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Modeling of extreme responses in presence nonlinear, temporal, spatial and interaction effects can be accomplished with mixed models. In addition, smoothing spline through mixed model and Bayesian approach together provide convenient framework for inference of extreme values. In this article, by representing as a mixed model, smoothing spline is used to assess nonlinear covariate effect on extreme values. For this reason, we assume that extreme responses given covariates and random effects are independent with generalized extreme value distribution. Then by using MCMC techniques in Bayesian framework, location parameter of distribution is estimated as a smooth function of covariates. Finally, the proposed model is employed to model the extreme values of ozone data.

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Geostatistical spatial count data in finite populations can be seen in many applications, such as urban management and medicine. The traditional model for analyzing these data is the spatial logit-binomial model. In the most applied situations, these data have overdispersion alongside the spatial variability. The binomial model is not the appropriate candidate to account for the overdispersion. The proper alternative is a beta-binomial model that has sufficient flexibility to account for the extra variability due to the possible overdispersion of counts. In this paper, we describe a Bayesian spatial beta-binomial for geostatistical count data by using a combination of the integrated nested Laplace approximation and the stochastic partial differential equations methods. We apply the methodology for analyzing the number of people injured/killed in car crashes in Mashhad, Iran. We further evaluate the performance of the model using a simulation study.

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In some instances, the occurrence of an event can be influenced by its spatial location, giving rise to spatial survival data. The accurate and precise estimation of parameters in a spatial survival model poses a challenge due to the complexity of the likelihood function, highlighting the significance of employing a Bayesian approach in survival analysis. In a Bayesian spatial survival model, the spatial correlation between event times is elucidated using a geostatistical model. This article presents a simulation study to estimate the parameters of classical and spatial survival models, evaluating the performance of each model in fitting simulated survival data. Ultimately, it is demonstrated that the spatial survival model exhibits superior efficacy in analyzing blood cancer data compared to conventional models.

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Spatial regression models are used to analyze quantitative spatial responses based on linear and non-linear relationships with explanatory variables. Usually, the spatial correlation of responses is modeled with a Gaussian random field based on a multivariate normal distribution. However, in practice, we encounter skewed responses, which are analyzed using skew-normal distributions. Closed skew-normal distribution is one of the extended families of skew-normal distributions, which has similar properties to normal distributions. This article presents a hierarchical Bayesian analysis based on a flexible subclass of closed skew-normal distributions. Given the time-consuming nature of Monte Carlo methods in hierarchical Bayes analysis, we have opted to use the variational Bayes approach to approximate the posterior distribution. This decision was made to expedite the analysis process without compromising the accuracy of our results. Then, the proposed model is implemented and analyzed based on the real earthquake data of Iran.