Search Results (14)

New copula families are constructed based on orthogonality in $L^2(0,1)$. Subclasses of idempotent copulas with square integrable densities are derived. It is shown that these copulas generate exchangeable Markov chains that behave as independent and identically distributed random variables conditionally on the initial variable. We prove that the extracted family of copulas is the only set of symmetric idempotent copulas with square integrable densities. We extend these copula families to asymmetric copulas with square integrable densities having special dependence properties. One of our extensions includes the Farlie–Gumbel–Morgenstern (FGM) copula family. The mixing properties of Markov chains generated by these copulas are established. The Spearman’s correlation coefficient ${\rho }_S$ is provided for each of these copula families. Some graphs are also provided to illustrate the properties of the copula densities. Full article

There are many families of bivariate distributions with given marginals. Most families, such as the Farlie–Gumbel–Morgenstern (FGM) and the Ali–Mikhail–Haq (AMH), are absolutely continuous, with an ordinary probability density. In contrast, there are few families with a singular part or a positive mass on a curve. We define a general condition useful to detect the singular part of a distribution. By continuous extension of the bivariate diagonal expansion, we define and study a wide family containing these singular distributions, obtain the probability density, and find the canonical correlations and functions. The set of canonical correlations is described by a continuous function rather than a countable sequence. An application to statistical inference is given. Full article

This study uses an effective, recently extended Farlie–Gumbel–Morgenstern (EFGM) family to derive the distribution of concomitants of K-record upper values (CKRV). For this CKRV, the negative cumulative residual extropy (NCREX), weighted NCREX (WNCREX), negative cumulative extropy (NCEX), and weighted NCEX (WNCEX) are theoretically and numerically examined. This study presents several beautiful symmetrical and asymmetric relationships that these inaccuracy measurements satisfy. Additionally, empirical estimations are provided for these measures, and their visualizations enable users to verify their accuracy. Full article

In this paper, we refined the concept of past extropy measure for concomitants of order statistics from Farlie-Gumbel-Morgenstern family. In addition, cumulative past extropy measure and dynamic cumulative past extropy measure for concomitant of rth order statistic are also conferred and their properties are studied. The problem of estimating the cumulative past extropy is investigated using empirical technique. The validity of the proposed estimator has been emphasized using simulation study. Full article

A novel four-parameter lifetime Lomax model is presented and investigated within the scope of this paper. The failure rate of the innovative model can be “monotonically decreasing failure rate,” “monotonically increasing failure rate,” or “constant failure rate,” and the density of the model can be “asymmetric right skewed,” “symmetric,” “asymmetric left skewed,” or “uniform density”. The new density is expressed as a blend of the Lomax densities that have been multiplied by an exponent. New bivariate Lomax types were created for our research. The maximum likelihood technique was utilized. We performed simulated experiments for the purpose of evaluating the finite sample behavior of maximum likelihood estimators, using “biases” and “mean squared errors” as our primary metrics of analysis. The novel distribution was evaluated based on a number of pertinent Lomax models, including Lomax extensions that were generated on the basis of odd log-logistic, Kumaraswamy, beta, gamma, and Topp–Leone families, among others. The newly developed extension demonstrated its relevance by predicting the service and failure times of datasets pertaining to aircraft windshields. 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

In this work, we reveal some distributional traits of concomitants of order statistics (COSs) arising from the extended Farlie–Gumbel–Morgenstern (FGM) bivariate distribution, which was developed and studied in recent work. The joint distribution and product moments of COSs for this family are discussed. Moreover, some useful recurrence relations between single and product moments of concomitants are obtained. In addition, the asymptotic behavior of the concomitant’s rank for order statistics (OSs) is studied. The information measures, differential entropy, Kullback–Leibler (KL) distance, Fisher information number (FIN), and cumulative past inaccuracy (CPI) are theoretically and numerically studied. Full article

In this work, we reveal some distributional characteristics of concomitants of generalized order statistics (GOS) with parameters that are pairwise different, arising from iterated Farlie–Gumbel–Morgenstern (IFGM) family of bivariate distributions. Additionally, the joint distribution and product moments of concomitants of GOS for this family are discussed. Moreover, some well-known information measures, i.e., extropy, cumulative residual extropy (CRJ), and negative cumulative extropy (NCJ), are derived. Applications of these results are given for order statistics, record values, and progressive type-II censored order statistics with uniform marginals distributions. Additionally, the issue of estimating the CRJ and NCJ is looked into, utilizing the empirical technique and the concomitant of GOS. Finally, bivariate real-world data sets have been analyzed for illustrative purposes, and the performance of the proposed method is quite satisfactory. Full article

In this paper we recall, extend and compute some information measures for the concomitants of the generalized order statistics (GOS) from the Farlie–Gumbel–Morgenstern (FGM) family. We focus on two types of information measures: some related to Shannon entropy, and some related to Tsallis entropy. Among the information measures considered are residual and past entropies which are important in a reliability context. Full article

In this paper, a new parametric compound G family of continuous probability distributions called the Poisson generalized exponential G (PGEG) family is derived and studied. Relevant mathematical properties are derived. Some new bivariate G families using the theorems of “Farlie-Gumbel-Morgenstern copula”, “the modified Farlie-Gumbel-Morgenstern copula”, “the Clayton copula”, and “the Renyi’s entropy copula” are presented. Many special members are derived, and a special attention is devoted to the exponential and the one parameter Pareto type II model. The maximum likelihood method is used to estimate the model parameters. A graphical simulation is performed to assess the finite sample behavior of the estimators of the maximum likelihood method. Two real-life data applications are proposed to illustrate the importance of the new family. Full article

A new family of probability distributions is defined and applied for modeling symmetric real-life datasets. Some new bivariate type G families using Farlie–Gumbel–Morgenstern copula, modified Farlie–Gumbel–Morgenstern copula, Clayton copula and Renyi’s entropy copula are derived. Moreover, some of its statistical properties are presented and studied. Next, the maximum likelihood estimation method is used. A graphical assessment based on biases and mean squared errors is introduced. Based on this assessment, the maximum likelihood method performs well and can be used for estimating the model parameters. Finally, two symmetric real-life applications to illustrate the importance and flexibility of the new family are proposed. The symmetricity of the real data is proved nonparametrically using the kernel density estimation method. Full article

In this paper, we propose a new three-parameter lifetime distribution for modeling symmetric real-life data sets. A simple-type Copula-based construction is presented to derive many bivariate- and multivariate-type distributions. The failure rate function of the new model can be “monotonically asymmetric increasing”, “increasing-constant”, “monotonically asymmetric decreasing” and “upside-down-constant” shaped. We investigate some of mathematical symmetric/asymmetric properties such as the ordinary moments, moment generating function, conditional moment, residual life and reversed residual functions. Bonferroni and Lorenz curves and mean deviations are discussed. The maximum likelihood method is used to estimate the model parameters. Finally, we illustrate the importance of the new model by the study of real data applications to show the flexibility and potentiality of the new model. The kernel density estimation and box plots are used for exploring the symmetry of the used data. Full article

In this article, we introduced a new extension of the binomial-exponential 2 distribution. We discussed some of its structural mathematical properties. A simple type Copula-based construction is also presented to construct the bivariate- and multivariate-type distributions. We estimated the model parameters via the maximum likelihood method. Finally, we illustrated the importance of the new model by the study of two real data applications to show the flexibility and potentiality of the new model in modeling skewed and symmetric data sets. Full article

In this paper, we introduce a new univariate version of the Lomax model as well as a simple type copula-based construction via Morgenstern family and via Clayton copula for introducing a new bivariate and a multivariate type extension of the new model. The new density has a strong physical interpretation and can be a symmetric function and unimodal with a heavy tail with positive skewness. The new failure rate function can be “upside-down”, “decreasing” with many different shapes and “decreasing-constant”. Some mathematical and statistical properties of the new model are derived. The model parameters are estimated using different estimation methods. For comparing the estimation methods, Markov Chain Monte Carlo (MCMC) simulations are performed. The applicability of the new model is illustrated via four real data applications, these data sets are symmetric and right skewed. We constructed a modified Chi-Square goodness-of-fit test based on Nikulin-Rao-Robson test in the case of complete and censored sample for the new model. Different simulation studies are performed along applications on real data for validation propose. Full article