Two-parameter Exponential Distribution Research Articles

New CUSUM and EWMA charts with simple post signal diagnostics for two‐parameter exponential distribution

AbstractThe two‐parameter exponential distribution (TPED) is often used to model time‐between‐events data. In this paper, we propose CUmulative SUM and exponentially weighted moving average charts for simultaneously monitoring the parameters (location and scale) of the TPED. A key feature of the proposed charts is their straightforward post‐signal diagnostics. Monte Carlo simulations are used to estimate the zero‐state and steady‐state average run‐length (ARL) profiles of the proposed charts. The ARL performances of existing and proposed charts are assessed in terms of expected weighted run‐length and relative mean index. It is found that the proposed charts outperform the existing charts. A real dataset is used to illustrate the implementation of the proposed charts.

A new EWMA chart for simultaneously monitoring the parameters of a shifted exponential distribution

In various scenarios where products and services are accompanied by warranties to ensure their reliability over a specified time, the two-parameter (shifted) exponential distribution serves as a fundamental model for time-to-event data. In modern production process, the products often come with warranties, and their quality can be manifested by the changes in the scale and origin parameters of a shifted exponential (SE) distribution. This paper introduces the Max-EWMA chart, employing maximum likelihood estimators and exponentially weighted moving average (EWMA) statistics, to jointly monitor SE distribution parameters. Additionally, we extend two additional charts, namely the Max-DEWMA and Max-TEWMA charts to enhance early-stage shift detection. Performance evaluations under zero-state and steady-state conditions compare these charts with the existing Max-CUSUM chart in terms of expected value and standard deviation of the run length (RL) distribution. Our findings reveal that among the Max-EWMA schemes, the Max-EWMA SE chart outperforms the others in terms of steady-state performance, while the Max-TEWMA chart surpasses the Max-EWMA and Max-DEWMA SE charts in respect to zero-state performance. Moreover, the proposed Max-EWMA schemes demonstrate advantages over Max-CUSUM, especially for small to moderate smoothing constants. We also provide an illustrative example to demonstrate the implementation of the proposed schemes.

Life Function Analysis of a Two-Parameter Exponential Distribution Using Robust Rank Regression

In this research, a proposed method is presented to estimate the life function of the two-parameter exponential distribution by using the robust rank regression on the dependent variable, which depends on the robust variance-covariance matrix (the FMCD and OGK method) and the robust mean vector through which the rank regression coefficients are estimated using the method of ordinary least squares and employing it in estimating the scale and location parameters of the exponential distribution and thus analyzing the life function on its basis. A comparison was also made between the proposed method and the classical method of rank regression on the dependent variable of life data at different sample sizes (from 10 to 25) to show the efficiency of the proposed method based on the mean square error of the estimated parameters, through a program prepared by the researcher in MATLAB language dedicated to this purpose. The practical application results concluded that the proposed robust method of rank regression on the dependent variable in estimating the parameters of the exponential distribution was more efficient than the classical method. Paper type: Research paper.

Adaptive EWMA control chart for monitoring two-parameter exponential distribution with type-II right censored data

Two-parameter exponential distributions have a widespread application in insurance, medical and industrial production. However, incomplete samples are often obtained due to experimental conditions. In this paper, a new adaptive exponentially weighted moving average control chart, called AEWMA-LR control chart, is proposed for monitoring the quality characteristics of two-parameter exponential distributions based on type-II censored data. The proposed monitoring scheme aims to achieve an adaptive effect by combining exponentially weighted moving averages and adjusting the weight factors according to the relative magnitude of the estimated location and scale parameter shifts. We explored the effects of smooth parameters, sample size, and censored parameters on the performance of AEWMA-LR control charts through a Monte Carlo simulation study. Then, a comparative analysis was conducted with existing EWMA-LR and EWMA-Max-MLE control charts based on the two-parameter exponential model. The simulation results of zero-state average run length (ARL) and conditional expected delay (CED) show that the proposed monitoring scheme outperforms the existing control charts. Finally, to demonstrate the usefulness of the AEWMA-LR control chart in real-world applications, we give three instances of two-parameter exponential distributions based on type-II right censored data.

Confidence intervals for a ratio of percentiles of location-scale distributions

The problem of estimating a ratio of percentiles of two independent location-scale distributions is considered. A fiducial approach is proposed and described in details for the normal, lognormal, two-parameter exponential and Weibull distributions. For the normal case, the fiducial confidence intervals (CIs) turn out to be exact when the variances are equal. Procedures for constructing CIs for ratio of percentiles involving two-parameter exponential distributions and Weibull distributions are given with computational details. The fiducial methods can be readily extended to the case where the samples are type II censored. The methods are illustrated using real-world data sets.

Optimal Statistical Estimation and Dynamic Adaptive Control of Airline Seat Protection Levels for Several Nested Fare Classes under Parametric Uncertainty of Customer Demand Models

Assigning seats in the same compartment to different fare classes of passengers is a major problem of airline seat allocation. Airlines sell the same seat at different prices according to the time at which the reservation is made and other conditions. Thus the same seat can be sold at different prices. The problem is to find an optimal policy that maximizes total expected revenue. To solve the above problem, this paper presents the novel computational approach to optimization and dynamic adaptive prediction of airline seat protection levels for multiple nested fare classes of single-leg flights under parametric uncertainty. It is assumed that time T (before the flight is scheduled to depart) is divided into h periods, namely a full fare period and h-1 discounted fare periods. The fare structure is given. An airplane has a seat capacity of N. For the sake of simplicity, but without loss of generality, we consider (for illustration) the case of nonstop flight with two fare classes (business and economy). The proposed airline's inventory management policy is based on the use of the proposed computational models. These models emphasize pivotal quantities and ancillary statistics relevant for obtaining statistical predictive limits for anticipated quantities under parametric uncertainty and are applicable whenever the statistical problem is invariant under a group of transformations that acts transitively on the parameter space. The proposed technique is based on a probability transformation and pivotal quantity averaging. It is conceptually simple and easy to use. Finally, we give illustrative examples, where the proposed analytical methodology is illustrated in terms of the two-parameter exponential distribution. Applications to other log-location-scale distributions could follow directly.

Alternative confidence interval estimation for the mean and coefficient of variation in a two-parameter exponential distribution

This paper presents interval estimation for the population mean and coefficient of variation in a two-parameter exponential distribution. The new generalized pivot, profile likelihood function and likelihood ratio statistic are derived and used to construct the confidence intervals. A highlight of this paper is that the generalized and likelihood ratio confidence intervals for the mean and coefficient of variation perform well in terms of coverage probability in many cases. Finally, two real-data applications on the air pollution of particulate matter (PM2.5) and the renewable energy through wind power of Thailand are used for illustration purposes.

One-Stage Multiple Comparisons of the Mean Lifetimes of k Treatments with the Average for Exponential Distributions under Heteroscedasticity

Making use of uniformly minimum-variance unbiased estimators for the parameters of two-parameter exponential distributions and the distribution of pivotal quantities, we propose one-stage multiple comparison procedures for k mean lifetimes with the average under heteroscedasticity. The multiple comparison procedures include one-sided and two-sided confidence intervals. These intervals can be applied to identify which treatment’s mean lifetime is better than the average or worse than the average in terms of the mean lifetimes of all treatments. Critical values are obtained in order to assure users that the given confidence coefficient has been reached; they are organized in table format for practical and convenient use. An example is provided to demonstrate the proposed techniques, wherein the mean survival times of four different lung cancer categories are compared with the average.

Comparison of the several two-parameter exponential distributed group means in the presence of outliers

The two-parameter exponential distribution is often used to model the lifetime of a product. The comparison of the mean lifetimes of several products is a main concern in reliability applications. In this study, the performance of the methods to compare the mean lifetimes of several products based on generalized p-value, parametric bootstrap, and fiducial approach are compared in the presence of outliers. The results of Monte-Carlo simulations clearly indicate that there is no uniformly powerful test. The parametric bootstrap test is superior to the others except in the case of the lower number of groups and the presence of outliers. An illustrative example of testing the equality lifetimes of a component is given to perform the proposed tests. The considered tests are implemented in an R package doex.

Estimation of process performance index for the two-parameter exponential distribution with measurement error

Measurement errors are inevitable in practice, but they are not considered in the existing process performance index. Therefore, we propose an estimation method of process performance index for the two-parameter exponential distribution with measurement errors to fill this gap. In this paper, the relationship between the unobservable actual value and measurement value is considered as full error model, and the maximum likelihood estimation method is considered to obtain the unknown parameters. In addition, we also use the Bootstrap method to construct confidence intervals of process performance index. The performance of the proposed estimation is investigated in terms of bias, mean square error (MSE) and average interval length. Simulation results show that the proposed estimator outperforms other estimators. Finally, an example of the mileage data of the military personnel carrier is given to illustrate the implementation of the proposed estimation method.

A Study of Shrinkage Estimators for Reliability Function and Variance Properties

A variety of shrinkage methods for estimating unknown parameters has been considered. We derive and compare the shrinkage estimators for the reliability function of the two-parameter exponential distribution. Simulation experiments are used to study the performances of these estimators.

Interval estimation of the two-parameter exponential constant stress accelerated life test model under Type-II censoring

ABSTRACT In this paper, interval estimation of the two-parameter exponential distribution based on constant-stress accelerated life test data under Type-II censoring is studied. The life-stress models of the scale and location parameters are assumed to be log-linear. The maximum likelihood estimators of the model parameters are obtained. The exact and approximate confidence intervals for the parameters in the life-stress model of the scale parameter are developed. The generalized confidence intervals for the parameters in the life-stress model of the location parameter, and mean and reliability function at the designed stress level are derived. The performance of the proposed generalized confidence intervals is assessed by the Monte Carlo simulation. Finally, a real example is used to illustrate our methods.

Interval Estimation for the Two-Parameter Exponential Distribution Based on the Upper Record Values

Using the data for upper record values, the interval estimation for the scale parameter of two-parameter exponential distribution is presented. In addition, two methods for the joint confidence region of two parameters are proposed. In terms of confidence region area, the simulation comparison of two methods of the confidence region is performed in this paper. The criterion of minimum confidence region area is used to obtain the optimal method of the confidence region. To illustrate our proposed interval estimation methods, one biometrical example is used and the corresponding confidence interval length and confidence region area are also calculated. Our research topic is related to the asymmetrical probability distributions and applications across disciplines.

The Bayesian and Bayesian shrinkage estimators under square error and Al-Bayyati loss functions with right censoring scheme

In this paper, the Bayesian and Bayesian shrinkage estimators of the scale parameter of two-parameter exponential distribution are provided based on the square error and Al-Bayyati loss functions and right censoring. Both informative (inverse-Gamma) and non-informative (quasi-density) priors are considered for the scale parameter of two-parameter exponential distribution. The estimators from different loss functions are evaluated against each other via the risk and risk efficiency measures. Based on the simulation scheme and real data set, we compare the proposed estimators under both loss functions with risk and risk efficiency measures, which lead to efficiency of the Bayesian shrinkage estimates under informative prior. In real data analysis, Bayesian and Bayesian shrinkage estimation of the scale parameter for the flood stages data set is considered with both square error and Al-Bayyati loss functions under the right censoring scheme.

Bounded risk per unit cost index constraint for sequential estimation of the mean in a two-parameter exponential distribution

In this article, the two-stage sequential sampling procedure is proposed to estimate the mean of an exponential distribution under the modified square error loss function. The main aim of the article is to consider the associated risk per unit cost function by bounding it from above with a fixed preassigned positive number, ω. We provide the exact distribution of the total sample size, explicit formulas for the mathematical expectation of the rth power of the total sample size, the expected value and mean squared error of the maximum likelihood estimate of the scale, and mean parameters of the exponential distribution under the two-stage sequential procedure. The performances of the proposed methodologies are investigated with the help of simulations. In the end, we supplement with extensive sets of data analysis via computer simulations, validating that the two-stage sequential method performs very well. Finally, using a real data set, the procedure is clearly illustrated.

Bootstrap Confidence Intervals for Common Signal-to-noise Ratio of Two-parameter Exponential Distributions

Signal-to-noise ratio (SNR) is a reciprocal of coefffficient of variation. The SNR is a measure of mean relative to the variability. Confidence procedures for common SNR of two-parameter exponential distributions were developed using generalized confidence interval (GCI) approach, large sample (LS) approach, adjusted method of variance estimates recovery (Adjusted MOVER) approach, and bootstrap approaches based on standard bootstrap (SB) and parametric bootstrap (PB). The performances of all approaches are measured by coverage probability and average length. Simulation studies show that all approaches have the coverage probabilities below the nominal confidence level of 0.95 when the common SNR is negative value. However, the coverage probabilities of all approaches are greater than the nominal confidence level of 0.95 when the common SNR is positive value. Moreover, the LS and AM approaches are the conservative confidence intervals. In addition, the GCI and PB approaches provide the confidence intervals with coverage probabilities close to the nominal confidence level of 0.95 when the sample sizes are large and the common SNR is positive value. The GCI and PB approaches are recommended to estimate the confidence intervals for the common SNR of two-parameter exponential distributions. Finally, all proposed approaches are employed in the data of the survival days of lung cancer patients for a demonstration.

Interval Estimation for the Two-Parameter Exponential Distribution under Progressive Type II Censoring on the Bayesian Approach

Under progressive type II censoring, the credible interval estimation and the credible region for parameters of two-parameter exponential distribution based on the Bayesian approach are presented in this paper. Two methods of Bayesian credible region are proposed under a given confidence level. We also presented the predictive interval of the future observation under this type of censoring. In order to compare the performance of our proposed Bayesian credible interval and region with the existing non-Bayesian methods, we conduct a simulation study by the Monte Carlo method to find the corresponding coverage probabilities. This research is related to the topic of asymmetrical probability distributions and applications across disciplines. Finally, one engineering example is used to demonstrate the Bayesian credible interval estimation methods proposed in this paper.

Prediction of the exponential fractional upper record-values

Abstract In this paper, two linear predictors of the fractional kth upper record-value based on two-parameter exponential distribution are proposed. Moreover, a free scale-location predictive interval is constructed for the future fractional kth upper record-value. The prediction results are formulated in a general set-up relying on two fractional kth upper record-values. Some important distributional properties for each point predictor are revealed. Furthermore, the mean square error and Pitman’s measure of closeness are used to compare the point predictors. Finally, a simulation study is carried out and a real data set is analyzed to explore the efficiency of the suggested methods.

Prediction of future generalized order statistics based on two-parameter exponential distribution for large samples

ABSTRACT Exact and asymptotic distributional properties are discussed in detail for two mean-squared error consistent point predictors of future-generalized order statistics (GOSs) based on two-parameter exponential distribution. These predictors work even if some observed data were missing. For each point predictor, the asymptotic distribution of the normalized difference between the future GOS and its point predictor is derived, when the scale parameter is known or unknown. It is revealed that the asymptotic distributions of these normalized differences are equal when the scale parameter is known. Two asymptotic prediction intervals of the future GOS are constructed whenever the scale parameter is known or unknown. Furthermore, two tests of outliers are proposed relying on the point predictors. Finally, a simulation study is conducted and a real data set is analyzed for illustrative purposes.

Bayesian Interval Estimation for the Two-Parameter Exponential Distribution Based on the Right Type II Censored Sample

The Bayesian interval estimation of the scale parameter for two-parameter exponential distribution is proposed based on the right type II censored sample. Under this type of censoring, two methods of Bayesian joint confidence region of the two parameters are also proposed. The simulation results show that the Bayesian method has a higher coverage probability than the existing method, so the Bayesian method is recommended for use. This research is related to the topic of asymmetrical probability distributions and applications across disciplines. The predictive interval of the future observation based on the right type II censored sample is also provided. One biometrical example is given to illustrate the proposed methods for the Bayesian interval estimations and prediction interval.