First-order Autoregressive Process Research Articles

Almost Sure Central Limit Theorem for Error Variance Estimator in Pth-Order Nonlinear Autoregressive Processes

In this paper, under some suitable assumptions, using the Taylor expansion, Borel–Cantelli lemma and the almost sure central limit theorem for independent random variables, the almost sure central limit theorem for error variance estimator in the pth-order nonlinear autoregressive processes with independent and identical distributed errors was established. Four examples, first-order autoregressive processes, self-exciting threshold autoregressive processes, threshold-exponential AR progresses and multilayer perceptrons progress, are given to verify the results.

Evaluation of metrics and thresholds for use in national-scale river harmful algal bloom assessments

The spatiotemporal distribution of harmful algal blooms (HABs) in rivers remains poorly understood, and there is an urgent need to develop a consistent set of metrics to better document HAB occurrences and forecast future events. Using data from seven sites in the Illinois River Basin, we computed metrics focused on HAB conditions related to excess algal growth and hypoxia. Daily mean chlorophyll and dissolved oxygen (DO) concentrations, gross primary productivity (GPP), and net ecosystem productivity (NEP) rates, focused on water quality status, identifying the timing of the transition from a clear-water to an algal dominated state. Early warning indicators (EWIs), the first-order autoregressive process (Ar1) and standard deviation (SD) of chlorophyll concentrations, focused on future events, forecasting blooms. Metrics were compared to either literature-derived or statistical-based thresholds and were normalized by total number of daily samples for an exceedance rate. Exceedances of a daily mean chlorophyll concentration averaged 50 % across all sites using a 10 µg L−1 threshold but increasing the threshold to 50 μg L−1 reduced the average exceedance rate to 5 %. The average exceedance rate for GPP (∼8 g O2 m2d−1 threshold) was 15 %, similar to the daily amplitude DO concentration (∼3 mg L−1 threshold), but the average for NEP (0 g O2 m2 d−1 threshold) was higher, at 28 %. The number of days with at least 1 continuous DO concentration below the threshold of 5, 3, or 2 mg L−1, had basin wide exceedance rates of 9 %, 3 %, and 2 %, respectively. Thresholds for EWIs, Ar1 and SD, were exceeded at 5 of the 7 sites with high chlorophyll concentrations and GPP rates. The correlation between proxies for algal biomass (chlorophyll concentration) and productivity (GPP) was strongest for sites in the middle region of the basin, with R2 values between 0.54 and 0.74. Although, cyanotoxin concentrations are the most commonly used metrics by states to define an inland water HAB, there is a paucity of publicly available data. The wider availability of chlorophyll and oxygen data combined with the results from this study suggest that biomass and productivity state and event-based metrics may be a promising way to assess and predict the vulnerability of rivers to some of the deleterious effects of HABs at broad spatial scales.

The implications of economic freedom and gender ideologies on women's opportunity-to-necessity entrepreneurship

PurposeThis study explores the relationship between economic freedom and gender ideologies on the allocation of women’s opportunity-to-necessity entrepreneurship across countries. Opportunity entrepreneurship is typically understood as one’s best option for work, whereas necessity entrepreneurship describes the choice as driven by no better option for work. Specifically, we examine how economic freedom (i.e. each country’s policies that facilitate voluntary exchange) and gender ideologies (i.e. each country’s propensity for gendered separate spheres) affect the distribution of women’s opportunity-to-necessity entrepreneurship across countries.Design/methodology/approachWe construct our sample by matching data from the following country-level sources: the Global Entrepreneurship Monitor’s Adult Population Survey (APS), the Fraser Institute’s Economic Freedom Index (EFI), the European/World Value Survey’s Integrated Values Survey (IVS) gender equality index, and other covariates from the IVS, Varieties of Democracy (V-dem) World Bank (WB) databases. Our final sample consists of 729 observations from 109 countries between 2006 and 2018. Entrepreneurial activity motivations are measured by the ratio of the percentage of women’s opportunity-driven total nascent and early-stage entrepreneurship to the percentage of female necessity-driven total nascent and early-stage entrepreneurship at the country level. Due to a first-order autoregressive process and heteroskedastic cross-sectional dependence in our panel, we estimate a fixed-effect regression with robust standard errors clustered by country.FindingsAfter controlling for multiple macro-level factors, we find two interesting findings. First, economic freedom positively affects the ratio of women’s opportunity-to-necessity entrepreneurship. We find that the size of government, sound money, and business and credit regulations play the most important role in shaping the distribution of contextual motivations over time and between countries. However, this effect appears to benefit efficiency and innovation economies more than factor economies in our sub-sample analysis. Second, gender ideologies of political equality positively affect the ratio of women’s opportunity-to-necessity entrepreneurship, and this effect is most pronounced for efficiency economies.Originality/valueThis study offers one critical contribution to the entrepreneurship literature by demonstrating how economic freedom and gender ideologies shape the distribution of contextual motivation for women’s entrepreneurship cross-culturally. We answer calls to better understand the variation within women’s entrepreneurship instead of comparing women’s and men’s entrepreneurial activity. As a result, our study sheds light on how structural aspects of societies shape the allocation of women’s entrepreneurial motivations through their institutional arrangements.

Randomness Test of Thinning Parameters for the NBRCINAR(1) Process

Non-negative integer-valued time series are usually encountered in practice, and a variety of integer-valued autoregressive processes based on various thinning operators are commonly used to model these count data with temporal dependence. In this paper, we consider a first-order integer-valued autoregressive process constructed by the negative binomial thinning operator with random coefficients, to address the problem of constant thinning parameters which might not always accurately represent real-world settings because of numerous external and internal causes. We estimate the model parameters of interest by the two-step conditional least squares method, obtain the asymptotic behaviors of the estimators, and furthermore devise a technique to test the constancy of the thinning parameters, which is essential for determining whether or not the proposed model should consider the parameters’ randomness. The effectiveness and dependability of the suggested approach are illustrated by a series of thorough simulation studies. Finally, two real-world data analysis examples reveal that the suggested approach is very useful and flexible for applications.

INAR(1) process with weighted negative binomial Lindley distributed innovations and applications to criminal and COVID-19 data

In this study, we introduce a pliant stationary first-order integer-valued autoregressive (INAR) process with weighted negative binomial Lindley innovations. The main properties of the model are derived. The methods of conditional maximum likelihood, conditional least square and Yule-Walker are used for estimating the process parameters, while the efficiency of these three methods is evaluated through a simulation study. Finally, the practical aspect of the proposed INAR(1) process is discussed on two time series of the monthly number of criminal mischief reports in Pittsburgh and the daily COVID-19 death cases in Paraguay.

On Sample Size Needed for Block Bootstrap Confidence Intervals to Have Desired Coverage Rates

Block bootstrap is widely used in constructing confidence intervals for parameters estimated from stationary time series. Theoretically, the method should provide valid confidence intervals as the length of the time series goes to infinity. In practice, however, it is necessary to know how large of a finite sample is required for block bootstrap confidence intervals to work well. This study aims to answer this question in a simple simulation setting where the data are generated from a first-order autoregressive process. The empirical coverage rates of several commonly used bootstrap confidence intervals for the mean, standard deviation, and the lag-1 autocorrelation coefficient are compared. A quite large sample is found necessary for the intervals to have the right coverage rates even when estimating a simple parameter like the mean. Some block bootstrap methods could fail when estimating the lag-1 autocorrelation. It is surprising that the coverage property even deteriorates as the sample size increases with some commonly used block bootstrap confidence intervals including the percentile intervals and bias-corrected intervals. KEYWORDS: Autocorrelation; Bias-Correction; Centering; Dependent Data; Percentile; Resampling; Simulation; Time Series

A Time-Varying Mixture Integer-Valued Threshold Autoregressive Process Driven by Explanatory Variables.

In this paper, a time-varying first-order mixture integer-valued threshold autoregressive process driven by explanatory variables is introduced. The basic probabilistic and statistical properties of this model are studied in depth. We proceed to derive estimators using the conditional least squares (CLS) and conditional maximum likelihood (CML) methods, while also establishing the asymptotic properties of the CLS estimator. Furthermore, we employed the CLS and CML score functions to infer the threshold parameter. Additionally, three test statistics to detect the existence of the piecewise structure and explanatory variables were utilized. To support our findings, we conducted simulation studies and applied our model to two applications concerning the daily stock trading volumes of VOW.

Estimating the effect of latent time-varying count exposures using multiple lists.

A major challenge in longitudinal built-environment health studies is the accuracy of commercial business databases that are used to characterize dynamic food environments. Different databases often provide conflicting exposure measures on the same subject due to different source credibilities. As on-site verification is not feasible for historical data, we suggest combining multiple databases to correct the bias in health effect estimates due to measurement error in any 1 datasource. We propose a joint model for the time-varying health outcomes, observed count exposures, and latent true count exposures. Our model estimates the time-specific quality of sources and incorporates time dependence of true count exposure by Poisson integer-valued first-order autoregressive process. We take a Bayesian nonparametric approach to flexibly account for location-specific exposures. By resolving the discordance between different databases, our method reduces the bias in the longitudinal health effect of the true exposures. Our method is demonstrated with childhood obesity data in California public schools with respect to convenience store exposures in school neighborhoods from 2001 to 2008.

Bivariate Poisson Generalized Lindley Distributions and the Associated BINAR(1) Processes

This paper proposes new bivariate distributions based on the Poisson generalized Lindley distribution as marginal. These models include the basic bivariate Poisson generalized Lindley (BPGL) and the Sarmanov-based bivariate Poisson generalized Lindley (SPGL) distributions. Subsequently, we introduce the BPGL and SPGL distributions as joint innovation distributions in a novel bivariate first-order integer-valued autoregressive process (BINAR(1)) based on binomial thinning. The model parameters in the BPGL and SPGL distributions are estimated using the method of maximum likelihood (ML) while we apply the conditional maximum likelihood (CML) for the BINAR(1) process. We conduct some simulation experiments to assess the small and large sample performances. Further, we implement the new BINAR(1)s to the Pittsburgh crime series data and they show better fitting criteria than other competing BINAR(1) models in the literature.

Bayesian Analysis of Threshold Autoregressive Model with First Order Autoregressive Innovations

Financial assets exhibit dramatic changes in behaviour. This work examined a two-regime Threshold autoregressive (TAR) models when the innovations follow a first-autoregressive order process. The Bayesian method is proposed to build in the linear first-order autoregressive process with identical distributed innovations. The practical usefulness of this method is demonstrated with simulated and real-life data using U.S.A quarterly real GDP as an example. In simulation experiments and real life example, an increase in first order process parameter, ρ value leads to better estimates in the proposed model. Also, the proposed model was compared with TAR model where the disturbance term does not exhibit regime switching. The proposed model performed well than the traditional TAR model using the simulated and real life data. An increase in first order process parameter, ρ will lead to better estimates and forecast. Hence, the proposed model performed well.

A new First-Order mixture integer-valued threshold autoregressive process based on binomial thinning and negative binomial thinning

In this paper, we introduce a new first-order mixture integer-valued threshold autoregressive process, based on the binomial and negative binomial thinning operators. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares (CLS) and conditional maximum likelihood (CML) estimators are derived and the asymptotic properties of the estimators are established. The inference for the threshold parameter is obtained based on the CLS and CML score functions. Moreover, the Wald test is applied to detect the existence of the piecewise structure. Simulation studies are considered, along with an application: the number of criminal mischief incidents in the Pittsburgh dataset

Multivariate threshold integer-valued autoregressive processes with explanatory variables

To capture the multivariate count time series showing piecewise phenomena, we introduce a class of first-order multivariate threshold integer-valued autoregressive process. The component of autoregressive coefficient is driven by explanatory variables via a logistic link function. Basic probabilistic and statistical properties of the model are discussed. The model parameters are estimated by means of conditional least squares and conditional maximum likelihood methods. A new algorithm to estimate the threshold parameter of the model is also provided. Moreover, the nonlinearity test of the model and existence test of explanatory variables are well addressed using a Wald-type test. Forecasting problem and model extension problem are also considered. Finally, some numerical results of the estimates and a real data example are presented.

On the discrete analogue of the Teissier distribution and its associated INAR(1) process

The continuous Teissier distribution was proposed by the French biologist Georges Teissier in 1934. This paper introduces the one-parameter discrete analogue distribution and studies some of its statistical properties, focussing the attention on the computer generation of pseudo-random data and the parameter estimation problem. More precisely, the new discrete distribution is suitable to deal with both underdispersed and overdispersed count data, the quantile function can be expressed in closed form in terms of the Lambert W function and the failure rate function is increasing. Monte Carlo simulation experiments revealed that estimation methods such as maximum likelihood, least squares and quantile least squares may produce estimates far away from the true parameter value. This drawback can be overcome due to the existence of an analytical expression for the quantile function and then accurate estimates are obtained by the maximum likelihood method even for small sample sizes. Additionally, the new distribution is also used to derive a first-order integer-valued autoregressive process. Different real data sets are used to illustrate that the proposed distribution provides a better fit than other alternative models and also outperforms other competitive processes when time series of counts are considered.

Transmit Antenna Subset Selection in Cooperative SM-MIMO Vehicular System

This paper proposes a transmit antenna subset selection (TASS) in decode-and-forward (DF) cooperative spatial modulation multiple-input multiple-output (CSM-MIMO) vehicular system to enhance the communication performance. Specifically, the first-order autoregressive (FOAR) process is taken to model the time-varying vehicular fading channels. The transmit antenna (TA) subsets at the source and the relay that maximize the minimum squared Euclidean distance (MSED) can be respectively selected, which requires no complete channel state information (CSI). A closed-form analytical average bit error rate (ABER) of the CSM-MIMO vehicular system with the proposed TASS is derived. Numerical results demonstrate the correctness of the analytical ABER performance. The proposed TASS in the CSM-MIMO vehicular system achieves superior ABER performance than the existing CSM-MIMO system and the traditional TASS schemes.

New bivariate Poisson extended exponential distributions and associated BINAR(1) processes with applications

This study proposes two types of bivariate Poisson extended exponential distributions: the basic bivariate Poisson extended exponential distribution and the Sarmanov-based bivariate Poisson extended exponential distribution. The two bivariate Poisson extended exponential distributions are then introduced as joint innovation distributions in a bivariate first-order integer-valued autoregressive process based on binomial thinning. The model parameters are estimated using maximum likelihood in basic bivariate Poisson extended exponential and Sarmanov-based bivariate Poisson extended exponential distributions. Conditional maximum likelihood is applied to the bivariate first-order integer-valued autoregressive process. Simulation experiments are used to evaluate the performance of small and large samples. In addition, the newly developed bivariate first-order integer-valued autoregressive models are then applied to the Pittsburgh crime series data and candies data. We show that they fit better than existing bivariate first-order integer-valued autoregressive models described in the literature.

Estimation of fine-resolution PM2.5 concentrations using the INLA-SPDE method

Estimating spatio-temporal PM2.5 concentrations is a method that can be used to impute missing observations; which means that it may have important applications in the assessment of health effects. Based on observations gathered from 119 monitoring stations located within Henan Province and its surrounding areas, China, this study constructed a spatio-temporal dynamic model and used the Integrated Nested Laplace Approximation-Stochastic Partial Differential Equation (INLA-SPDE) method to estimate daily PM2.5 concentrations in Henan Province. Local geographical variables, such as elevation, road density, and land use, were integrated as spatial covariates. Meanwhile, meteorological variables, such as precipitation, air pressure, relative humidity, temperature, and wind speed, were integrated as spatio-temporal covariates. In addition, a first-order autoregressive process and a spatially correlated random effect were explicitly specified to capture the spatio-temporal dependence of PM2.5 concentrations. The validation results showed that the predictions were in good agreement with the observations (10-fold cross validated R2: 0.9407; Root mean square error: 10.7135 μg/m3), and the coverage probability of the 95% confidence interval was 96.04%. This study confirmed that the INLA-SPDE method can effectively model the spatio-temporal variability in PM2.5 concentrations. We also derived estimations of PM2.5 concentrations with a high spatio-temporal resolution, which should improve the assessment of related health effects.

A New INAR(1) Model for ℤ-Valued Time Series Using the Relative Binomial Thinning Operator

Abstract A new first-order integer-valued autoregressive process (INAR(1)) with extended Poisson innovations is introduced based on a signed version of the thinning operator, called relative binomial thinning operator, which can be considered as an extension of standard binomial thinning operator introduced by Steutel, F.W. and van Harn, K. (1979. Discrete analogues of self-decomposability and stability. Ann. Probab. 7: 893–899). It is appropriate for modeling Z $\mathbb{Z}$ -valued time series and either positive or negative correlations. Some properties of the process are established. Conditional least squares, Yule–Walker and conditional maximum likelihood methods are considered for the parameter estimation of the model. Moreover, simulation experiments are carried out to attest to the performance of the estimation methods. The applicability of the proposed model is investigated through a practical data set of the Saudi stock market.

Two Features of the GINAR(1) Process and Their Impact on the Run-Length Performance of Geometric Control Charts.

The geometric first-order integer-valued autoregressive process (GINAR(1)) can be particularly useful to model relevant discrete-valued time series, namely in statistical process control. We resort to stochastic ordering to prove that the GINAR(1) process is a discrete-time Markov chain governed by a totally positive order 2 (TP2) transition matrix.Stochastic ordering is also used to compare transition matrices referring to pairs of GINAR(1) processes with different values of the marginal mean. We assess and illustrate the implications of these two stochastic ordering results, namely on the properties of the run length of geometric charts for monitoring GINAR(1) counts.

Probabilistic Modeling for Cementitious Materials Based on Data of Nanoindentation

By introducing probabilistic modeling approaches, the interface transition zone (ITZ) and the bulk paste (BP) of concrete are investigated on the basis of the nanoindentation results, especially with respect to the relationship between the microstructure and mechanical properties of cementitious materials. The investigation of the probability density function (PDF) of the random field for nanoindentation properties revealed that the same properties of the ITZ and the BP usually yield the same PDF, which was elaborated from different points of view. Specifically, a log-normal distribution was best for nanoindentation hardness, whereas the Weibull distribution and gamma distribution were much more suitable for the nanoindentation modulus of both the ITZ and the BP. According to the comparative study of the correlation structure, both the ITZ and BP obeyed the exponential correlation structure associated with a first-order autoregressive process, and basically exhibited a similar scale of fluctuation. Furthermore, the scales of fluctuation were found to be directly related to the clinker size and the distance between clinkers. Our work provides a new approach to stochastically modeling cementitious materials, where the content of hydration products controls the mean values of nano-properties, the indentation property dominates the PDF, and the nano-topological structure governs the correlation structure.

UAV-Assisted Wireless Information and Power Transfer for Self-Sustained IoT Communications

This article investigates the performance of an unmanned aerial vehicle (UAV)-assisted wireless power transfer (WPT)-enabled communications for energy-constrained wireless devices. Herein, a UAV serves as a radio frequency (RF) energy harvesting (EH) transmitter for a pair of Internet-of-Things devices (IoDs). It also provides relay cooperation to IoDs using decode-and-forward (DF) operation. The composite fading channels associated with the UAV and IoDs are modeled by considering the probability of line-of-sight (LoS)/non-LoS (NLoS) and shadowing components, assuming log-normal distribution. Besides, the small-scale fading scenarios for LoS and NLoS communications are assumed to follow Nakagami- <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> and Rayleigh distributions, respectively. In addition, considering relative mobility between the UAV and IoDs, we exploit the first-order autoregressive process to model the fading channels during the information transmission (IT) phase. For such a realistic system setup, we derive the expressions for the outage probability (OP) at the IoDs using Gauss-Hermite quadrature under different shadowing and fading channel combinations. The system throughput and energy efficiency are also quantified using the derived OP expressions. The analytical findings are verified through numerical and simulation results. Our results highlight the impact of various system/channel parameters on the system’s performance and provide significant insights into the system’s behavior.