An Improved Estimation of Parameter of Morgenstern-Type Bivariate Exponential Distribution Using Ranked Set Sampling

In this chapter, the authors suggest some improved versions of estimators of Morgenstern type bivariate exponential distribution (MTBED) based on the observations made on the units of ranked set sampling (RSS) regarding the study variable Y, which is correlated with the auxiliary variable X, where (X,Y) follows a MTBED. In this chapter, they firstly suggested minimum mean squared error estimator for estimation of 𝜃2 based on censored ranked set sample and their special case; further, they have suggested minimum mean squared error estimator for best linear unbiased estimator of 𝜃2 based on censored ranked set sample and their special cases; they also suggested minimum mean squared error estimator for estimation of 𝜃2 based on unbalanced multistage ranked set sampling and their special cases. Efficiency comparisons are also made in this work.

Reliability assessment of the standby system with dependent components by bivariate exponential distributions

In the reliability analysis of systems, all system components are often assumed independent and failure of any component does not depend on any other component. One of the reasons for doing so is that considerations of calculation and elegance typically pull in simplicity. But in real-world applications, there are very complex systems with lots of subsystems and a choice of multiple components that may interact with each other. Therefore, components of the system can be affected by the occurrence of a failure in any of the components. The purpose of this paper is to give an explicit formula for the computation of the reliability of a system with two parallel active components and one spare component. It is assumed that parallel components are dependent and operate simultaneously. Two distributions of Freund’s bivariate exponential and Marshall–Olkin bivariate exponential are used to model dependency between components. The results show that the reliability of the system with Freund’s bivariate exponential distribution has lower reliability. The circumstances that lead to them, namely load-sharing in the case of Freund, results in lower reliability. Finally, a numerical example is solved to evaluate the proposed model and sensitivity analysis is performed on the system reliability function. The obtained results show that because the proposed model is influenced by the dependency, compared to traditional models, it has the characteristic of leading to reduced time to (first) failure for achieving specified reliability.

The relation between defending, (dis)liking, and the classroom bullying norm: A cross-sectional social network approach in late childhood

This study investigates the extent to which defending victims of bullying depends on liking and disliking and its relation with the classroom bullying norm (descriptive and popularity) in a sample of 1,272 students (50.8% boys) in 48 fifth-grade classrooms. Social network analysis with bivariate exponential random graph modelings showed that children are more likely to defend victims whom they like, who like them, and who are liked by the same classmates than victims who they dislike, who dislike them, and with whom they share antipathies by and to the same classmates. In addition, the analysis showed that bullying norms had an inconclusive effect on the relation between defending and (dis)liking.

A New Family of Bivariate Exponential Distributions with Negative Dependence Based on Counter-Monotonic Shock Method

We introduce a new family of bivariate exponential distributions based on the counter-monotonic shock model. This family of distribution is easy to simulate and includes the Fréchet lower bound, which allows to span all degrees of negative dependence. The construction and distributional properties of the proposed bivariate distribution are presented along with an estimation of the parameters involved in our model based on the method of moments. A simulation study is carried out to evaluate the performance of the suggested estimators. An extension to the general model describing both negative and positive dependence is sketched in the last section of the paper.

COMPARISON OF EXPONENTIAL COVARIANCE FUNCTIONS FOR BIVARIATE GEOSTATISTICAL DATA

In the analysis of multivariate spatial random elds, it is essential to dene a covariance structure that adequately models the relationship between the variables under study. We propose a covariance structure with exponential correlation function for bivariate random elds, the SEC model. We compare the SEC model fits with the bivariate separable exponential model and the bivariate exponential model with constraints, which are particular cases of the full bivariate Matern model, presented in the literature. A simulation study assess characteristics of the proposed model. The model is tted to a weather data set from Brazil, bearing in mind the importance of analyzing climate data to predict adverse environmental conditions. Predictive measures are used to compare the models under study. The satisfactory results compared to the models considered and the simpler structure makes the SEC model an alternative for the analysis of bivariate spatial elds.

A new extreme value copula and new families of univariate distributions based on Freund’s exponential model

The use of the exponential distribution and its multivariate generalizations is extremely popular in lifetime modeling. Freund’s bivariate exponential model (1961) is based on the idea that the remaining lifetime of any entity in a bivariate system is shortened when the other entity defaults. Such a model can be quite useful for studying systemic risk, for instance in financial systems. Guzmics and Pflug (2019) revisited Freund’s model, deriving the corresponding bivariate copula and examined some characteristics of it; furthermore, we opened the door for a multivariate setting. Now we present further investigations in the bivariate model: we compute the tail dependence coefficients, we examine the marginal and joint distributions of the componentwise maxima, which leads to an extreme value copula, which – to the best of our knowledge – has not been investigated in the literature yet. The original bivariate model of Freund has been extended to more variables by several authors. We also turn to the multivariate setting, and our focus is different from that of the previous generalizations, and therefore it is novel: examining the distribution of the sum and of the average of the lifetime variables (provided that the shock parameters are all the same) leads to new families of univariate distributions, which we call Exponential Gamma Mixture Type I and Type II (EGM) distributions. We present their basic properties, we provide asymptotics for them, and finally we also provide the limiting distribution for the EGM Type II distribution.

A Note on the Time to Failure of a Two-Unit Parallel Redundant System with Deterioration on a Lattice

In this paper, we consider a two-unit parallel redundant system with deterioration on a lattice, where each unit has multi-stage deterioration levels, say, n levels. The transition from one deterioration level to the subsequent level occurs following the well-known Marshall-Olkin bivariate exponential distribution. We derive the closed form of the Laplace transform of the time to system failure in the two-unit parallel redundant system with deterioration on n×n lattice without repair and simultaneous failure, as well as the simple system on 3×3 lattice.

Multivariate Inverted Kumaraswamy Distribution: Derivation and Estimation

Industrial revolution leads to the manufacturing of multicomponent products; to guarantee the sufficiency of the product and consumer satisfaction, the producer has to study the lifetime of the products. This leads to the use of bivariate and multivariate lifetime distributions in reliability engineering. The most popular and applicable is Marshall–Olkin family of distributions. In this paper, a new bivariate lifetime distribution which is the bivariate inverted Kumaraswamy (BIK) distribution is found and its properties are illustrated. Estimation using both maximum likelihood and Bayesian approaches is accomplished. Using different selection criteria, it is found that BIK provides the best performance compared with other bivariate distributions like bivariate exponential and bivariate inverse Weibull distributions. As a generalization, the multivariate inverted Kumaraswamy (MIK) distribution is derived. Few studies have been conducted on the multivariate Marshall–Olkin lifetime distributions. To the best of our knowledge, none of them handle estimation process. In this paper, we developed an algorithm to show how to estimate the unknown parameters of MIK using both maximum likelihood and Bayesian approaches. This algorithm could be applied in estimating other Marshall–Olkin multivariate lifetime distributions.