Multivariate Exponential Model Research Articles

Inference for Multicomponent Systems With Dependent Failures

Multicomponent systems may experience failures with correlations amongst failure times of groups of components, and some subsets of components may experience common cause, simultaneous failures. We present a novel, general approach to model construction and inference in multicomponent systems incorporating these correlations in an approach that is tractable even in very large systems. In our formulation, the system is viewed as being made up of independent overlapping subsystems (IOS). In these systems, components are grouped together into overlapping subsystems, and further into nonoverlapping subunits. Each subsystem has an independent failure process, and each component's failure time is the time of the earliest failure in all of the subunits of which it is a part. We apply this method to observations of an IOS model based on a multicomponent system accumulating damage due to a series of shocks, and with no repair/rectification actions. The model associates individual shock processes with each subsystem, and includes the Marshall–Olkin multivariate exponential model as a special case. We present approaches to simulation and to the estimation of the parameters of the model, given component failure data for various system configurations (series, parallel, and other arrangements).

Equivariant estimation for the parameters of location-scale multivariate exponential models and its application in reliability analysis

ABSTRACTBy considering an absolutely continuous location-scale multivariate exponential model (Weier and Basu, 1980), we obtain minimum risk equivariant estimator(s) of the parameter(s). Given a location-scale multivariate exponential random vector, it is shown that the normalized spacings associated with the random vector are independent standard exponential. The distribution of the complete sufficient statistic is derived. We derive the performance measures of standby, parallel, and series systems and also obtain the minimum risk equivariant estimator of the mean time before failure of the three systems. Some of the results of this article are extensions of those of Chandrasekar and Sajesh (2010).

Multicomponent Systems With Multiplicative Aging and Dependent Failures

We extend our earlier studies of a multicomponent system accumulating damage due to a series of fatal and nonfatal shocks. The model introduces statistical dependence among the system components by associating individual shock processes with potentially overlapping subsystems made up of groupings of components. We construct an aging and statistical dependence model where damage accumulates multiplicatively with each shock. We derive a representation of the system's joint survival function, and show that the Marshall-Olkin multivariate exponential model can be obtained as a special case of this model. We propose an approach to the simulation of the performance of the system, and provide several illustrative examples. We conclude by identifying possible further extensions of this model.

Relationship between sweet orange yield and intensity of Citrus Variegated Chlorosis

Citrus Variegated Chlorosis (CVC) is currently present in approximately 40% of citrus plants in Brazil and causes an annual loss of around 120 million US dollars to the Brazilian citrus industry. Despite the fact that CVC has been present in Brazil for over 20 years, a relationship between disease intensity and yield loss has not been established. In order to achieve this, an experiment was carried out in a randomized block design in a 3 × 2 factorial scheme with 10‐year‐old Natal sweet orange. The following treatments were applied: irrigation with 0, 50 or 100% of the evapotranspiration of the crop, combined with natural infection or artificial inoculation with Xylella fastidiosa, the causal agent of CVC. The experiment was evaluated during three seasons. A negative exponential model was fitted to the relationships between yield versus CVC severity and yield versus Area Under Disease Progress Curve (AUDPC). In addition, the relationship between yield versus CVC severity and canopy volume was fitted by a multivariate exponential model. The use of the AUDPC variable showed practical limitations when compared with the variable CVC severity. The parameter values in the relationship of yield–CVC severity were similar for all treatments unlike in the multivariate model. Consequently, the yield–CVC intensity relationship (with 432 data points) could be described by one single model: y = 114·07 exp(−0·017 x), where y is yield (symptomless fruit weight in kg) and x is disease severity (R2 = 0·45; P < 0·01).

Two MEWMA Charts for Gumbel's Bivariate Exponential Distribution

Data described by the exponential distribution are commonly encountered in manufacturing processes, reliability analysis, and human-service management. Time between events (TBE) charts have been suggested to monitor exponential data. However, existing studies on TBE charts are limited to univariate cases assuming there is only one process characteristic of interest. In this paper, two multivariate exponential weighted moving average (MEWMA) charts are proposed for the simultaneous monitoring of the mean vector of Gumbel's bivariate exponential (GBE) TBE model: One based on the raw observations and the other based on the transformed data. A numerical example is given to illustrate the implementation of the two MEWMA charts. We compare the average run-length performances of the two proposed charts with the following individual TBE charts pairs: the paired individual t charts, the paired individual exponentially weighted moving-average (EWMA) charts on raw data, and the paired individual EWMA charts on transformed data. The results of the comparative studies show that our MEWMA charts outperform all the other charts. The proposed MEWMA charts can be easily extended to higher dimensions. Some brief discussions concerning monitoring Gumbel's multivariate exponential model with more than two variables are also included.

Lack of Regional Selectivity During the Progression of Parkinson Disease

Dopamine terminal loss in the putamen of patients with Parkinson disease (PD) shows a regional heterogeneity, reflecting selective vulnerability of degenerating neurons to mechanisms of cell death. If the same pathogenic mechanisms are responsible for the onset and progression of PD, the regional selectivity of dopamine cell loss will be the same throughout the course of the disorder. To investigate the regional selectivity of dopamine terminal loss during the progression of PD. We studied 67 patients with PD and 20 healthy subjects using positron emission tomography with [(11)C](+/-)dihydrotetrabenazine (DTBZ). Regional values of DTBZ binding potential (calculated as maximum specific binding [B(max)] divided by the equilibrium dissociation constant K(d)) against disease duration in the putamen of PD patients were best described by a multivariate exponential model with distinct parallel asymptotic values that were significantly (P<.001) different across 4 regions of the putamen. The extent of loss of DTBZ binding potential with disease progression during the clinical stage of PD (early vs late PD) was similar between the anterior (-33%, using early PD as the baseline) and posterior (-29%) putamen. In contrast, the extent of loss of DTBZ binding potential in early PD, which reflects the cumulated loss of DTBZ binding potential from the onset of the disorder (in healthy subjects vs those with early PD), was significantly (P<.001) lower in the posterior (-58%, using healthy subjects as the baseline) than the anterior (-42%) putamen. To the extent that DTBZ positron emission tomography provides an accurate estimate of loss of dopamine neurons, our findings suggest that the mechanisms responsible for the progression of PD may not be the same as those responsible for its onset.

Unbiased Estimation for a Multivariate Exponential whose Components have a Common Shift

It is shown that for independent and identically distributed random vectors, for which the components are independent and exponentially distributed with a common shift, we can construct unbiased estimators of their density, derived from the Uniform Minimum Variance Unbiased Estimator (UMVUE) of their distribution function. As direct applications of the UMVUEs of the density functions we present a Chi-square goodness of fit test of the model, and give two tables of the UMVUEs of some commonly used functions of the unknown parameters of the multivariate exponential model considered in this paper.

Multivariate life testing in variably scaled environments.

This paper examines modeling and inference questions for experiments in which different subsets of a set of kappa possibly dependent components are tested in r different environments. In each environment, the failure times of the set of components on test is assumed to be governed by a particular type of multivariate exponential (MVE) distribution. For any given component tested in several environments, it is assumed that its marginal failure rate varies from one environment to another via a change of scale between the environments, resulting in a joint MVE model which links in a natural way the applicable MVE distributions describing component behavior in each fixed environment. This study thus extends the work of Proschan and Sullo (1976) to multiple environments and the work of Kvam and Samaniego (1993) to dependent data. The problem of estimating model parameters via the method of maximum likelihood is examined in detail. First, necessary and sufficient conditions for the identifiability of model parameters are established. We then treat the derivation of the MLE via a numerically-augmented application of the EM algorithm. The feasibility of the estimation method is demonstrated in an example in which the likelihood ratio test of the hypothesis of equal component failure rates within any given environment is carried out.

Some inference results in an absolutely continuous multivariate exponential model of Block

In this paper, we obtain MLEs of the parameters and of large sample test for testing independence and symmetry of k components in the k + 1 parameter version of an absolutely continuous multivariate exponential distribution (ACMVED) of Block (1975).

Some inference results in several symmetric multivariate exponential models

In this paper, we consider multivariate exponential models with identical marginals. We obtain MLEs of the parameters and large sample tests for mutual independence of k components in the multivariate exponential models proposed by Weinman (1966), Marshall-Olkin (1967) and Block (1975).

Heavy traffic results for single-server queues with dependent (EARMA) service and interarrival times

Models are given for sequences of correlated exponential interarrival and service times for a single-server queue. These multivariate exponential models are formed as probabilistic linear combinations of sequences of independent exponential random variables and are easy to generate on a computer. Limiting results for customer waiting time under heavy traffic conditions are obtained for these queues. Heavy traffic results are useful for analyzing the effect of correlated interarrival and service times in queues on such quantities as queue length and customer waiting time. They can also be used to check simulation results.

Heavy traffic results for single-server queues with dependent (EARMA) service and interarrival times

Models are given for sequences of correlated exponential interarrival and service times for a single-server queue. These multivariate exponential models are formed as probabilistic linear combinations of sequences of independent exponential random variables and are easy to generate on a computer. Limiting results for customer waiting time under heavy traffic conditions are obtained for these queues. Heavy traffic results are useful for analyzing the effect of correlated interarrival and service times in queues on such quantities as queue length and customer waiting time. They can also be used to check simulation results.