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31 pages, 1168 KiB

Open AccessArticle

A Seasonal Transmuted Geometric INAR Process: Modeling and Applications in Count Time Series

by Aishwarya Ghodake, Manik Awale, Hassan S. Bakouch, Gadir Alomair and Amira F. Daghestani

Mathematics 2025, 13(15), 2334; https://doi.org/10.3390/math13152334 - 22 Jul 2025

Viewed by 313

In this paper, the authors introduce the transmuted geometric integer-valued autoregressive model with periodicity, designed specifically to analyze epidemiological and public health time series data. The model uses a transmuted geometric distribution as a marginal distribution of the process. It also captures varying [...] Read more.

In this paper, the authors introduce the transmuted geometric integer-valued autoregressive model with periodicity, designed specifically to analyze epidemiological and public health time series data. The model uses a transmuted geometric distribution as a marginal distribution of the process. It also captures varying tail behaviors seen in disease case counts and health data. Key statistical properties of the process, such as conditional mean, conditional variance, etc., are derived, along with estimation techniques like conditional least squares and conditional maximum likelihood. The ability to provide k-step-ahead forecasts makes this approach valuable for identifying disease trends and planning interventions. Monte Carlo simulation studies confirm the accuracy and reliability of the estimation methods. The effectiveness of the proposed model is analyzed using three real-world public health datasets: weekly reported cases of Legionnaires’ disease, syphilis, and dengue fever. Full article

(This article belongs to the Special Issue Applied Statistics in Real-World Problems)

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23 pages, 401 KiB

Open AccessFeature PaperArticle

Combining Generalized Linear Autoregressive Moving Average and Bootstrap Models for Analyzing Time Series of Respiratory Diseases and Air Pollutants

by Ana Julia Alves Camara, Valdério Anselmo Reisen, Glaura Conceicao Franco and Pascal Bondon

Mathematics 2025, 13(5), 859; https://doi.org/10.3390/math13050859 - 5 Mar 2025

Viewed by 779

Abstract

The generalized linear autoregressive moving-average model (GLARMA) has been used in epidemiology to evaluate the impact of pollutants on health. These effects are quantified through the relative risk (RR) measure, which inference can be based on the asymptotic properties of the maximum likelihood [...] Read more.

The generalized linear autoregressive moving-average model (GLARMA) has been used in epidemiology to evaluate the impact of pollutants on health. These effects are quantified through the relative risk (RR) measure, which inference can be based on the asymptotic properties of the maximum likelihood estimator. However, for small series, this can be troublesome. This work studies different types of bootstrap confidence intervals (CIs) for the RR. The simulation study revealed that the model parameter related to the data’s autocorrelation could influence the intervals’ coverage. Problems could arise when covariates present an autocorrelation structure. To solve this, using the vector autoregressive (VAR) filter in the covariates is suggested. Full article

(This article belongs to the Special Issue Advances in Time Series Analysis and Forecasting)

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15 pages, 1718 KiB

Open AccessArticle

The Negative Binomial INAR(1) Process under Different Thinning Processes: Can We Separate between the Different Models?

by Dimitris Karlis, Naushad Mamode Khan and Yuvraj Sunecher

Stats 2024, 7(3), 793-807; https://doi.org/10.3390/stats7030048 - 27 Jul 2024

Viewed by 1413

Abstract

The literature on discrete valued time series is expanding very fast. Very often we see new models with very similar properties to the existing ones. A natural question that arises is whether the multitude of models with very similar properties can really have [...] Read more.

The literature on discrete valued time series is expanding very fast. Very often we see new models with very similar properties to the existing ones. A natural question that arises is whether the multitude of models with very similar properties can really have a practical purpose or if they mostly present theoretical interest. In the present paper, we consider four models that have negative binomial marginal distributions and are autoregressive in order 1 behavior, but they have a very different generating mechanism. Then we try to answer the question whether we can distinguish between them with real data. Extensive simulations show that while the differences are small, we still can discriminate between the models with relatively moderate sample sizes. However, the mean forecasts are expected to be almost identical for all models. Full article

(This article belongs to the Special Issue Modern Time Series Analysis II)

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16 pages, 480 KiB

Open AccessArticle

Higher-Order INAR Model Based on a Flexible Innovation and Application to COVID-19 and Gold Particles Data

by Fatimah E. Almuhayfith, Anuresha Krishna, Radhakumari Maya, Muhammad Rasheed Irshad, Hassan S. Bakouch and Munirah Almulhim

Axioms 2024, 13(1), 32; https://doi.org/10.3390/axioms13010032 - 31 Dec 2023

Viewed by 2200

Abstract

INAR models have the great advantage of being able to capture the conditional distribution of a count time series based on their past observations, thus allowing it to be tailored to meet the unique characteristics of count data. This paper reviews the two-parameter [...] Read more.

INAR models have the great advantage of being able to capture the conditional distribution of a count time series based on their past observations, thus allowing it to be tailored to meet the unique characteristics of count data. This paper reviews the two-parameter Poisson extended exponential (PEE) distribution and its corresponding INAR(1) process. Then the INAR of order p (INAR(p)) model that incorporates PEE innovations is proposed, its statistical properties are presented, and its parameters are estimated using conditional least squares and conditional maximum likelihood estimation methods. Two practical data sets are analyzed and compared with competing INAR models in an effort to gauge the performance of the proposed model. It is found that the proposed model performs better than the competitors. Full article

(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)

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17 pages, 396 KiB

Open AccessArticle

A Two-Step Estimation Method for a Time-Varying INAR Model

by Yuxin Pang, Dehui Wang and Mark Goh

Axioms 2024, 13(1), 19; https://doi.org/10.3390/axioms13010019 - 27 Dec 2023

Cited by 1 | Viewed by 1679

Abstract

This paper proposes a new time-varying integer-valued autoregressive (TV-INAR) model with a state vector following a logistic regression structure. Since the autoregressive coefficient in the model is time-dependent, the Kalman-smoothed method is applicable. Some statistical properties of the model are established. To estimate [...] Read more.

This paper proposes a new time-varying integer-valued autoregressive (TV-INAR) model with a state vector following a logistic regression structure. Since the autoregressive coefficient in the model is time-dependent, the Kalman-smoothed method is applicable. Some statistical properties of the model are established. To estimate the parameters of the model, a two-step estimation method is proposed. In the first step, the Kalman-smoothed estimation method, which is suitable for handling time-dependent systems and nonstationary stochastic processes, is utilized to estimate the time-varying parameters. In the second step, conditional least squares is used to estimate the parameter in the error term. This proposed method allows estimating the parameters in the nonlinear model and deriving the analytical solutions. The performance of the estimation method is evaluated through simulation studies. The model is then validated using actual time series data. Full article

(This article belongs to the Special Issue Time Series Analysis: Research on Data Modeling Methods)

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14 pages, 311 KiB

Open AccessArticle

New One-Parameter Over-Dispersed Discrete Distribution and Its Application to the Nonnegative Integer-Valued Autoregressive Model of Order One

by Muhammed Rasheed Irshad, Sreedeviamma Aswathy, Radhakumari Maya and Saralees Nadarajah

Mathematics 2024, 12(1), 81; https://doi.org/10.3390/math12010081 - 26 Dec 2023

Cited by 5 | Viewed by 1576

Abstract

Count data arise in inference, modeling, prediction, anomaly detection, monitoring, resource allocation, evaluation, and performance measurement. This paper focuses on a one-parameter discrete distribution obtained by compounding the Poisson and new X-Lindley distributions. The probability-generating function, moments, skewness, kurtosis, and other properties are [...] Read more.

Count data arise in inference, modeling, prediction, anomaly detection, monitoring, resource allocation, evaluation, and performance measurement. This paper focuses on a one-parameter discrete distribution obtained by compounding the Poisson and new X-Lindley distributions. The probability-generating function, moments, skewness, kurtosis, and other properties are derived in the closed form. The maximum likelihood method, method of moments, least squares method, and weighted least squares method are used for parameter estimation. A simulation study is carried out. The proposed distribution is applied as the innovation in an INAR(1) process. The importance of the proposed model is confirmed through the analysis of two real datasets. Full article

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29 pages, 2475 KiB

Open AccessArticle

Consistent Model Selection Procedure for Random Coefficient INAR Models

by Kaizhi Yu and Tielai Tao

Entropy 2023, 25(8), 1220; https://doi.org/10.3390/e25081220 - 16 Aug 2023

Cited by 1 | Viewed by 1527

Abstract

In the realm of time series data analysis, information criteria constructed on the basis of likelihood functions serve as crucial instruments for determining the appropriate lag order. However, the intricate structure of random coefficient integer-valued time series models, which are founded on thinning [...] Read more.

In the realm of time series data analysis, information criteria constructed on the basis of likelihood functions serve as crucial instruments for determining the appropriate lag order. However, the intricate structure of random coefficient integer-valued time series models, which are founded on thinning operators, complicates the establishment of likelihood functions. Consequently, employing information criteria such as AIC and BIC for model selection becomes problematic. This study introduces an innovative methodology that formulates a penalized criterion by utilizing the estimation equation within conditional least squares estimation, effectively addressing the aforementioned challenge. Initially, the asymptotic properties of the penalized criterion are derived, followed by a numerical simulation study and a comparative analysis. The findings from both theoretical examinations and simulation investigations reveal that this novel approach consistently selects variables under relatively relaxed conditions. Lastly, the applications of this method to infectious disease data and seismic frequency data produce satisfactory outcomes. Full article

(This article belongs to the Special Issue Information Theoretic Criteria: New Theoretical Developments and Applications)

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30 pages, 1502 KiB

Open AccessArticle

An Observation-Driven Random Parameter INAR(1) Model Based on the Poisson Thinning Operator

by Kaizhi Yu and Tielai Tao

Entropy 2023, 25(6), 859; https://doi.org/10.3390/e25060859 - 27 May 2023

Cited by 3 | Viewed by 1876

Abstract

This paper presents a first-order integer-valued autoregressive time series model featuring observation-driven parameters that may adhere to a particular random distribution. We derive the ergodicity of the model as well as the theoretical properties of point estimation, interval estimation, and parameter testing. The [...] Read more.

This paper presents a first-order integer-valued autoregressive time series model featuring observation-driven parameters that may adhere to a particular random distribution. We derive the ergodicity of the model as well as the theoretical properties of point estimation, interval estimation, and parameter testing. The properties are verified through numerical simulations. Lastly, we demonstrate the application of this model using real-world datasets. Full article

(This article belongs to the Special Issue Discrete-Valued Time Series)

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25 pages, 424 KiB

Open AccessArticle

Ruin Analysis on a New Risk Model with Stochastic Premiums and Dependence Based on Time Series for Count Random Variables

by Lihong Guan and Xiaohong Wang

Entropy 2023, 25(4), 698; https://doi.org/10.3390/e25040698 - 21 Apr 2023

Cited by 1 | Viewed by 2577

Abstract

In this paper, we propose a new discrete-time risk model of an insurance portfolio with stochastic premiums, in which the temporal dependence among the premium numbers of consecutive periods is fitted by the first-order integer-valued autoregressive (INAR(1)) process and the temporal dependence among [...] Read more.

In this paper, we propose a new discrete-time risk model of an insurance portfolio with stochastic premiums, in which the temporal dependence among the premium numbers of consecutive periods is fitted by the first-order integer-valued autoregressive (INAR(1)) process and the temporal dependence among the claim numbers of consecutive periods is described by the integer-valued moving average (INMA(1)) process. To measure the risk of the model quantitatively, we study the explicit expression for a function whose solution is defined as the Lundberg adjustment coefficient and give the Lundberg approximation formula for the infinite-time ruin probability. In the case of heavy-tailed claim sizes, we establish the asymptotic formula for the finite-time ruin probability via the large deviations of the aggregate claims. Two numerical examples are provided in order to illustrate our theoretical findings. Full article

(This article belongs to the Special Issue Discrete-Valued Time Series)

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25 pages, 926 KiB

Open AccessArticle

Zero-and-One Integer-Valued AR(1) Time Series with Power Series Innovations and Probability Generating Function Estimation Approach

by Vladica S. Stojanović, Hassan S. Bakouch, Eugen Ljajko and Najla Qarmalah

Mathematics 2023, 11(8), 1772; https://doi.org/10.3390/math11081772 - 7 Apr 2023

Cited by 3 | Viewed by 2386

Abstract

Zero-and-one inflated count time series have only recently become the subject of more extensive interest and research. One of the possible approaches is represented by first-order, non-negative, integer-valued autoregressive processes with zero-and-one inflated innovations, abbr. ZOINAR(1) processes, introduced recently, around the year 2020 [...] Read more.

Zero-and-one inflated count time series have only recently become the subject of more extensive interest and research. One of the possible approaches is represented by first-order, non-negative, integer-valued autoregressive processes with zero-and-one inflated innovations, abbr. ZOINAR(1) processes, introduced recently, around the year 2020 to the present. This manuscript presents a generalization of ZOINAR processes, given by introducing the zero-and-one inflated power series (ZOIPS) distributions. Thus, the obtained process, named the ZOIPS-INAR(1) process, has been investigated in terms of its basic stochastic properties (e.g., moments, correlation structure and distributional properties). To estimate the parameters of the ZOIPS-INAR(1) model, in addition to the conditional least-squares (CLS) method, a recent estimation technique based on probability-generating functions (PGFs) is discussed. The asymptotic properties of the obtained estimators are also examined, as well as their Monte Carlo simulation study. Finally, as an application of the ZOIPS-INAR(1) model, a dynamic analysis of the number of deaths from the disease COVID-19 in Serbia is considered. Full article

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15 pages, 500 KiB

Open AccessEditor’s ChoiceArticle

Time Series of Counts under Censoring: A Bayesian Approach

by Isabel Silva, Maria Eduarda Silva, Isabel Pereira and Brendan McCabe

Entropy 2023, 25(4), 549; https://doi.org/10.3390/e25040549 - 23 Mar 2023

Cited by 1 | Viewed by 1968

Abstract

Censored data are frequently found in diverse fields including environmental monitoring, medicine, economics and social sciences. Censoring occurs when observations are available only for a restricted range, e.g., due to a detection limit. Ignoring censoring produces biased estimates and unreliable statistical inference. The [...] Read more.

Censored data are frequently found in diverse fields including environmental monitoring, medicine, economics and social sciences. Censoring occurs when observations are available only for a restricted range, e.g., due to a detection limit. Ignoring censoring produces biased estimates and unreliable statistical inference. The aim of this work is to contribute to the modelling of time series of counts under censoring using convolution closed infinitely divisible (CCID) models. The emphasis is on estimation and inference problems, using Bayesian approaches with Approximate Bayesian Computation (ABC) and Gibbs sampler with Data Augmentation (GDA) algorithms. Full article

(This article belongs to the Special Issue Discrete-Valued Time Series)

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12 pages, 2979 KiB

Open AccessArticle

Bayesian Analysis of Spatial Model for Frequency of Tornadoes

by Haitao Zheng, Yi Zhang, Qiaoju Chen, Qingshan Yang, Guoqing Huang, Dahai Wang and Ruili Liu

Atmosphere 2023, 14(3), 472; https://doi.org/10.3390/atmos14030472 - 27 Feb 2023

Viewed by 1773

Abstract

Frequency analysis of tornadoes is very important for risk analysis and disaster control. In this paper, the annual frequency of tornadoes that occurred in the United States from 1967 to 2016 is analyzed. The simple analysis shows that frequencies of tornadoes of different [...] Read more.

Frequency analysis of tornadoes is very important for risk analysis and disaster control. In this paper, the annual frequency of tornadoes that occurred in the United States from 1967 to 2016 is analyzed. The simple analysis shows that frequencies of tornadoes of different sites are spatially correlated and over-dispersed. To explain the two characteristics of the data, the Bayesian hierarchical model is proposed. For comparison purposes, the Bayesian model with negative binomial distribution, Poisson distribution, Polya distribution, and first-order, non-negative, integer-valued autoregressive model with Bell innovations(BL-INAR(1)) are considered to fit the frequency of tornado. The distribution parameters of all sites are assumed to be spatially correlated, and the corresponding Bayesian hierarchical models were established. MCMC (Markov Chain Monte Carlo) method is applied to parameter estimations and relative statistical inference. By comparison of the analysis results, the negative binomial distribution is recommended to analyze the overdispersion and spatial correlation among the sites of the data. The comparison between the simulated frequencies based on the proposed model and the actual frequencies also verifies that the proposed method is a better model for the data. Full article

(This article belongs to the Special Issue Advances in Computational Wind Engineering and Wind Energy)

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21 pages, 482 KiB

Open AccessFeature PaperArticle

Partial Autocorrelation Diagnostics for Count Time Series

by Christian H. Weiß, Boris Aleksandrov, Maxime Faymonville and Carsten Jentsch

Entropy 2023, 25(1), 105; https://doi.org/10.3390/e25010105 - 4 Jan 2023

Cited by 10 | Viewed by 2599

Abstract

In a time series context, the study of the partial autocorrelation function (PACF) is helpful for model identification. Especially in the case of autoregressive (AR) models, it is widely used for order selection. During the last decades, the use of AR-type count processes, [...] Read more.

In a time series context, the study of the partial autocorrelation function (PACF) is helpful for model identification. Especially in the case of autoregressive (AR) models, it is widely used for order selection. During the last decades, the use of AR-type count processes, i.e., which also fulfil the Yule–Walker equations and thus provide the same PACF characterization as AR models, increased a lot. This motivates the use of the PACF test also for such count processes. By computing the sample PACF based on the raw data or the Pearson residuals, respectively, findings are usually evaluated based on well-known asymptotic results. However, the conditions for these asymptotics are generally not fulfilled for AR-type count processes, which deteriorates the performance of the PACF test in such cases. Thus, we present different implementations of the PACF test for AR-type count processes, which rely on several bootstrap schemes for count times series. We compare them in simulations with the asymptotic results, and we illustrate them with an application to a real-world data example. Full article

(This article belongs to the Special Issue Discrete-Valued Time Series)

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18 pages, 1153 KiB

Open AccessArticle

Effect of Fuzzy Time Series on Smoothing Estimation of the INAR(1) Process

by Mahmoud El-Morshedy, Mohammed H. El-Menshawy, Mohammed M. A. Almazah, Rashad M. El-Sagheer and Mohamed S. Eliwa

Axioms 2022, 11(9), 423; https://doi.org/10.3390/axioms11090423 - 24 Aug 2022

Cited by 4 | Viewed by 2012

Abstract

In this paper, the effect of fuzzy time series on estimates of the spectral, bispectral and normalized bispectral density functions are studied. This study is conducted for one of the integer autoregressive of order one (INAR(1)) models. The model of interest here is [...] Read more.

In this paper, the effect of fuzzy time series on estimates of the spectral, bispectral and normalized bispectral density functions are studied. This study is conducted for one of the integer autoregressive of order one (INAR(1)) models. The model of interest here is the dependent counting geometric INAR(1) which is symbolized by (DCGINAR(1)). A realization is generated for this model of size n = 500 for estimation. Based on fuzzy time series, the forecasted observations of this model are obtained. The estimators of spectral, bispectral and normalized bispectral density functions are smoothed by different one- and two-dimensional lag windows. Finally, after the smoothing, all estimators are studied in the case of generated and forecasted observations of the DCGINAR(1) model. We investigate the contribution of the fuzzy time series to the smoothing of these estimates through the results. Full article

(This article belongs to the Special Issue Statistical Methods and Applications)

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22 pages, 481 KiB

Open AccessArticle

A New Bivariate INAR(1) Model with Time-Dependent Innovation Vectors

by Huaping Chen, Fukang Zhu and Xiufang Liu

Stats 2022, 5(3), 819-840; https://doi.org/10.3390/stats5030048 - 19 Aug 2022

Cited by 7 | Viewed by 2286

Abstract

Recently, there has been a growing interest in integer-valued time series models, especially in multivariate models. Motivated by the diversity of the infinite-patch metapopulation models, we propose an extension to the popular bivariate INAR(1) model, whose innovation vector is assumed to be time-dependent [...] Read more.

Recently, there has been a growing interest in integer-valued time series models, especially in multivariate models. Motivated by the diversity of the infinite-patch metapopulation models, we propose an extension to the popular bivariate INAR(1) model, whose innovation vector is assumed to be time-dependent in the sense that the mean of the innovation vector is linearly increased by the previous population size. We discuss the stationarity and ergodicity of the observed process and its subprocesses. We consider the conditional maximum likelihood estimate of the parameters of interest, and establish their large-sample properties. The finite sample performance of the estimator is assessed via simulations. Applications on crime data illustrate the model. Full article

(This article belongs to the Section Time Series Analysis)

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