Bivariate Exponential Distribution Research Articles

The impact of the correlation coefficient of interarrival and service times on queueing performance: The [formula omitted] case

This paper is concerned on an M/M/1 queue with correlated interarrival and service times. In particular, we assume that the interarrival time and service time of a customer have a bivariate exponential distribution. By utilizing a Markov modulated fluid flow (MMFF) process associated with the age process of the customer in service, we obtain a number of queueing quantities in closed form. Using the solutions, the impact of the correlation coefficient of the interarrival and service times on a variety of queueing quantities is explored quantitatively and qualitatively. Specifically, we establish a monotonic relationship between the decay rates of queueing quantities and the correlation coefficient of the interarrival and service times. We also show that the decay rates are increasing in the correlation coefficient. In addition, queues with the maximum/minimum correlation coefficient are analyzed. Two examples are presented to demonstrate the importance of using the correlation coefficient in queueing performance/economic analysis.

Phase II control chart for monitoring Gumbel’s bivariate exponential distribution

Schemes for monitoring two or more time between events (TBE) data are becoming increasingly significant in statistical process monitoring. In practice, parameters must be estimated prior to the start of process monitoring. In this paper, we first propose a multivariate time between events (MTBE) control chart based on the one-sample log-likelihood ratio test for monitoring the Gumbel’s bivariate exponential (GBE) distribution, and then we analyze the effect of parameter estimation on this chart. To avoid the effect of parameter estimation, we then propose a method based on the two-sample log-likelihood ratio test for monitoring the shifts occurring in either the scale parameter or the dependence parameter or both for the GBE distribution process. The proposed method is designed with two-sample log-likelihood ratio test for univariate marginal distribution function and Gumbel copula function. A Max-type function-based approach is implemented for joint monitoring of margin distribution and copula-related structure. In contrast to the traditional schemes used to monitor bivariate processes, the strength of our proposed technique lies in the ability to identify exactly which component of the scale parameter or dependence parameter is responsible for the source of the signal. The performance of the proposed chart is analyzed via Monte Carlo simulations. Finally, two examples are mentioned to illustrate the implementation process of the proposed scheme.

Robust estimation of dependent competing risk model under interval monitoring and determining optimal inspection intervals

AbstractRecently, a growing interest is evident in modelling dependent competing risks in lifetime prognosis problems. In this work, we propose to model the dependent competing risks by Marshal‐Olkin bivariate exponential distribution. The observable data consists of a number of failures due to different causes across different time intervals. The failure count data is common in instances like one‐shot devices where the state of the subjects is inspected at different inspection times rather than the exact failure times. The point estimation of the lifetime distribution in the presence of competing risk has been studied through a divergence‐based robust estimation method called minimum density power divergence estimation (MDPDE) with and without constraint. The optimal value of the tuning parameter has been obtained. The testing of the hypothesis is performed based on a Wald‐type test statistic. The influence function is derived for the point estimator and the test statistic, reflecting the degree of robustness. Another key contribution of this work is determining the optimal inspection times based on predefined objectives. This article presents the determination of multi‐criteria‐based optimal design. Population‐based heuristic algorithm nondominated sorting‐based multiobjective Genetic algorithm is exploited to solve this optimization problem.

Bayesian Inference for a Hidden Truncated Bivariate Exponential Distribution with Applications

In many real-life scenarios, one variable is observed only if the other concomitant variable or the set of concomitant variables (in the multivariate scenario) is truncated from below, above, or from a two-sided approach. Hidden truncation models have been applied to analyze data when bivariate or multivariate observations are subject to some form of truncation. While the statistical inference for hidden truncation models (truncation from above) under the frequentist and the Bayesian paradigms has been adequately discussed in the literature, the estimation of a two-sided hidden truncation model under the Bayesian framework has not yet been discussed. In this paper, we consider the Bayesian inference for a general two-sided hidden truncation model based on the Arnold–Strauss bivariate exponential distribution. In addition, a Bayesian model selection approach based on the Bayes factor to select between models without truncation, with truncation from below, from above, and two-sided truncation is also explored. An extensive simulation study is carried out for varying parameter choices under the conjugate prior set-up. For illustrative purposes, a real-life dataset is re-analyzed to demonstrate the applicability of the proposed methodology.

An EM algorithm for absolutely continuous Marshall-Olkin bivariate Pareto distribution with location and scale

Block and Basu bivariate exponential distribution is one of the most popular absolutely continuous bivariate distributions. In this article, we have considered a Block–Basu type bivariate Pareto distribution. Here in the standard manner, first Marshall-Olkin type singular bivariate distribution has been constructed, and then by taking away the singular component similar to the Block and Basu model, an absolute continuous BB-BVPA model has been constructed. Further, the location and scale parameters also have been introduced. Therefore, the model has seven parameters. We drive different properties of this absolutely continuous distribution. Since the maximum likelihood estimators of the parameters cannot be expressed in a closed form, we propose to use an EM algorithm to compute the estimators of the model parameters. Some simulation experiments have been performed for illustrative purposes. We fit the model to rainfall data in the context of landslide risk estimation. The methods are implemented in the publicly available R package bvpa.

USE OF RANKED SET SAMPLING IN THE ESTIMATION OF PARAMETERS 𝝁𝟐 AND 𝝈𝟐 OF MORGENSTERN TYPE BIVARIATE EXPONENTIAL DISTRIBUTION

In this article we have derived the best linear unbiased estimator (BLUE) of the parameters 2 and 2 involved in the Morgenstern type bivariate exponential distribution by ranked set sampling (RSS). Also we have evaluated the BLUE's of 2 and 2 involved in the Morgenstern type bivariate exponential distribution by concomitants of ranked set sampling for some specific values of the sample size n. The efficiency comparison of the proposed estimators with that of the estimators using concomitants of order statistics and modified ranked set sampling is also made.

Modeling the Optimal Combination of Proportional and Stop-Loss Reinsurance with Dependent Claim and Stochastic Insurance Premium

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.

Inferences Based on Correlated Randomly Censored Gumbel’s Type-I Bivariate Exponential Distribution

Inferences Based on Correlated Randomly Censored Gumbel’s Type-I Bivariate Exponential Distribution

A New Stochastic Order of Multivariate Distributions: Application in the Study of Reliability of Bridges Affected by Earthquakes

In this article, we introduce and study a new stochastic order of multivariate distributions, namely, the conditional likelihood ratio order. The proposed order and other stochastic orders are analyzed in the case of a bivariate exponential distributions family. The theoretical results obtained are applied for studying the reliability of bridges affected by earthquakes. The conditional likelihood ratio order involves the multivariate stochastic ordering; it resembles the likelihood ratio order in the univariate case but is much easier to verify than the likelihood ratio order in the multivariate case. Additionally, the likelihood ratio order in the multivariate case implies this ordering. However, the conditional likelihood ratio order does not imply the weak hard rate order, and it is not an order relation on the multivariate distributions set. The new conditional likelihood ratio order, together with the likelihood ratio order and the weak hazard rate order, were studied in the case of the bivariate Marshall–Olkin exponential distributions family, which has a lack of memory type property. At the end of the paper, we also presented an application of the analyzed orderings for this bivariate distributions family to the study of the effects of earthquakes on bridges.

Reliability Estimation for Dependent Left-Truncated and Right-Censored Competing Risks Data with Illustrations

In this paper, a competing risks model with dependent causes of failure is considered under left-truncated and right-censoring scenario. When the dependent failure causes follow a Marshall–Olkin bivariate exponential distribution, estimation of model parameters and reliability indices are proposed from classic and Bayesian approaches, respectively. Maximum likelihood estimators and approximate confidence intervals are constructed, and conventional Bayesian point and interval estimations are discussed as well. In addition, E-Bayesian estimators are proposed and their asymptotic behaviors have been investigated. Further, another objective-Bayesian analysis is also proposed when a noninformative probability matching prior is used. Finally, extensive simulation studies are carried out to investigate the performance of different methods. Two real data examples are presented to illustrate the applicability.

Mixture Model on Development of Bivariate Product Distribution and its Properties

In the study, some bivariate distributions were developed from mixture model offspring, using the Independent (Product) distribution approach. These developments are categorized under the IID and IInD: where the Bivariate Exponential distribution, Bivariate Lindley distribution and Bivariate Juchez distribution are constructed as IIDs; and Bivariate Exponential-Lindley distribution, Bivariate Exponential-Juchez distribution and Bivariate Lindley-Juchez distribution as (IInDs). The properties of these distributions which involve: the shape of the bivariate PDFs, moments, moment generating function, mean, covariance and coefficient of correlation, maximum likelihood estimator, reliability analysis, renewal property and probability patterns; are studied across the distributions. Finally, under renewal properties, functions are derived which can model two-dimensional queuing and renewal processes, for events where the arrival and service times are dependent.

Reliability analysis of a two-unit‏ ‏standby system under Marshall–Olkin dependency

PurposeOne of the common approaches to improve systems reliability is using standby redundancy. Although many works are available in the literature on the applications of standby redundancy, the system components are assumed to be independent of each other. But, in reality, the system components can be dependent on one another, causing the failure of each component to affect the failure rate of the remaining active components. In this paper, a standby two-unit system is considered, assuming a dependency between the switch and its associated active component.Design/methodology/approachThis paper assumes that the failures between the switch and its associated active component follow the Marshall–Olkin exponential bivariate exponential distribution. Then, the reliability analysis of the system is done using the continuous-time Markov chain method.FindingsThe derived equations application to determine the system steady-state availability, system reliability and sensitivity analysis on the mean time to failure is demonstrated using a numerical illustration.Originality/valueAll previous models assumed independency between the switch and the associated active unit in the standby redundancy approach. In this paper, the switch and its associated component are assumed to be dependent on each other.

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In this article we have derived the best linear unbiased estimators (BLUE’s) of the parameters µ2 and involved σ2 in the Morgenstern type bivariate exponential distribution (MTBED) by concomitants of record values. Efficiency comparisons are also made on the proposed estimators with that of competing unbiased estimators for µ2 and σ2 involved in MTBED.

Enhancing the detection power of multivariate time between events control charts for Gumbel’s bivariate exponential distribution

One of the most important goals of statistical process control is quickly detecting any changes in the processes and consequently reducing process variability. Control charts are graphical tools that are widely used to achieve these goals. In this regard, monitoring methods based on the time between events (TBEs) have received much attention in recent years. These methods are widespread for monitoring high-quality processes and temporal quality characteristics. Although univariate TBE control charts have been extensively studied in the literature, multivariate TBE control charts need more investigation. In this paper, the performance of the multivariate exponentially weighted moving average (MEWMA), paired individual exponentially weighted moving average (PEWMA), and multivariate cumulative sum (MCUSUM) control charts for the bivariate TBEs which follow Gumbel's bivariate exponential (GBE) distribution are compared based on the average time to signal (ATS) metric. Also, the multivariate rate (MRA) control chart is proposed for enhancing the performance of the multivariate TBE control charts. Real-world examples are used to illustrate the applicability of the proposed monitoring scheme. The results of the performance evaluation show that the proposed approach can considerably enhance the performance of the multivariate TBE control charts.

On the Representation of Correlated Exponential Distributions by Phase Type Distributions

In this paper we present results for bivariate exponential distributions which are represented by phase type distributions. The paper extends results from previous publications [3, 11] on this topic by introducing new representations that require a smaller number of phases to reach some correlation coefficient and introduces different ways to describe correlation between exponentially distributed random variables.

A Variable Sampling Interval Multivariate Exponentially Weighted Moving Average Control Chart for Monitoring the Gumbel’s Bivariate Exponential Data

The general assumption for designing a multivariate control chart is that the multiple variables are independent and normally distributed. This assumption may not be tenable in many practical situations, because multiple variables with dependency often need to be monitored simultaneously to ensure the process is in-control. The Gumbel’s Bivariate Exponential (GBE) distribution is considered a better model for skewed data with dependency in reliability analysis. In this paper, a Multivariate Exponentially Weighted Moving Average (MEWMA) control chart with Variable Sampling Interval (VSI) feature is developed to monitor the mean vector of GBE model. Monte-Carlo simulations are used to calculate the ATS (Average Time to Signal) values of the proposed VSI MEWMA GBE chart for three different types of shifts. Meanwhile, some tables are provided to show the ATS performances of the proposed chart with different designed parameters. Furthermore, both zero- and steady-state ATS performances of the proposed VSI MEWMA GBE chart are compared with those of the FSI (Fix Sampling Interval) MEWMA chart. Comparative results show that the proposed chart is superior to its FSI counterpart in monitoring the GBE data. In addition, a numerical example is provided to show that the proposed chart performs well in monitoring GBE data.

Robust Tests for the Mean Difference in Paired Data using Jackknife Resampling Technique

The paired sample t-test is a type of classical test statistics that is used to test the difference between two means in paired data, but it is not robust against the violation of the normality assumption. In this paper, some alternative robust tests are suggested by combining the Jackknife resampling with each of the Wilcoxon signed-rank test for small sample size and Wilcoxon signed-rank test for large sample size, using normal approximation. The Monte Carlo simulation experiments were employed to study the performance of the test statistics of each of these tests depending on the type one error rates and the power rates of the test statistics. All these tests were applied on different sample sizes generated from three distributions, represented by Bivariate normal distribution, contaminated Bivariate normal distribution, and Bivariate exponential distribution.

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.

Robust Tests for the Mean Difference in Paired Data by Using Bootstrap Resampling Technique

The paired sample t-test for testing the difference between two means in paired data is not robust against the violation of the normality assumption. In this paper, some alternative robust tests have been suggested by using the bootstrap method in addition to combining the bootstrap method with the W.M test. Monte Carlo simulation experiments were employed to study the performance of the test statistics of each of these three tests depending on type one error rates and the power rates of the test statistics. The three tests have been applied on different sample sizes generated from three distributions represented by Bivariate normal distribution, Bivariate contaminated normal distribution, and the Bivariate Exponential distribution.

On Discriminant Analysis with some Bivariate Exponential Distributions

Discriminant analysis has found relevance in many areas of science and engineering. Its purpose is to assign an observation x to one of two or more groups on the basis of the value of x. Many problems of discriminant analysis have assumed that the underlying distribution is the multivariate normal in which x is assumed to have means