In the conventional methods of flood frequency analysis, the flood peak variable is just considered and assumed that this variable follows some specific parametric distribution function. This assumption would restrict us and lead us to the limited available information to evaluate the flood risk. It is well known that a flood event has three variables of flood peak, volume and duration which are random in nature and are mutually dependent. In this research, the concept of the Copula function is briefly introduced and then used to modeling the dependency structure of the flood variables of the Karun River at the Ahvaz hydrometric station and then estimate their joint probability distribution. We use three well-known and appropriate copulas, including Ali–Mikhail–Haq, Cook–Johnson and Gumbel–Hougaard which belong to the Archimedean class of copulas, to modeling the joint probability distribution of the flood variables, where the marginal distributions of the flood’s variables are selected from the parametric and non-parametric distributions. The Gumbel–Hougaard family led to better modeling of different combination of flood’s variables based on goodness of fit criteria.The selected copula is used to estimate conditional cumulative distribution function and joint return periods which lead to better estimation of flood risk.

This paper explores an extension of the bivariate Schur-constant model, introducing an additional parameter to its associated Archimedean copula for greater flexibility. We analyze the dependence properties of the proposed model and illustrate our findings with several examples. Furthermore, provide a likelihood ratio test to compare the performance of the extended Archimedean copula with that of the traditional Archimedean subfamily. Two real-data analysis are also included.

The aim of the present study was to analyze the correlation and simulation of the values of reference evapotranspiration of meteorological stations in the Lut Desert (Bam, Birjand, and Tabas) in the period 1984-2019 using common copulas and corresponding wind speed on a monthly scale. By examining the correlation between the pair-variables of wind speed and reference evapotranspiration in Bam, Birjand, and Tabas stations, the correlation was evaluated at about 0. 7, 0. 76, and 0. 80, respectively. By examining the structure of twodimensional copulas and their conditional density based on different criteria, the Galambos copula was selected as the best copula for all stations. By selecting the best marginal distribution and the best copula, the frequency analysis, joint return period, and copula-based simulation were presented. By presenting the joint probability curves, the probabilities of both wind speed and reference evapotranspiration at each station were presented simultaneously in the form of typical curves, which can provide very useful information about the probabilistic behavior of the studied data. Finally, the reference evapotranspiration values were simulated using a copulabased model and wind speed values. The simulation results of the mentioned values showed that the correlation of the pair-variables was appropriate. The accuracy, efficiency and error rate of simulating the values of reference evapotranspiration values were evaluated using various statistical tests. Successful simulation of this parameter is in fact a reason for accurate selection of the copula structure, which indicated a maximum error rate (RMSE) of 0. 1 mm per day and 98% efficiency for all stations. By application of this algorithm in simulating the values of reference evapotranspiration and presenting its estimated relationship in the region, the amount of reference evapotranspiration can be easily simulated with available meteorological data and without much computational complexity.

Copulas are essential probability tools for characterizing the joint distribution of random variables. In this article, we contribute to the topic by studying special bivariate copulas. They have the property of being defined with piecewise components and are designed for the analysis of data that have dependence structures with distinct substructures in square zones. The theoretical properties of the copulas are studied, with emphasis on their mathematical validity and some dependence measures. In particular, it is shown that the Kendall tau coefficient has a simple expression that is governed by several parameters, demonstrating the flexibility of the approach. In addition, a real data example is provided to demonstrate the applicability of the copulas. Fair comparisons with other standard copulas motivate their use in other practical scenarios.

Lifetime performance index is widely used as process capability index to evaluate the performance and potential of a process. In manufacturing industries, the lifetime of a product is considered to be conforming if it exceeds a given lower threshold value, so nonconforming products are those that fail to exceed this value. Nonconformities are so important that affect the safe or effective use of the products. This article deals with the processes that the products' lifetime is related to a two-component system, distributed as Farlie-Gumbel-Morgenstern (FGM) copula-based bivariate exponential and presen ts the probability of non-conforming products. Also, bootstrap upper confidence bounds are constructed and their performance are investigated in simulation study. In addition, Monte Carlo scheme is applied to do hypothesis testing on it. Finally, two example sets are presented to demonstrate the application of the proposed index.

This study investigates the relationship between two key economic indicators: income and wealth. Despite apparent similarities in the behavior of these two variables, numerous instances highlight their differences and lack of complete correlation. The main objective of this paper is to analyze the dependency structure between individuals’ income and wealth using advanced statistical tools. To model the dependency between income and wealth, the copula function—an innovative tool in probability theory—has been employed. In this regard, various copula functions are examined, and the Clayton copula family along with the Farlie-Gumbel-Morgenstern (FGM) copula family are identified as the most suitable choices for the studied data. Additionally, the study introduces a new bivariate index based on the Total Time on Test (TTT) transform in the bivariate case, utilizing copula functions. This index is applied to real-world data on the income and wealth of Iranian households, and the results demonstrate a significant relationship between the two variables at the societal level. Therefore, it can be concluded that using this index in economic data reveals that the proposed bivariate TTT index can effectively represent the dependency structure between income and wealth. Furthermore, the use of Clayton and FGM copulas also shows a good fit with the empirical data.

Dust storm is a stochastical event that depends on several parameters, therefore, multivariate analysis of this event is really important. In this study, importance and shortage of multivariate analysis of natural disasters such as dust storm is investigated. For this purpose, Copula theory is used for bivariate analysis of dust storm. Copulas function are useful tools for multivariate frequency analysis of natural disasters. One of the main advantages of Copula theory is that there is no limit to select the type of marginal distribution. In order to perform bivariate analysis of dust storm, severe dust storm was selected based on definition of WMO from 1982 to 2014 in Yazd province. 34 dust storm events were extracted and maximum wind speed and geopotential heights were determined corresponding to stormy days. Finally, the bivariate return period was calculated based on maximum wind speed and geopotential height using the t-student Copula as the best function. Also, univariate return period of dust storm was calculated based on maximum wind speed and geopotential height, separately for comparison with bivariate return period. The results showed bivariate analysis of return period of dust storm is more accurate than univariate return period.

One of the useful distributions in modeling mortality (or failure) data is the univariate Gompertz--Makeham distribution. To examine the relationship between the two variables, the extended bivariate Gompertz--Makeham distribution is introduced, and its properties are provided. Also, some reliability indices, including aging intensity and stress-strength reliability, are calculated for the proposed model. Here, a new copula function is constructed based on the extended bivariate Gompertz--Makeham  distribution. Some of its features including dependency properties, such as dependence structure, some  measures of dependence, and tail dependence,  are studied.The estimation of the  parameters of new copula is presented, and at the end, a simulation study and a performance analysis based on the real data are presented.  So, by analyzing the mortality data due to COVID-19, the appropriateness of the proposed model is examined.