الفهرس 
Chapter 1يوجد فقط 14 صفحة متاحة للعرض العام
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المستخلص
”Time series analysis is a notable tool for analyzing time-dependent data. Interest in modelling discrete-valued time series models has grown significantly in recent decades, leading to the introduction of various models tailored for stationary processes with discrete marginal distributions. One driving factor behind this trend is the common occurrence of time series data characterized by small integer values. For instance, the number of customers in a department store per day, monthly number of cases of some disease, the number of thunderstorms in a day, the weekly sales of a specific soap product in a supermarket, the monthly records of crimes and offenses cases, the daily COVID-19 count cases and deaths, the number of different IP addresses, and the daily number of road traffic accidents. The goal is to develop a model for time series data of integers to can mimic the attributes of the continuous-valued autoregressive (AR) model. The integer-valued autoregressive (INAR) with Poisson marginal was introduced by McKenzie (1985) and Al-Osh and Alzaid (1987), utilizing the binomial thinning operator that Steutel and van Harn (1979) proposed. While the Poisson INAR(1) model exhibits the property of equi-dispersion, it falls short in accurately representing many real-world datasets characterized by over-dispersion or under-dispersion. Consequently, the Poisson INAR(1) model does not always serve as an adequate framework for modeling time series data with integer values. In light of this, there has been a growing development about various models that better account for the phenomena of over-dispersion or under-dispersion in recent literature. Modifications to the INAR(1) model have been made by altering the distribution of the innovation term of this model by discrete marginals with various probabilistic properties. Qi et al. (2019) implemented INAR(1) with innovations following zero and one Poisson distributions; Sharafi et al. (2023) employed zero-modified Poisson-Lindley innovations within the model; and Bourguignon et al. (2019) expanded the model to include innovations following double Poisson and generalized Poisson distributions, serving datasets exhibiting over-dispersion or under-dispersion. Another common approach involves altering the thinning operator within the INAR(1) framework. Weiss (2018) reviewed various thinning operators, including random coefficient thinning, iterated thinning, and quasi-binomial thinning, among others. Ristić et al. (2009) proposed the negative binomial thinning operator, utilizing it to develop an over-dispersed integer-valued time series model with a geometric marginal distribution. In the recent years, several first-order models for non-negative integer-valued time series have been developed. However, in certain situations, truncated models, which are integer-valued time series excluding zeros, prove to be more suitable. In the field of count data time series, truncated models have not been extensively explored, and there are few published papers to this subject. As aforementioned, count time series is a recurring subject in the scientific literature due to its applicability to several real-life situations observed and collected with the time. Motivated by the count time series’ application, this thesis introduces some flexible integer-valued time series models with statistical analysis to those models. This thesis is divided into five Chapters, a brief review of them can be seen below. In Chapter 1, we summarize some basic terms, fundamental concepts, notations, and definitions of time series, such as stationarity, the autocovariance function, the autocorrelation function, linear time series models, autoregressive models (AR), moving average models (MA), the mixed autoregressive moving average model (ARMA). A brief presentation of non-negative integer valued time series models of order one (INAR(1)) and thinning operators are reviewed. We summarize some types of thinning operators based on independent and dependent counting series that used for modelling non-negative data counts. Consequently, we provide a summary and review of some INAR(1) models based on the considered thinning like INAR(1) and mixture Pegram thinning with Poisson, geometric, negative binomial marginals. Parameter estimation (such as Yule-Walker, conditional least squares, maximum likelihood), identification, assessing adequacy and diagnostic tools of the model are reviewed. In Chapter 2, we propose a first-order, non-negative, integer-valued autoregressive (INAR(1)) model with one-misrecorded Poisson (OMP) marginal via a combination of the generalized binomial thinning and mixture Pegram operators for zero inflated and one deflated count time series. The suggested model is suitable for multimodal, equi- and over-dispersed data modeling. It contains two particular cases: the mixture of Pegram and thinning of a first-order integer-valued autoregressive (MPT(1)) with Poisson and one-misrecorded Poisson. The distribution of the innovation term is derived as a mixture of degenerate distributions at 0, 1, and two Poisson distributions with certain parameters. Regression and several statistical properties of the proposed model are discussed. We investigate the distribution of the runs (the lengths of zeros and ones). The parameters of the model are estimated using the ML, MYW, and MCLS methods. The estimation of the parameters, their behavior, and their performance are presented through a simulation study. Two practical data sets on the monthly cases of criminal records and weekly sales are applied to check the proposed process’s performance against other relevant INAR(1) models, showing its capabilities in the challenging case of over-dispersed count data. Furthermore, the proposed model discusses data forecasting. Under-dispersion and over-dispersion in count time series data are frequently observed. As a result, in Chapter 3, we introduce a first-order, non-negative integer-valued autoregressive process with intervened geometric innovations (INARIG(1)) based on the binomial thinning operator. It contains the zero-truncated geometric process as a submodel. This model can be used to describe zero-truncated count time series with all modes of dispersion. The main statistical and conditional properties of the proposed process are obtained. The model’s parameters are estimated using the conditional maximum-likelihood, conditional least-squares, and Yule-Walker methodologies, and the asymptotic properties of the estimators are also obtained. In order to evaluate the consistency and efficiency of the estimators under the three estimating methodologies, Monte Carlo simulation experiments are conducted. The suggested model is fitted to a time series of the daily death counts from COVID-19 and a weekly sale of soap to demonstrate its potential for challenging over-dispersed and under-dispersed count data. The INARIG(1) model is suitable for modelling COVID-19 and weekly sales data, according to model adequacy verification with Pearson residuals, predictive capability, etc. For the mentioned data sets, the INARIG(1) model takes into account a number of forecasting techniques, including the classical, round classical, coherent median, and modified sieve bootstrap methods. Chapter 4: This Chapter introduces a first-order integer-valued autoregressive (INAR(1)) process with size-biased Poisson-Lindley innovations, utilizing a mix of the Pegram and binomial thinning operators. This model is apt for modelling zero-truncated count time series with over-, equi-, and under-dispersion, and can be represented as a random coefficient process. The main statistical and conditional properties of the proposed process are obtained. A stationary solution and its uniqueness are discussed, along with an investigation into the strict stationarity and ergodicity of the process. The model’s parameters are estimated using the conditional maximum likelihood and Yule-Walker methodologies. In order to evaluate the consistency and efficiency of the estimators under the two estimating methodologies, Monte Carlo simulations are conducted. To assess the effectiveness of the proposed process, we utilize two truncated data sets at zero, focusing on daily COVID-19 death counts from Kenya and El Salvador. A comparison is made between the suggested model and several competing INAR(1) models, showcasing its capability to handle both over-dispersed and under-dispersed count data effectively. The proposed model is suitable for modeling the considered data sets according to a set of goodness-of-fit measures. Based on the modified sieve bootstrapping and rounding the classical predictors, we provide integer forecasts of the future death of COVID-19. In Chapter 5, the main results that are gained in the Chapters 1-4 are summarized and concluded. Some further development, recommendations, and challenges for future work are introduced with some hints.