Gumbel's Bivariate Exponential Distribution Research Articles

Multivariate time between events control charts for Gumbel's bivariate exponential distribution with estimated parameters

The traditional approach of statistical process control is to estimate the in-control parameters from reference samples obtained in Phase I and then apply them to Phase II monitoring, which affects the performance of the monitoring scheme. We analyzed the effect of parameter estimation on three multivariate time between events control charts for monitoring the Gumbel's bivariate exponential distribution, namely the joint multivariate exponentially weighted moving average and multivariate rate, the joint multivariate cumulative sum and multivariate rate, and the joint paired individual exponentially weighted moving average and multivariate rate control charts. The simulation results show that the performance of the control charts is severely affected if control limits with known parameters are used, especially when only a few historical sample data are available. Mainly, the steady-state average time to signal was used as an evaluation index to analyze the affected degree of the three control charts, and the control limits were corrected. In addition, we evaluate the effect of model misspecification on the control charts. Finally, an example is given to illustrate the application of the proposed method.

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.

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.

Distribution of a linear function of correlated ordered variables

In this paper, we have considered the problem of finding the distribution of a linear combination of the minimum and the maximum for a general bivariate distribution. The general results are used to obtain the required distribution in the case of bivariate normal, bivariate exponential 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. The distributions of the minimum and maximum are obtained as special cases.

Probabilistic structure of storm surface runoff considering the dependence between average intensity and storm duration of rainfall events

The probabilistic nature of rainfall and surface runoff events forms the basis of hydrologic design. Two defining characteristics of rainfall events, average intensity and duration, are selected to construct the probabilistic structure of storms using Gumbel's bivariate exponential distribution. This bivariate stochastic rainfall model is then utilized to derive the probabilistic structure of surface runoff using the Soil Conservation Service rainfall runoff model. The zero‐runoff probability, mean, and standard deviation of rainfall excess are evaluated. Copulas serve as an alternative way to derive zero‐runoff probability. The results indicate that the dependence between average intensity and duration has a significant effect on statistical properties of rainfall excess. The advantage of using copulas for complicated multivariate stochastic models is demonstrated. The proposed model is associated with land use and soil type information from the geographic information system to produce maps of statistical properties of rainfall excess. A case study of model application is performed for the state of Indiana, United States. Results show that while the mean and standard deviation of rainfall excess increase with curve number, these quantities are very sensitive to the dependence between average intensity and duration of storm events.

An application of Gumbel's bivariate exponential distribution in estimation of warranty cost of motor cycles

In this article we present an application of Gumbel's bivariate exponential distribution model in the context of estimating warranty costs of motor cycles under a new warranty policy. The problem in question is as follows: Under the present two‐dimensional warranty policy, repair costs (termed as warranty costs) of a motorcycle during the age of first six months or within the usage of 8,000 kilometers are borne by the company. To enhance customer satisfaction, the company wanted to bear the repair costs up to an age of one year or a usage of 12,000 kilometers. The problem is to estimate the expected hike in warranty costs if the warranty policy were revised as mentioned above. Using the past data, the problem is solved by studying the underlying renewal process. Gumbel's bivariate exponential distribution function is found to be useful in approximating the renewal function. Some practical difficulties posed by the past data in the analysis are highlighted and tackled in an interesting way.

Product Moments of Multivariate Random Vectors

We derive a formula for the product moment in terms of the joint survival function when (X 1,…,X p) is a non-negative random vector. In the course of the derivation, we present an independent approach for deriving the formula for (which is already known in the literature). These formulas will be useful tools for deriving explicit expressions for the product moments for the various bivariate and multivariate distributions in statistics specified by the joint tails of the cumulative distribution function. We illustrate their use by deriving product moments of general order for the Gumbel's bivariate exponential distribution, the bivariate Weibull distribution, Marshall and Olkin's bivariate exponential distribution, Muliere and Scarsini's bivariate Pareto distribution, and the conditionally specified bivariate Pareto distribution.

A note on characterizations based on truncated expectations

Abstract Many characterization results of the bivariate exponential distribution and the bivariate geometric distribution have been proved in the literature. Recently Nair and Nair (1988b, Ann. Inst. Statist. Math. 40 (2), 267–271) obtained a characterization result of the Gumbel bivariate exponential distribution and a bivariate geometric distribution based on truncated moments. In this note, we extend the results of Nair and Nair (1988b) to obtain a general result, characterizing these two bivariate distributions based on the truncated expectation of a function h, satisfying some mild conditions.

On Concomitants of Order Statistics from Morgenstern Family

In the present paper the distribution theory of concomitants of order statistics from the Morgenstern family of distribution is investigated. An application of the results in providing some quick estimates of the parameters in the Gumbel's bivariate exponential distribution is also discussed.

Comparison of Some Stress-Strength Reliability Estimators

This paper compares the existing estimators of stress-strength reliability under independent exponential model and also under the dependent set up when stress and strength follow Gumbel's bivariate exponential distribution.

Estimation of Flood Forecasting Errors and Flow-Duration Joint Probabilities of Exceedance

The conceptual analogy between the impulse response function and the unit hydrograph is applied for the identification of a transfer function model for real-time flood forecasting. The real-time corrections of flood forecasts are obtained from probability distributions of observed and forecasted (without real-time corrections) flows approximated by exponential distributions. The method adds considerable smoothing to error terms. The flood event is defined as the amount of total flow in excess of an alarm level of the flood over its duration. The correlation between the flood flow and its duration is considered to estimate the joint probabilities of exceedance (JPE) of designed flood flow over specified duration using the joint distribution of the flow and duration, given by the Gumbel's bivariate exponential distribution (second type). It is discussed that the univariate analysis (i.e., flow alone) underestimates the exceedance probability of flood flow. The study shows that the JPE increases with the increase of correlation between the flood flow and its duration. The radar-derived rainfall data in the Irwell basin, U.K., are applied in a case study. The method of JPE can be extended to link flood forecasting and flow-duration profile for effective management of flood risk areas.

Two characterisations of a gamma mixture distribution

Two characterisations are obtained for a gamma mixture distribution. The first is a generalisation of a result of Engel, Zijlstra and Philips [4] and the second based on Gumbel's bivariate exponential distribution. The two characterisatio are of direct relevance to some practical problems.

On characterizing the bivariate exponential and geometric distributions

In this note, a characterization of the Gumbel's bivariate exponential distribution based on the properities of the conditional moments is discussed. The result forms a sort of bivariate analogue of the characterization of the univariate exponential distribution given by Sahobov and Geshev (1974) (cited in Lau and Rao ((1982), Sankhyā Ser. A, 44, 87)). A discrete version of the property provides a similar conclusion relating to a bivariate geometric distribution.

A Characterization of the Bivariate Exponential Distribution

AbstractIn this article a characterisation of the Gumbel's Bivariate Exponential Distribution is established on the basis of the properties of the conditional expectation of the component variables. The characterising property is proposed as the definition of lack of memory in the bivariate case.

A Characteristic Property of Gumbel's Bivariate Exponential Distribution

A Characteristic Property of Gumbel's Bivariate Exponential Distribution