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It was proved about 60 years ago that if a continuous random variable X has an increasing failure rate  then its order statistics will also be increasing failure rate, and this problem remained unproved for the discrete case until recently a proof method using an integral inequality was provided. In this article, we present a completely different method to solve this problem.

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Events in one financial institution can affect other institutions. For this reason, systemic risk is of interest to risk analysts, and the most important methods of measuring it are the CoVaR and CoES. If there is a dependence between the returns of two financial institutions, Copula functions can be used to examine the structure of the dependence between them. Since return data are often  are unstable  over time, ARMA-GARCH time series models can be used to model variability. In this paper, CoVaR is evaluated for four copula functions, and then CoES are estimated based on that in ARMA-GARCH models with GED  distributions. Then, these two measures are calculated with the returns of  Tejarat and Mellat banks.

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This paper presents a nonparametric Bayesian method for estimating nonstationary covariance structures in big spatial datasets. The approach extends the Vecchia approximation and assumes conditional independence among ordered data points, leading to a sparse precision matrix and sparse Cholesky decomposition. This enables modeling an $n$-dimensional Gaussian process as a sequence of Bayesian linear regressions. Data ordering via maximum minimum distance improves model performance. Applying the grouping algorithm to ordered data removes weak dependencies and defines a block-sparse covariance structure, significantly reducing computational burden and enhancing accuracy. Simulations and real data analysis show that posterior samples from the proposed method yield narrower uncertainty intervals than those from ungrouped approaches.