This paper discusses the estimation of parameters in the zero-inflated Poisson (ZIP) model by the method of moments. The method of moments estimators (MMEs) are analytically compared with the maximum likelihood estima...This paper discusses the estimation of parameters in the zero-inflated Poisson (ZIP) model by the method of moments. The method of moments estimators (MMEs) are analytically compared with the maximum likelihood estimators (MLEs). The results of a modest simulation study are presented.展开更多
Zero-Inflated Poisson model has found a wide variety of applications in recent years in statistical analyses of count data, especially in count regression models. Zero-Inflated Poisson model is characterized in this p...Zero-Inflated Poisson model has found a wide variety of applications in recent years in statistical analyses of count data, especially in count regression models. Zero-Inflated Poisson model is characterized in this paper through a linear differential equation satisfied by its probability generating function [1] [2].展开更多
The Goel-Okumoto software reliability model, also known as the Exponential Nonhomogeneous Poisson Process,is one of the earliest software reliability models to be proposed. From literature, it is evident that most of ...The Goel-Okumoto software reliability model, also known as the Exponential Nonhomogeneous Poisson Process,is one of the earliest software reliability models to be proposed. From literature, it is evident that most of the study that has been done on the Goel-Okumoto software reliability model is parameter estimation using the MLE method and model fit. It is widely known that predictive analysis is very useful for modifying, debugging and determining when to terminate software development testing process. However, there is a conspicuous absence of literature on both the classical and Bayesian predictive analyses on the model. This paper presents some results about predictive analyses for the Goel-Okumoto software reliability model. Driven by the requirement of highly reliable software used in computers embedded in automotive, mechanical and safety control systems, industrial and quality process control, real-time sensor networks, aircrafts, nuclear reactors among others, we address four issues in single-sample prediction associated closely with software development process. We have adopted Bayesian methods based on non-informative priors to develop explicit solutions to these problems. An example with real data in the form of time between software failures will be used to illustrate the developed methodologies.展开更多
The Goel-Okumoto software reliability model is one of the earliest attempts to use a non-homogeneous Poisson process to model failure times observed during software test interval. The model is known as exponential NHP...The Goel-Okumoto software reliability model is one of the earliest attempts to use a non-homogeneous Poisson process to model failure times observed during software test interval. The model is known as exponential NHPP model as it describes exponential software failure curve. Parameter estimation, model fit and predictive analyses based on one sample have been conducted on the Goel-Okumoto software reliability model. However, predictive analyses based on two samples have not been conducted on the model. In two-sample prediction, the parameters and characteristics of the first sample are used to analyze and to make predictions for the second sample. This helps in saving time and resources during the software development process. This paper presents some results about predictive analyses for the Goel-Okumoto software reliability model based on two samples. We have addressed three issues in two-sample prediction associated closely with software development testing process. Bayesian methods based on non-informative priors have been adopted to develop solutions to these issues. The developed methodologies have been illustrated by two sets of software failure data simulated from the Goel-Okumoto software reliability model.展开更多
The delayed S-shaped software reliability growth model (SRGM) is one of the non-homogeneous Poisson process (NHPP) models which have been proposed for software reliability assessment. The model is distinctive because ...The delayed S-shaped software reliability growth model (SRGM) is one of the non-homogeneous Poisson process (NHPP) models which have been proposed for software reliability assessment. The model is distinctive because it has a mean value function that reflects the delay in failure reporting: there is a delay between failure detection and reporting time. The model captures error detection, isolation, and removal processes, thus is appropriate for software reliability analysis. Predictive analysis in software testing is useful in modifying, debugging, and determining when to terminate software development testing processes. However, Bayesian predictive analyses on the delayed S-shaped model have not been extensively explored. This paper uses the delayed S-shaped SRGM to address four issues in one-sample prediction associated with the software development testing process. Bayesian approach based on non-informative priors was used to derive explicit solutions for the four issues, and the developed methodologies were illustrated using real data.展开更多
In this work, some non-homogeneous Poisson models are considered to study the behaviour of ozone in the city of Puebla, Mexico. Several functions are used as the rate function for the non-homogeneous Poisson process. ...In this work, some non-homogeneous Poisson models are considered to study the behaviour of ozone in the city of Puebla, Mexico. Several functions are used as the rate function for the non-homogeneous Poisson process. In addition to their dependence on time, these rate functions also depend on some parameters that need to be estimated. In order to estimate them, a Bayesian approach will be taken. The expressions for the distributions of the parameters involved in the models are very complex. Therefore, Markov chain Monte Carlo algorithms are used to estimate them. The methodology is applied to the ozone data from the city of Puebla, Mexico.展开更多
Mpox remains a signi_cant public health challenge in endemic regions of Africa.Understanding its spatial distribution and identifying key drivers in high-risk countries is critical for guiding e_ective interventions.T...Mpox remains a signi_cant public health challenge in endemic regions of Africa.Understanding its spatial distribution and identifying key drivers in high-risk countries is critical for guiding e_ective interventions.This study applies a Zero-Inated Poisson(ZIP)model with spatial autocorrelation to estimate the adjusted relative risk(RR)of Mpox incidence across 24 African countries,strati_ed by Human Development Index(HDI)levels.The model accounts for overdispersion and excess zeros by incorporating spatial random e_ects and socio-environmental covariates,and was validated through model diagnostics and sensitivity analysis,demonstrating robustness of results.Spatial analysis revealed substantial heterogeneity in Mpox incidence,with elevated risk in the Democratic Republic of Congo(DRC),Nigeria,and Central African Republic(CAR)persisting after covariate adjustment(p<0.001).Higher HDI levels were inversely associated with Mpox risk,with HDI quintile Q4(very high HDI)showing a signi_cant reduction(aRR=0.431;95%CrI:0.099{0.724).Protective factors in low-risk areas included increased life expectancy at birth(aRR=0.768;95%CrI:0.688{0.892),higher educational attainment(aRR=0.774;95%CrI:0.680{0.921),nonlinear increases in gross national income(GNI)per capita,and a greater density of skilled health workers(aRR=0.788;95%CrI:0.701{0.934).Conversely,higher urban density was associated with increased Mpox risk,underscoring the inuence of population clustering on transmission dynamics.Notably,statistically signi_cant elevated-risk areas persisted in endemic countries of Western and Central Africa after covariate adjustment(p<0.001).In contrast,previously undetected risk emerged in parts of Southern and Eastern Africa post-adjustment,revealing latent patterns obscured in the crude analysis(p<0.001).Exceedance probability maps identi_ed countries with P(RR>1)>0.9 as priority areas for intensi_ed surveillance and targeted intervention.These patterns were not fully explained by the included covariates,suggesting the inuence of unmeasured factors such as environmental and climate variability,zoonotic reservoirs,or human{animal interactions.Further research is needed to deepen understanding of Mpox epidemiology and support locally tailored interventions.展开更多
Road crash prediction models are very useful tools in highway safety, given their potential for determining both the crash frequency occurrence and the degree severity of crashes. Crash frequency refers to the predict...Road crash prediction models are very useful tools in highway safety, given their potential for determining both the crash frequency occurrence and the degree severity of crashes. Crash frequency refers to the prediction of the number of crashes that would occur on a specific road segment or intersection in a time period, while crash severity models generally explore the relationship between crash severity injury and the contributing factors such as driver behavior, vehicle characteristics, roadway geometry, and road-environment conditions. Effective interventions to reduce crash toll include design of safer infrastructure and incorporation of road safety features into land-use and transportation planning;improvement of vehicle safety features;improvement of post-crash care for victims of road crashes;and improvement of driver behavior, such as setting and enforcing laws relating to key risk factors, and raising public awareness. Despite the great efforts that transportation agencies put into preventive measures, the annual number of traffic crashes has not yet significantly decreased. For in-stance, 35,092 traffic fatalities were recorded in the US in 2015, an increase of 7.2% as compared to the previous year. With such a trend, this paper presents an overview of road crash prediction models used by transportation agencies and researchers to gain a better understanding of the techniques used in predicting road accidents and the risk factors that contribute to crash occurrence.展开更多
A new two-parameter count distribution is derived starting with probabilistic arguments around the gamma function and the digamma function. This model is a generalization of the Poisson model with a noteworthy assortm...A new two-parameter count distribution is derived starting with probabilistic arguments around the gamma function and the digamma function. This model is a generalization of the Poisson model with a noteworthy assortment of qualities. For example, the mean is the main model parameter;any possible non-trivial variance or zero probability can be attained by changing the other model parameter;and all distributions are visually natural-shaped. Thus, exact modeling to any degree of over/under-dispersion or zero-inflation/deflation is possible.展开更多
Analysis of diarrhoea data in Malawi has been commonly done using classical methods. However, different approaches, such as Bayesian methods, have been introduced in literature. This study aimed at trying out semi-par...Analysis of diarrhoea data in Malawi has been commonly done using classical methods. However, different approaches, such as Bayesian methods, have been introduced in literature. This study aimed at trying out semi-parametric methods in comparison with classical ones, as well as how each isolates risk factors for child diarrhoea. This was done by fitting Logit, Poisson, and Bayesian models to 2006 Malawi Multiple Indicator Cluster Survey data. The comparison between Logit and Poisson models was done via chi-square＇s goodness-of-fit test. Confidence and Credible Intervals were used to compare Logit/Poisson and Bayesian model estimates. Modelling and inference in Bayesian method was done through MCMC techniques. The results showed agreement in significance and direction of estimates from Bayesian and Poisson/Logit models, but Poisson provided better fit than Logit model. Further, all the models identified child＇s age, breastfeeding status, region of stay and toilet-sharing status as significant factors for determining the child＇s risk. The models ruled out effects of mother＇s education, area of residence, and source of drinking water on the risk. Bayesian model separately proved significant closeness to lake/river factor. The findings imply that classical and semi-parametric models are equally helpful when estimating the child＇s risk to diarrhoea.展开更多
In this paper, a zero-and-one-inflated Poisson (ZOIP) model is studied. The maximum likelihoodestimation and the Bayesian estimation of the model parameters are obtained based on dataaugmentation method. A simulation ...In this paper, a zero-and-one-inflated Poisson (ZOIP) model is studied. The maximum likelihoodestimation and the Bayesian estimation of the model parameters are obtained based on dataaugmentation method. A simulation study based on proposed sampling algorithm is conductedto assess the performance of the proposed estimation for various sample sizes. Finally, two realdata-sets are analysed to illustrate the practicability of the proposed method.展开更多
文摘This paper discusses the estimation of parameters in the zero-inflated Poisson (ZIP) model by the method of moments. The method of moments estimators (MMEs) are analytically compared with the maximum likelihood estimators (MLEs). The results of a modest simulation study are presented.
文摘Zero-Inflated Poisson model has found a wide variety of applications in recent years in statistical analyses of count data, especially in count regression models. Zero-Inflated Poisson model is characterized in this paper through a linear differential equation satisfied by its probability generating function [1] [2].
文摘The Goel-Okumoto software reliability model, also known as the Exponential Nonhomogeneous Poisson Process,is one of the earliest software reliability models to be proposed. From literature, it is evident that most of the study that has been done on the Goel-Okumoto software reliability model is parameter estimation using the MLE method and model fit. It is widely known that predictive analysis is very useful for modifying, debugging and determining when to terminate software development testing process. However, there is a conspicuous absence of literature on both the classical and Bayesian predictive analyses on the model. This paper presents some results about predictive analyses for the Goel-Okumoto software reliability model. Driven by the requirement of highly reliable software used in computers embedded in automotive, mechanical and safety control systems, industrial and quality process control, real-time sensor networks, aircrafts, nuclear reactors among others, we address four issues in single-sample prediction associated closely with software development process. We have adopted Bayesian methods based on non-informative priors to develop explicit solutions to these problems. An example with real data in the form of time between software failures will be used to illustrate the developed methodologies.
文摘The Goel-Okumoto software reliability model is one of the earliest attempts to use a non-homogeneous Poisson process to model failure times observed during software test interval. The model is known as exponential NHPP model as it describes exponential software failure curve. Parameter estimation, model fit and predictive analyses based on one sample have been conducted on the Goel-Okumoto software reliability model. However, predictive analyses based on two samples have not been conducted on the model. In two-sample prediction, the parameters and characteristics of the first sample are used to analyze and to make predictions for the second sample. This helps in saving time and resources during the software development process. This paper presents some results about predictive analyses for the Goel-Okumoto software reliability model based on two samples. We have addressed three issues in two-sample prediction associated closely with software development testing process. Bayesian methods based on non-informative priors have been adopted to develop solutions to these issues. The developed methodologies have been illustrated by two sets of software failure data simulated from the Goel-Okumoto software reliability model.
文摘The delayed S-shaped software reliability growth model (SRGM) is one of the non-homogeneous Poisson process (NHPP) models which have been proposed for software reliability assessment. The model is distinctive because it has a mean value function that reflects the delay in failure reporting: there is a delay between failure detection and reporting time. The model captures error detection, isolation, and removal processes, thus is appropriate for software reliability analysis. Predictive analysis in software testing is useful in modifying, debugging, and determining when to terminate software development testing processes. However, Bayesian predictive analyses on the delayed S-shaped model have not been extensively explored. This paper uses the delayed S-shaped SRGM to address four issues in one-sample prediction associated with the software development testing process. Bayesian approach based on non-informative priors was used to derive explicit solutions for the four issues, and the developed methodologies were illustrated using real data.
文摘In this work, some non-homogeneous Poisson models are considered to study the behaviour of ozone in the city of Puebla, Mexico. Several functions are used as the rate function for the non-homogeneous Poisson process. In addition to their dependence on time, these rate functions also depend on some parameters that need to be estimated. In order to estimate them, a Bayesian approach will be taken. The expressions for the distributions of the parameters involved in the models are very complex. Therefore, Markov chain Monte Carlo algorithms are used to estimate them. The methodology is applied to the ozone data from the city of Puebla, Mexico.
基金funded by the Canadian Institutes of Health Research(CIHR)under the Mpox and other zoonotic threats Team Grant(FRN.187246)W.A.W acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada(NSERC)Discovery Grant(Appl No.:RGPIN-2023-05100).
文摘Mpox remains a signi_cant public health challenge in endemic regions of Africa.Understanding its spatial distribution and identifying key drivers in high-risk countries is critical for guiding e_ective interventions.This study applies a Zero-Inated Poisson(ZIP)model with spatial autocorrelation to estimate the adjusted relative risk(RR)of Mpox incidence across 24 African countries,strati_ed by Human Development Index(HDI)levels.The model accounts for overdispersion and excess zeros by incorporating spatial random e_ects and socio-environmental covariates,and was validated through model diagnostics and sensitivity analysis,demonstrating robustness of results.Spatial analysis revealed substantial heterogeneity in Mpox incidence,with elevated risk in the Democratic Republic of Congo(DRC),Nigeria,and Central African Republic(CAR)persisting after covariate adjustment(p<0.001).Higher HDI levels were inversely associated with Mpox risk,with HDI quintile Q4(very high HDI)showing a signi_cant reduction(aRR=0.431;95%CrI:0.099{0.724).Protective factors in low-risk areas included increased life expectancy at birth(aRR=0.768;95%CrI:0.688{0.892),higher educational attainment(aRR=0.774;95%CrI:0.680{0.921),nonlinear increases in gross national income(GNI)per capita,and a greater density of skilled health workers(aRR=0.788;95%CrI:0.701{0.934).Conversely,higher urban density was associated with increased Mpox risk,underscoring the inuence of population clustering on transmission dynamics.Notably,statistically signi_cant elevated-risk areas persisted in endemic countries of Western and Central Africa after covariate adjustment(p<0.001).In contrast,previously undetected risk emerged in parts of Southern and Eastern Africa post-adjustment,revealing latent patterns obscured in the crude analysis(p<0.001).Exceedance probability maps identi_ed countries with P(RR>1)>0.9 as priority areas for intensi_ed surveillance and targeted intervention.These patterns were not fully explained by the included covariates,suggesting the inuence of unmeasured factors such as environmental and climate variability,zoonotic reservoirs,or human{animal interactions.Further research is needed to deepen understanding of Mpox epidemiology and support locally tailored interventions.
文摘Road crash prediction models are very useful tools in highway safety, given their potential for determining both the crash frequency occurrence and the degree severity of crashes. Crash frequency refers to the prediction of the number of crashes that would occur on a specific road segment or intersection in a time period, while crash severity models generally explore the relationship between crash severity injury and the contributing factors such as driver behavior, vehicle characteristics, roadway geometry, and road-environment conditions. Effective interventions to reduce crash toll include design of safer infrastructure and incorporation of road safety features into land-use and transportation planning;improvement of vehicle safety features;improvement of post-crash care for victims of road crashes;and improvement of driver behavior, such as setting and enforcing laws relating to key risk factors, and raising public awareness. Despite the great efforts that transportation agencies put into preventive measures, the annual number of traffic crashes has not yet significantly decreased. For in-stance, 35,092 traffic fatalities were recorded in the US in 2015, an increase of 7.2% as compared to the previous year. With such a trend, this paper presents an overview of road crash prediction models used by transportation agencies and researchers to gain a better understanding of the techniques used in predicting road accidents and the risk factors that contribute to crash occurrence.
文摘A new two-parameter count distribution is derived starting with probabilistic arguments around the gamma function and the digamma function. This model is a generalization of the Poisson model with a noteworthy assortment of qualities. For example, the mean is the main model parameter;any possible non-trivial variance or zero probability can be attained by changing the other model parameter;and all distributions are visually natural-shaped. Thus, exact modeling to any degree of over/under-dispersion or zero-inflation/deflation is possible.
文摘Analysis of diarrhoea data in Malawi has been commonly done using classical methods. However, different approaches, such as Bayesian methods, have been introduced in literature. This study aimed at trying out semi-parametric methods in comparison with classical ones, as well as how each isolates risk factors for child diarrhoea. This was done by fitting Logit, Poisson, and Bayesian models to 2006 Malawi Multiple Indicator Cluster Survey data. The comparison between Logit and Poisson models was done via chi-square＇s goodness-of-fit test. Confidence and Credible Intervals were used to compare Logit/Poisson and Bayesian model estimates. Modelling and inference in Bayesian method was done through MCMC techniques. The results showed agreement in significance and direction of estimates from Bayesian and Poisson/Logit models, but Poisson provided better fit than Logit model. Further, all the models identified child＇s age, breastfeeding status, region of stay and toilet-sharing status as significant factors for determining the child＇s risk. The models ruled out effects of mother＇s education, area of residence, and source of drinking water on the risk. Bayesian model separately proved significant closeness to lake/river factor. The findings imply that classical and semi-parametric models are equally helpful when estimating the child＇s risk to diarrhoea.
基金The research is supported by the Natural Science Foundation of China(Nos.11271136,81530086,11671303,11201345)the 111 Project of China(No.B14019)+5 种基金the Natural Science Foundation of Zhejiang Province(No.LY15G010006)the China Postdoctoral Science Foundation(No.2015M572598)National Natural Science Foundation of China(CN)[grant number 11671303],[grant number 11201345]:Ministry of Education of the People’s Republic of China(CN)[grant number B14019]China Postdoctoral Science Foundation(CN)[grant number 2015M572598]National Natural Science Foundation of China(CN)[grant number 11271136],[grant number 81530086]Natural Science Foundation of Zhejiang Province(CN)[grant number LY15G010006].
文摘In this paper, a zero-and-one-inflated Poisson (ZOIP) model is studied. The maximum likelihoodestimation and the Bayesian estimation of the model parameters are obtained based on dataaugmentation method. A simulation study based on proposed sampling algorithm is conductedto assess the performance of the proposed estimation for various sample sizes. Finally, two realdata-sets are analysed to illustrate the practicability of the proposed method.
