consider an epidemic change-point detection and propose a test statistic based on the QMLE of the parameter. Under the null hypothesis of a constant parameter (no change), t…
Remark.9 in Bose-Dasgupta-Rubin (2002) review states that when a non-negative integer-valued infinitely divisible law has an atom at unity then its support cannot have any gaps. Here one has two questions. (i) Why there are no gaps and (ii) Can there be gaps if the condition is not satisfied. Our investigation with these questions in mind centers on the implications of having and not having atoms at zero and unity. We give two examples/ constructions, which show that the remark needs modification and we modify it.
This paper introduces a Nearly Unstable INteger-valued AutoRegressive Conditional Heteroskedasticity (NU-INARCH) process for dealing with count time series data. It is proved that a proper normalization of the NU-INARCH process endowed with a Skorohod topology weakly converges to a Cox-Ingersoll-Ross diffusion. The asymptotic distribution of the conditional least squares estimator of the correlation parameter is established as a functional of certain stochastic integrals. Numerical experiments based on Monte Carlo simulations are provided to verify the behavior of the asymptotic distribution u…
Estimation of the covariance matrix of asset returns is crucial to portfolio construction. As suggested by economic theories, the correlation structure among assets differs between emerging markets and developed countries. It is therefore imperative to make rigorous statistical inference on correlation matrix equality between the two groups of countries. However, if the traditional vector-valued approach is undertaken, such inference is either infeasible due to limited number of countries comparing to the relatively abundant assets, or invalid due to the violations of temporal independence ass…
We provide a probabilistic and infinitesimal view of how the principal component analysis procedure (PCA) can be generalized to analysis of nonlinear manifold valued data. Starting with the probabilistic PCA interpretation of the Euclidean PCA procedure, we show how PCA can be generalized to manifolds in an intrinsic way that does not resort to linearization of the data space. The underlying probability model is constructed by mapping a Euclidean stochastic process to the manifold using stochastic development of Euclidean semimartingales. The construction uses a connection and bundles of covar…
Tensor-valued data benefits greatly from dimension reduction as the reduction in size is exponential in the number of modes. To achieve maximal reduction without loss in information, our objective in this work is to give an automated procedure for the optimal selection of the reduced dimensionality. Our approach combines a recently proposed data augmentation procedure with the higher-order singular value decomposition (HOSVD) in a tensorially natural way. We give theoretical guidelines on how to choose the tuning parameters and further inspect their influence in a simulation study. As our prim…
The probability density function (PDF) and cumulative distribution function of the sum of L independent but not necessarily identically distributed Gamma variates, applicable to the output statistics of maximal ratio combining (MRC) receiver operating over Nakagami-m fading channels or in other words to the statistical analysis of the scenario where the sum of squared Nakagami-m distributions are user-of-interest, is presented in closed-form in terms of well-known Meijer's G function and easily computable Fox's H-bar function for integer valued and non-integer valued m fading parameters. Furth…
