Bivariate Exponential Distribution Research Articles

A family of bivariate exponential distributions and their copulas

In the present paper we derive a family of bivariate exponential distributions based on an extended lack of memory property of a class of univariate distributions. These bivariate exponential laws are time transformed exponential models possessing Archimedean copulas. The bivariate aging properties and various dependence relationships are characterized in terms of the univariate aging concepts of the baseline distributions from which the bivariate models are generated.

Bayes estimation of Moran–Downton bivariate exponential distribution based on censored samples

In this paper, we discuss a Bayesian estimation procedure for the parameters in a Moran–Downton bivariate exponential distribution based on complete and censored samples. A Markov-chain Monte Carlo method is used to obtain the Bayes estimates of the parameters. An intensive simulation experiment is conducted to study the performance of the proposed Bayesian estimation procedure. Discussions and suggestions are provided based on the simulation results. A numerical example is presented to illustrate the Bayesian estimation procedure developed here and some concluding remarks are provided.

Efficient simulation of complete and censored samples from common bivariate exponential distributions

Let $$(X_{i:n},Y_{[i:n]})$$ be the vector of the $$i$$ th $$X$$ -order statistic and its concomitant observed in a random sample of size $$n$$ where the marginal distribution of $$X$$ is absolutely continuous. We describe some general algorithms for simulation of complete and Type II censored samples $$\{(X_{i:n}, Y_{[i:n]}), 1 \le i \le r \le n\}$$ from such bivariate distributions. We study in detail several algorithms for simulating complete and censored samples from Downton, Marshall---Olkin, Gumbel (Type I) and Farlie-Gumbel-Morgenstern bivariate exponential distributions. We show that the conditioning method in conjunction with an efficient simulation of exponential order statistics that exploits the independence of spacings provides the best method with substantial savings over the basic method. Efficient simulation is essential for investigating the finite-sample distributional properties of functions of order statistics and their concomitants.

Optimal allocation for estimating the correlation coefficient of Morgenstern type bivariate exponential distribution by ranked set sampling with concomitant variable

Ranked set sample is applicable whenever ranking of a set of sample units can be done easily by a judgement method of the study variable or of the auxiliary variable. This paper considers ranked set sample based on the auxiliary variable X which is correlated with the study variable Y, where (X, Y) follows Morgenstern type bivariate exponential distribution. The authors discuss the optional allocation for unbiased estimators of the correlation coefficient ρ of the random variables X and Y when the auxiliary variable X is used for ranking the sample units and the study variable Y is measured for estimating the correlation coefficient. This paper first gives a class of unbiased estimators of ρ when the mean θ of the study variable Y is known and obtains an essentially complete subclass of this class. Further, the optimal allocation of the unbiased estimators is found in this subclass and is proved to be Bayes, admissible, and minimax. Finally, the unbiased estimator of ρ under the optimal allocation in the case of known θ is reformed for estimating ρ in the case of unknown θ, and the reformed estimator is shown to be strongly consistent.

ON A CLASS OF GENERALIZED MARSHALL–OLKIN BIVARIATE DISTRIBUTIONS AND SOME RELIABILITY CHARACTERISTICS

We consider here a general class of bivariate distributions from reliability point of view, and refer to it as generalized Marshall–Olkin bivariate distributions. This class includes as special cases the Marshall–Olkin bivariate exponential distribution and the class of bivariate distributions studied recently by Sarhan and Balakrishnan [25]. For this class, the reliability, survival, hazard, and mean residual life functions are all derived, and their monotonicity is discussed for the marginal as well as the conditional distributions. These functions are also studied for the series and parallel systems based on this bivariate distribution. Finally, the Clayton association measure for this bivariate model is derived in terms of the hazard gradient.

Estimation and Confidence Intervals for Clock Offset in Networks with Bivariate Exponential Delays

There is a large and increasing literature on statistical modeling-based estimation of the offset between two clocks. Recent work has focused on the construction of confidence intervals for offset. However, in most of this work it has been assumed that the network delays that occur during the synchronization process are independent. The network delays are often modeled as independent exponential random variables. Thus, we introduce the use of a bivariate exponential distribution to capture the anticipated correlation between the network delays and derive a maximum likelihood estimator and a confidence interval procedure for the offset parameter. We then illustrate how use of the independent model for network delays can lead to improper inference about the offset parameter.

基于F-G-M Copula的一个二元概率模型

本文通过对F-G-M二元指数分布的弱成分进行改进，得到了一个基于F-G-M Copula的二元概率模型，利用这个模型，讨论了弱成分改进后的串联系统可靠度。 In this paper, a bivariate probability model based on the F-G-M copula is derived by improving the weaker component of the F-G-M bivariate exponential distribution. The reliability of the series system after improving the weaker conponent is computed by using this model.

A New Warranty Policy With Failure Times and Warranty Servicing Times

We develop a new warranty policy with respect to the failure time and warranty servicing time, where those two variables are statistically correlated in bivariate distributions. Based on the developed approaches, we investigate the properties of the bivariate function, and obtain the number of warranty services in a warranty period using the field data. The warranty service includes the repair service, and replacement service. An approach of analyzing the warranty cost is proposed for a product in which the failure time and warranty servicing time are used simultaneously to determine the eligibility of a warranty claim. In the vast literature of the warranty modeling, the two-dimensional warranty is commonly consisting of the usage, and the age. In this study, we focus on the failure time and warranty servicing time as two factors under the warranty policy. Using the field data, the parameters are calculated, and the warranty model is investigated using a bivariate exponential distribution (BED) which was the best fit distribution based on the field data. Numerical examples are discussed to demonstrate the applicability of the methodology derived in the paper. The results of our research in this paper indicate that more sophisticated bivariate distributions other than the BED is worth the effort, as a future research problem, or using other data sets in a general industrial application setting.

Bayes estimation for the Marshall–Olkin bivariate Weibull distribution

In this paper, we consider the Bayesian analysis of the Marshall–Olkin bivariate Weibull distribution. It is a singular distribution whose marginals are Weibull distributions. This is a generalization of the Marshall–Olkin bivariate exponential distribution. It is well known that the maximum likelihood estimators of the unknown parameters do not always exist. The Bayes estimators are obtained with respect to the squared error loss function and the prior distributions allow for prior dependence among the components of the parameter vector. If the shape parameter is known, the Bayes estimators of the unknown parameters can be obtained in explicit forms under the assumptions of independent priors. If the shape parameter is unknown, the Bayes estimators cannot be obtained in explicit forms. We propose to use the importance sampling method to compute the Bayes estimators and also to construct associated credible intervals of the unknown parameters. The analysis of one data set is performed for illustrative purposes. Finally we indicate the analysis of data sets obtained from series and parallel systems.

On Some Properties of Bivariate Exponential Distributions

We show that the two bivariate exponential distributions constructed in Bladt and Nielsen[ 5 ] have the maximum and minimum correlation coefficients for any given order. We also generalize their constructions to the case where the matrix representations of the two (marginal) exponential distributions have different orders and show that the new constructions also have the maximum and minimum correlation coefficients. Our main tool is a majorization result for a special set of PH-generators.

Estimation of Freund model under censored data

Abstract We consider a life testing experiment in which several two-component shared parallelsystems are put on test, and the test is terminated at a predesigned experiment time.In this thesis, the maximum likelihood estimators for parameters of Freund’s bivariateexponential distribution under the system level life testing are obtained. Results ofcomparative studies based on Monte Carlo simulation are presented.Keywords: Failure rate, Freund’s bivariate exponential model, maximum likelihoodestimator, shared parallel systems. 1. Introduction On the life testing of the two-component system, the system works normally even thoughone of the two components has a failure. If each lifetime of the components is Xand Y,it is generally assumed that Xand Y are reciprocally dependent random variables. In thetwo-component system, if one of the components has a failure, it inuences the failure rateof the other. This is called the shared parallel system. It might apply to the study of enginefailures in two-engine planes, or to the performance of a person’s eyes, ears, kidneys, orother pared organs. That is, the lifetimes of each component of the shared parallel system areinterdependent. The lifetimes of the shared parallel system which consists of two componentshave a bivariate exponential distribution (BVE) and this BVE has been studied a lot so far.Let X and Y be respectively the random variables for lifetimes of component 1 andcomponent 2 in two-component shared parallel system, and and be respectively thefailure rates of component 1 and component 2 in the system. If the rst failure of thecomponent 1 occurs at x, the failure rate of the component 2 changes over

Location models with stochastic demand points

This paper presents two approaches for solving both the minimax and minisum location problems and as well as a bi-objective location problem. The bi-objective location problem is a combination of both minimax and minisum location problems with recti-linear distances and randomly distributed destinations. Wesolowsky. G.O. [Journal of Regional Science 18: 53–60, 1977] has considered the stochastic extension of Weber problem with rectilinear distance norm and suggested some iterative method after dividing the problem into two parts, one for the X coordinates and other for the Y coordinates, where these co-ordinates may be correlated. In the present investigation exact approaches for both Minisum and Minimax objectives with randomly distributed destinations have been considered separately with both bi-variate exponential and bi-variate uniform distributions and formulated as a non-linear programming problem. Also the area restriction concept has been introduced so that the facility to be located should be within certain restricted area. The consideration of area restrictions has been implemented by incorporating a convex polygon as the constraint set. It has been proved that the solution obtained by this method will give the global optimal solution. The second part of this investigation is the bi-objective problem. The aim of the bi-objective problem is to find the satisfying solution which is desired in many realistic situations. Situation in which the proposed model is applicable is the location of a new warehouse where the sources of goods shipped to the warehouse and also the supply points to which the goods will be shipped is not known in advance. Due to the volatile and competitive nature of the market the supply and the demand points or in other words the variable points are not known in advance but some probability distribution can be predicted. It is required to find the location of a new warehouse which is closer to the variable points and simultaneously the total shipping cost per month from the warehouse to the variable points is minimum. For both single as well as bi-objective problems rectilinear distance norm has been considered as it is more appropriate to the different realistic situations. Two types of compromise solution methods have been suggested to get a satisfactory solution of the bi-objective problem.

Freund 이변량 지수분포의 매개변수 추정과정 검토

본 연구에서는 연최대치 독립 호우사상의 결정에 사용되는 Freund 이변량 지수분포의 매개변수 추정과정을 구체적으로 검토하였다. 먼저, 모멘트법을 이용하는 경우를 구체적으로 검토하고, 그 결과를 최우도법을 적용한 결과와 비교하였다. 두 방법을 1961~2010년 서울지점의 시강우 자료에 적용하여 연최대치 독립 호우사상을 선정하고, 그 결과를 비교 검토하였다. 이러한 과정을 통해 얻은 결과는 다음과 같다. 첫째, 매개변수 추정방법으로 모멘트법을 적용하는 경우에는 두변량의 평균과 분산뿐만 아니라 상관계수도 고려해 주어야 하는 것으로 나타났다. 둘째, 최우도법은 두변량의 평균에 대한 재현성이 우수하고, 모멘트법은 분산의 경년변동을 잘 나타내는 것으로 나타났다. 셋째, 모멘트법과 최우도법을 통해 선정한 연최대치 독립 호우사상들은 대체로 유사한 것으로 나타났다. 다르게 선정된 호우사상은 최우도법의 경우에는 총 강우량이 큰 것, 모멘트법의 경우에는 강우강도가 큰 것으로 나타났다. This study reviewed the parameter estimation procedure of the Freund bivariate exponential distribution for the decision of the annual maximum rainfall event. The method of moments was reviewed first, whose results were compared with those from the method of maximum likelihood. Both methods were applied to the hourly rainfall data of the Seoul rain gauge station measured from 1961 to 2010 to select the annual maximum rainfall events, which were also compared each other. The results derived are as follows. First, when applying the method of moments for the parameter estimation, it was found necessary to consider the correlation coefficient between the two variables as well as the mean and variance. Second, the method of maximum likelihood was better to reproduce the mean, but the method of moments was better to reproduce the annual variation of the variance. Third, The annual maximum rainfall events derived were very similar in both cases. Among differently selected annual maximum rainfall events, those with the higher rainfall amount were selected by the method of maximum likelihood, but those with the higher rainfall intensity by the method of moments.

Fisher information in censored samples from Downton's bivariate exponential distribution

We develop a simple approach to finding the Fisher information matrix (FIM) for a single pair of order statistic and its concomitant, and Type II right, left, and doubly censored samples from an arbitrary bivariate distribution. We use it to determine explicit expressions for the FIM for the three parameters of Downton's bivariate exponential distribution for single pairs and Type II censored samples. We evaluate the FIM in censored samples for finite sample sizes and determine its limiting form as the sample size increases. We discuss implications of our findings to inference and experimental design using small and large censored samples and for ranked-set samples from this distribution.

Adaptive walks on correlated fitness landscapes with heterogeneous connectivities

We propose a model for studying the statistical properties of adaptive walks on correlatedfitness landscapes which are established in genotype spaces of complex structure. Thefitness distribution on the genotype space follows either the bivariate Gaussian distributionor the bivariate exponential distribution. In both cases the degree of correlation of thefitness landscape can be tuned by using a single parameter. To perform the adaptivewalks two distinct rules are applied: the random adaptation walk (RAW) and thegradient adaptation walk (GAW). While for the RAW the mean walk length, , is a monotonic increasing function of the connectivity of the genotype space, for theGAW is a one-humped function. The RAW produces longer adaptive walks compared to theGAW, though its performance is slightly poorer and thereby the local maxima reachedby the GAW algorithm are usually closer to the global optimum of the fitnesslandscape.

Reliability Analysis of A Two Non-Identical Unit System with Repair and Replacements having Correlated Failures and Repairs

A two non-identical unit parallel system model is investigated and analysed. The system consists of two non-identical units, priority and non-priority arranged in parallel configuration. The priority unit is given preference in repair over non-priority unit. The priority unit is repairable but non-priority unit is not repairable and to be replaced after a random period of operation. The non-priority unit gives a signal for its replacement before going into failure mode and may be replaced by a new unit just after giving signal or upon failure. The failure and repair times of the priority unit are taken to be correlated random variables having bivariate exponential distribution. The failure time and the time after which non-priority unit gives signal are exponential random variables with different parameters and the time for replacement of non-priority unit is taken as general. The reliability analysis of this model has been carried out by using regenerative point technique.

A Nonparametric Signed-Rank Control Chart for Bivariate Process Location

A nonparametric control chart based on a bivariate signed-rank test is developed for monitoring the changes in the location of a bivariate process. The average run length performance of the proposed nonparametric chart is investigated using a simulation study and is compared with a parametric control chart under bivariate normal and bivariate double exponential distributions. Further the performance of the proposed nonparametric chart is improved by using runs rules and the concept of a synthetic control chart.

A method of moments to estimate bivariate survival functions: the copula approach

In this paper we discuss the problem on parametric and non parametric estimation of the distributions generated by the Marshall-Olkin copula. This copula comes from the Marshall-Olkin bivariate exponential distribution used in reliability analysis. We generalize this model by the copula and different marginal distributions to construct several bivariate survival functions. The cumulative distribution functions are not absolutely continuous and they unknown parameters are often not be obtained in explicit form. In order to estimate the parameters we propose an easy procedure based on the moments. This method consist in two steps: in the first step we estimate only the parameters of marginal distributions and in the second step we estimate only the copula parameter. This procedure can be used to estimate the parameters of complex survival functions in which it is difficult to find an explicit expression of the mixed moments. Moreover it is preferred to the maximum likelihood one for its simplex mathematic form; in particular for distributions whose maximum likelihood parameters estimators can not be obtained in explicit form.

Regression mean residual life of a parallel system of dependent components

In this paper, we consider a system consisting of two dependent components and we are interested in the average remaining life of the component that fails last when (i) the first failure occurs at time t and (ii) the first failure occurs after time t. For both the cases, expressions are derived in the case of general bivariate normal distribution and a class of bivariate exponential distribution including bivariate exponential distribution of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution.

A Bayesian analysis for the Block and Basu bivariate exponential distribution in the presence of covariates and censored data

In this paper, we introduce a Bayesian Analysis for the Block and Basu bivariate exponential distribution using Markov Chain Monte Carlo (MCMC) methods and considering lifetimes in presence of covariates and censored data. Posterior summaries of interest are obtained using the popular WinBUGS software. Numerical illustrations are introduced considering a medical data set related to the recurrence times of infection for kidney patients and a medical data set related to bone marrow transplantation for leukemia.