Search Results (8)

Many applications in health research involve the analysis of multivariate distributions of random variables. In this paper, we review the basic theory of copulas to illustrate their advantages in deriving a joint distribution from given marginal distributions, with a specific focus on bivariate cases. Particular attention is given to the Archimedean family of copulas, which includes widely used functions such as Clayton and Gumbel–Hougaard, characterized by a single association parameter and a relatively simple structure. This work differs from previous reviews by providing a focused overview of applied studies in biomedical research that have employed Archimedean copulas, due to their flexibility in modeling a wide range of dependence structures. Their ease of use and ability to accommodate rotated forms make them suitable for various biomedical applications, including those involving survival data. We briefly present the most commonly used methods for estimation and model selection of copula’s functions, with the purpose of introducing these tools within the broader framework. Several recent examples in the health literature, and an original example of a pediatric study, demonstrate the applicability of Archimedean copulas and suggest that this approach, although still not widely adopted, can be useful in many biomedical research settings. Full article

This paper proposes a system made up of four subsystems connected in sequence. The first and third subsystems each have one unit, the second has two, and the fourth has three. Every subsystem operates in parallel and is governed by the K-Out-of-n:G rule. Nonetheless, each subsystem needs at least one operational unit in order for the system to work. While a unit’s failure has an exponential distribution, repair is simulated using a general distribution and a distribution from the Gumbel–Hougaard family of copula. This study’s primary objective is to assess and contrast the system performance while our system is running under these two different repair policies. The problem is solved by combining the supplementary variable technique with Laplace transforms. We use reliability metrics to assess system performance. The second objective of this study is to present a reduction approach plan aimed at improving the overall reliability metrics of our system. Full article

This research envisages a system composed of three subsystems connected in series. Each subsystem comprises three units connected in parallel. For the system to function, at least one unit per subsystem must remain operational. Unit failure is governed by an exponential distribution, while unit repair is governed by either a general distribution or a Gumbel–Hougaard family copula distribution. The primary goal of this research is to compare the overall performance of our system under these two different regimes for performing repairs. Laplace transforms and supplementary variable methods are employed in solving the system. Our metrics for evaluating system performance are the availability, reliability, mean time to failure, and cost. The second goal of this research is to showcase a strategy for reduction that enhances the overall efficiency and availability of our system. Full article

In reliability studies, we are interested in the behaviour of a system when it interacts with its surrounding environment. To assess the system’s behaviour in a reliability sense, we can take the system’s intrinsic quality as strength and the outcome of interactions as stress. Failure is observed whenever stress exceeds strength. Taking Y as a random variable representing the stress the system experiences and random variable X as its strength, the probability of not failing can be taken as a proxy for the reliability of the component and given as $P(Y<X)=1-P(X<Y)$. This way, in the present paper, it is considered that X and Y follow generalized extreme value distributions, which represent a family of continuous probability distributions that have been extensively applied in engineering and economic contexts. Our contribution deals with a more general scenario where stress and strength are not independent and copulas are used to model the dependence between the involved random variables. In such modelling framework, we explored the proper selection of copula models characterizing the dependence structure. The Gumbel–Hougaard, Frank, and Clayton copulas were used for modelling bivariate data sets. In each case, information criteria were considered to compare the modelling capabilities of each copula. Two economic applications, as well as an engineering one, on real data sets are discussed. Overall, an easy-to-use methodological framework is described, allowing practitioners to apply it to their own research projects. Full article

Wind energy is a kind of renewable energy that plays a significant role in remote areas for pumping water. The windmill is also used to generate electricity. The windmill is also known as a wind pump when it is used for pumping water. In this work, the authors proposed a complex hybrid model of an example of combined system (windmill, rechargeable battery and pumping system) to evaluate the system’s performance. System performance was affected by system degradation due to system failure. These factors also affected the profit of the user. Two types of repair facilities for continuous and satisfactory performance of the system were assumed. To illustrate the system modeling using the Gumbel–Hougaard family of the copula, numerical examples were used for the exploration of Markov results of the reliability measures and the profit of the system with the warranty period, with this also being demonstrated graphically. Full article

In the automotive industry, it is important to know whether the failure of some car parts may be related to the failure of others. This project studies warranty claims for five engine components obtained from a major car manufacturer with the purpose of modeling the joint distributions of the failure of two parts. The one-dimensional distributions of components are combined to construct a bivariate copula model for the joint distribution that makes it possible to estimate the probabilities of two components failing before a given time. Ultimately, the influence of the failure of one part on the operation of another related part can be described, predicted, and addressed. The performance of several families of one-parameter Archimedean copula models (Clayton, Gumbel–Hougaard, survival copulas) is analyzed, and Bayesian model selection is performed. Both right censoring and conditional approaches are considered with the emphasis on conditioning to the warranty period. Full article

The Northeast region of Brazil (NRB) is the most populous semiarid area in the world and is extremely susceptible to droughts. The severity and duration of these droughts depend on several factors, and they do not necessarily follow the same behavior. The aim of this work is to evaluate the frequency of droughts in the NRB and calculate the return period of each drought event using the copula technique, which integrates the duration and severity of the drought in the NRB in a joint bivariate distribution. Monthly precipitation data from 96 meteorological stations spatially distributed in the NRB, ranging from 1961 to 2017, are used. The copula technique is applied to the Standardized Precipitation Index (SPI) on the three-month time scale, testing three families of Archimedean copula functions (Gumbel–Hougaard, Clayton and Frank) to reveal which model is best suited for the data. Averagely, the most frequent droughts observed in the NRB are concentrated in the northern sector of the region, with an observed duration varying from three and a half to five and a half months. However, the eastern NRB experiences the most severe droughts, lasting for 14 to 24 months. The probability distributions that perform better in modeling the series of severity and duration of droughts are exponential, normal and lognormal. The observed severity and duration values show that, for average values, the return period across the region is approximately 24 months. Still in this regard, the southernmost tip of the NRB stands out for having a return period of over 35 months. Regarding maximum observed values of severity and duration, the NRB eastern strip has the longest return period (>60 months), mainly in the southeastern portion where a return period above 90 months was observed. The northern NRB shows the shortest return period (~45 months), indicating that it is the NRB sector with the highest frequency of intense droughts. These results provide useful information for drought risk management in the NRB. Full article

Flood events are characterized by flood peaks and volumes that can be mutually constructed using a copula function. The Indus basin system of Pakistan is periodically threatened by floods during monsoon seasons and thus causes huge losses to infrastructure as well as the community and economy. The design flood hydrograph (DFH) of suitable magnitude and degree is imperative for sheltering dams against the flood risk. The hydrological pair of flood peak and volume is required to be defined using a multivariate analysis method. In this paper, the joint probability function of the hydrological pair is employed to derive the DFH in the Indus basin system of Pakistan. Firstly, we compared the fitting performance of different probability distributions (PDs) as a marginal distribution. Next, we compared the Archimedean family of copulas to construct the bivariate joint distribution of flood peak and volume. Later, the equal frequency combination (EFC) method and most likely combination (MLC) method using “OR” joint return period (JRP_or), was involved to derive the design flood quantiles. Finally, we derived the DFH using the two combination methods based on Gumbel–Hougaard copula for different return periods. We presented the combination methods for updating the shape of the DFH in Pakistan. Our study will contribute towards the improvement of design standards of dams and environmental recovery in Pakistan. Full article