Search Results (29)

This paper addresses the limitations of existing bivariate generalized exponential (GE) distributions for modeling lifetime data, which often exhibit rigid dependence structures or non-GE marginals. To overcome these limitations, we introduce four new bivariate GE distributions based on the Farlie–Gumbel–Morgenstern, Gumbel–Barnett, Clayton, and Frank copulas, which allow for more flexible modeling of various dependence structures. We employ a Bayesian framework with Markov Chain Monte Carlo (MCMC) methods for parameter estimation. A simulation study is conducted to evaluate the performance of the proposed models, which are then applied to a real-world dataset of electrical treeing failures. The results from the data application demonstrate that the copula-based models, particularly the one derived from the Frank copula, provide a superior fit compared to existing bivariate GE models. This work provides a flexible and robust framework for modeling dependent lifetime data. Full article

The classical exponential model, despite its flexibility, fails to describe data with non-constant failure or between-event dependency. To overcome this limitation, two new bivariate lifetime distributions are introduced in this paper. The Farlie–Gumbel–Morgenstern (FGM)-based and Ali–Mikhail–Haq (AMH)-based modified Fréchet–exponential (MFE) models, by embedding the flexible MEF margin in the FGM and AMH copulas. The resulting distributions accommodate a wide range of positive or negative dependence while retaining analytical traceability. Closed-form expressions for the joint and marginal density, survival, hazard, and reliability functions are derived, together with product moments and moment-generating functions. Unknown parameters are estimated through the maximum likelihood estimation (MLE) and inference functions for margins (IFM) methods, with asymptotic confidence intervals provided for these parameters. An extensive Monte Carlo simulation quantifies the bias, mean squared error, and interval coverage, indicating that IFM retains efficiency while reducing computational complexity for moderate sample sizes. The models are validated using two real datasets, from the medical sector regarding the infection recurrence times of 30 kidney patients undergoing peritoneal dialysis, and from the economic sector regarding the growth of the gross domestic product (GDP). Overall, the proposed copula-linked MFE distributions provide a powerful and economical framework for survival analysis, reliability, and economic studies. Full article

The research focuses on probabilistic modeling of maximum rainfall in Zielona Góra, Poland, to improve urban drainage system design. The study utilizes archived pluviographic data from 1951 to 2020, collected at the IMWM-NRI meteorological station. These data include 10 min rainfall records and aggregated hourly and daily totals. The study employs various statistical distributions, including Fréchet, gamma, generalized exponential (GED), Gumbel, log-normal, and Weibull, to model rainfall intensity–duration–frequency (IDF) relationships. After testing the goodness of fit using the Anderson–Darling test, Bayesian Information Criterion (BIC), and relative residual mean square Error (rRMSE), the GED distribution was found to best describe rainfall patterns. A key outcome is the development of a new rainfall model based on the GED distribution, allowing for the estimation of precipitation amounts for different durations and exceedance probabilities. However, the study highlights limitations, such as the need for more accurate local models and a standardized rainfall atlas for Poland. 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 paper presents a new methodology for generating continuous statistical distributions, integrating the exponentiated odds ratio within the framework of survival analysis. This new method enhances the flexibility and adaptability of distribution models to effectively address the complexities inherent in contemporary datasets. The core of this advancement is illustrated by introducing a particular subfamily, the “Type 2 Gumbel Weibull-G family of distributions”. We provide a comprehensive analysis of the mathematical properties of these distributions, including statistical properties such as density functions, moments, hazard rate and quantile functions, Rényi entropy, order statistics, and the concept of stochastic ordering. To test the robustness of our new model, we apply five distinct methods for parameter estimation. The practical applicability of the Type 2 Gumbel Weibull-G distributions is further supported through the analysis of three real-world datasets. These real-life applications illustrate the exceptional statistical precision of our distributions compared to existing models, thereby reinforcing their significant value in both theoretical and practical statistical applications. Full article

This study uses a hybrid model of the exponential generalised auto-regressive conditional heteroscedasticity (eGARCH)-extreme value theory (EVT)-Gumbel copula model to investigate the dependence structure between Bitcoin and the South African Rand, and quantify the portfolio risk of an equally weighted portfolio. The Gumbel copula, an extreme value copula, is preferred due to its versatile ability to capture various tail dependence structures. To model marginals, firstly, the eGARCH(1, 1) model is fitted to the growth rate data. Secondly, a mixture model featuring the generalised Pareto distribution (GPD) and the Gaussian kernel is fitted to the standardised residuals from an eGARCH(1, 1) model. The GPD is fitted to the tails while the Gaussian kernel is used in the central parts of the data set. The Gumbel copula parameter is estimated to be $\alpha =1.007$, implying that the two currencies are independent. At 90%, 95%, and 99% levels of confidence, the portfolio’s diversification effects (DE) quantities using value at risk (VaR) and expected shortfall (ES) show that there is evidence of a reduction in losses (diversification benefits) in the portfolio compared to the risk of the simple sum of single assets. These results can be used by fund managers, risk practitioners, and investors to decide on diversification strategies that reduce their risk exposure. Full article

In this paper, we address the analysis of bivariate lifetime data from a length-biased exponential distribution observed under Type II progressive censoring with random removals, where the number of units removed at each failure time follows a binomial distribution. We derive the likelihood function for the progressive Type II censoring scheme with random removals and apply it to the bivariate length-biased exponential distribution. The parameters of the proposed model are estimated using both likelihood and Bayesian methods for point and interval estimators, including asymptotic confidence intervals and bootstrap confidence intervals. We also employ different loss functions to construct Bayesian estimators. Additionally, a simulation study is conducted to compare the performance of censoring schemes. The effectiveness of the proposed methodology is demonstrated through the analysis of two real datasets from the industrial and computer science domains, providing valuable insights for illustrative purposes. 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

Exploring the realm of extreme weather events is indispensable for various engineering disciplines and plays a pivotal role in understanding climate change phenomena. In this study, we examine the ability of 10 probability distribution functions—including exponential, normal, two- and three-parameter log-normal, gamma, Gumbel, log-Gumbel, Pearson type III, log-Pearson type III, and SQRT-ET max distributions—to assess annual maximum 24 h rainfall series obtained over a long period (1972–2022) from three nearby meteorological stations. Goodness-of-fit analyses including Kolmogorov–Smirnov and chi-square tests reveal the inadequacy of exponential and normal distributions in capturing the complexity of the data sets. Subsequent frequency analysis and multi-criteria assessment enable us to discern optimal functions for various scenarios, including hydraulic engineering and sediment yield estimation. Notably, the log-Gumbel and three-parameter log-normal distributions exhibit superior performance for high return periods, while the Gumbel and three-parameter log-normal distributions excel for lower return periods. However, caution is advised regarding the overuse of log-Gumbel, due to its high sensitivity. Moreover, as our study considers the application of mathematical and statistical methods for the detection of extreme events, it also provides insights into climate change indicators, highlighting trends in the probability distribution of annual maximum 24 h rainfall. As a novelty in the field of functional analysis, the log-Gumbel distribution with a finite sample size is utilised for the assessment of extreme events, for which no previous work seems to have been conducted. These findings offer critical perspectives on extreme rainfall modelling and the impacts of climate change, enabling informed decision making for sustainable development and resilience. Full article

The recent flooding events in the UAE have emphasized the need for a reassessment of flood frequencies to mitigate risks. The exponential urbanization and climatic changes in the UAE require a reform for developing and updating intensity–duration–frequency (IDF) curves. This study introduces a methodology to develop and update IDF curves for the UAE at a high spatial resolution using CHIRPS (Climate Hazards Group InfraRed Precipitation with Station) data. A bias correction was applied to the CHIRPS data, resulting in an improved capture of extreme events across the country. The Gumbel distribution was the most suitable theoretical distribution for the UAE, exhibiting a strong fit to the observed data. The study also revealed that the CHIRPS-derived IDF curves matched the shape of IDF curves generated using rain gauges. Due to orographic rainfall in the northeastern region, the IDF intensities were at their highest there, while the aridity of inland regions resulted in the lowest intensities. These findings enhance our understanding of rainfall patterns in the UAE and support effective water resource management and infrastructure planning. This study demonstrates the potential of the CHIRPS dataset for IDF curve development, emphasizes the importance of performing bias corrections, and recommends tailoring adjustments to the intended application. Full article

This article is an extension of the Chris-Jerry distribution (C-JD) in that a two-parameter Chris-Jerry distribution (TPCJD) is suggested and its characteristics are studied. Based on the determined domain of attraction and other major statistical properties, the proposed TPCJD seems to fit into the Gumbel domain. Additionally, it has been confirmed that the stress strength is reliable. The tail study suggests that the TPCJD’s substantial tail makes it suited for a range of applications. The study took into account the single acceptance sampling approach using both simulation and real-life situations. The parameters of the TPCJD were estimated by some classical and Bayesian approaches. The mean squared errors (MSE), linear-exponential, and generalized entropy loss functions were deployed to obtain the Bayesian estimators aided by the Markov chain Monte Carlo (MCMC) simulation. An analysis of lifetime data on two events justified the use of the proposed distribution after comparing the results with some standard lifetime models. Full article

Considering the wide applicability of two-parameter distributions in the frequency analysis of extreme events, this article presents new elements regarding the use of thirteen two-parameter probability distributions, using three parameter estimation methods. All the necessary elements for the application of these distributions are presented using the method of ordinary moments (MOM), the method of linear moments (L-moments) and the method of high order linear moments (LH-moments). Only these three methods are analyzed, because they are usually applied in the analysis regarding the regionalization of extreme events. As a case study, the frequency analysis of the maximum annual flows on the Siret River, Lungoci station, Romania, was made. For the recommended methods (L- and LH-moments), from the thirteen analyzed distributions, the log-normal distribution had the best results, with the theoretical values L-coefficient of variation and L-kurtosis (0.297, 0.192, 0.323, 0.185, 0.336, and 0.185) best approximating the corresponding values of the recorded data (0.339, 0.185, 0.233, 0.199, 0.198, and 0.205). Full article

In survival analyses, infections at the catheter insertion site among kidney patients using portable dialysis machines pose a significant concern. Understanding the bivariate infection recurrence process is crucial for healthcare professionals to make informed decisions regarding infection management protocols. This knowledge enables the optimization of treatment strategies, reduction in complications associated with infection recurrence and improvement of patient outcomes. By analyzing the bivariate infection recurrence process in kidney patients undergoing portable dialysis, it becomes possible to predict the probability, timing, risk factors and treatment outcomes of infection recurrences. This information aids in identifying the likelihood of future infections, recognizing high-risk patients in need of close monitoring, and guiding the selection of appropriate treatment approaches. Limited bivariate distribution functions pose challenges in jointly modeling inter-correlated time between recurrences with different univariate marginal distributions. To address this, a Copula-based methodology is presented in this study. The methodology introduces the Kavya–Manoharan transformation family as the lifetime model for experimental units. The new bivariate models accurately measure dependence, demonstrate significant properties, and include special sub-models that leverage exponential, Weibull, and Pareto distributions as baseline distributions. Point and interval estimation techniques, such as maximum likelihood and Bayesian methods, where Bayesian estimation outperforms maximum likelihood estimation, are employed, and bootstrap confidence intervals are calculated. Numerical analysis is performed using the Markov chain Monte Carlo method. The proposed methodology’s applicability is demonstrated through the analysis of two real-world data-sets. The first data-set, focusing on infection and recurrence time in kidney patients, indicates that the Farlie–Gumbel–Morgenstern bivariate Kavya–Manoharan–Weibull (FGMBKM-W) distribution is the best bivariate model to fit the kidney infection data-set. The second data-set, specifically that related to UEFA Champions League Scores, reveals that the Clayton Kavya–Manoharan–Weibull (CBKM-W) distribution is the most suitable bivariate model for fitting the UEFA Champions League Scores. This analysis involves examining the time elapsed since the first goal kicks and the home team’s initial goal. Full article

This paper investigates an optimal reinsurance policy using a risk model with dependent claim and insurance premium by assuming that the insurance premium is random. Their dependence structure is modeled using Sarmanov’s bivariate exponential distribution and the Farlie–Gumbel–Morgenstern (FGM) copula-based bivariate exponential distribution. The reinsurance premium paid by the insurer to the reinsurer is fixed and is charged by the expected value premium principle (EVPP) and standard deviation premium principle (SDPP). The main objective of this paper is to determine the proportion and retention limit of the optimal combination of proportional and stop-loss reinsurance for the insurer. Specifically, with a constrained reinsurance premium, we use the minimization of the Value-at-Risk (VaR) of the insurer’s net cost. When determining the optimal proportion and retention limit, we provide some numerical examples to illustrate the theoretical results. We show that the dependence parameter, the probability of claim occurrence, and the confidence level have effects on the optimal VaR of the insurer’s net cost. Full article

In this research, we design the Farlie–Gumbel–Morgenstern bivariate moment exponential distribution, a bivariate analogue of the moment exponential distribution, using the Farlie–Gumbel–Morgenstern approach. With the analysis of real-life data, the competitiveness of the Farlie–Gumbel–Morgenstern bivariate moment exponential distribution in comparison with the other Farlie–Gumbel–Morgenstern distributions is discussed. Based on the Farlie–Gumbel–Morgenstern bivariate moment exponential distribution, we develop the distribution theory of concomitants of order statistics and derive the best linear unbiased estimator of the parameter associated with the variable of primary interest (study variable). Evaluations are also conducted regarding the efficiency comparison of the best linear unbiased estimator relative to the respective unbiased estimator. Additionally, empirical illustrations of the best linear unbiased estimator with respect to the unbiased estimator are performed. Full article