One-parameter Exponential Distribution Research Articles

Joint reconstruction and prediction regions for unobserved exponential lifetimes under doubly Type II censoring

ABSTRACT The doubly Type II censoring scheme, in which some of the smallest and largest lifetimes are censored, is an important and widely used censoring scheme in life-testing experiments and reliability. In this paper, the problem of simultaneous reconstruction and prediction for unobserved lifetimes in a doubly Type II censored sample from the one-parameter exponential distribution is discussed. For this purpose, the joint reconstruction and prediction (JRP) regions are obtained based on two independent pivotal quantities. First, the balanced JRP regions for the unobserved lifetimes are derived, and then a constrained optimization problem is formulated and solved to find the JRP regions with the smallest area. For illustrative and comparative purposes, a simulation study and two illustrative examples are presented.

Using the exponential distribution as a synthetic curve volume of runoff

The paper analyzes the problem of determining the volume of runoff from the basin by means of runoff simulation methods using a one-parameter exponential distribution. Using the data of the hydrometeorological service on the maximum daily amount of precipitation of various types for some weather stations located throughout the territory of Ukraine, the dependencies of the probability distribution of daily precipitation layers were obtained, in the form of the total depth of precipitation and the total duration of precipitation. As a result of the approximation of the probability distribution of daily precipitation layers, the values of the frequency of events and the average value of the depth of daily precipitation layers for different regions of Ukraine and correlation coefficients were obtained. The use of the cumulative exponential distribution makes it possible to obtain a synthetic curve of the volume of runoff to determine the volume of annual surface runoff that is subject to treatment at treatment facilities.

English

The method used here focuses on the basic quantities and supporting statistics needed to construct prediction limits or optimal solutions for expected outcomes under the parametric uncertainty of applied stochastic models. It is applicable whenever the statistical problem is invariant under a group of transformations acting transitively on the parameter space. This method does not require any tabulation and is applicable whether the statistics are complete or Type II censored. The exact prediction limits of the order statistics associated with a sample of basic distributions can be found quickly and easily, making tables, simulations, Monte Carlo estimated percentiles, special computer programs, and approximations unnecessary. The proposed technique is based on the transformation of probabilities and the averaging of reference values. It is conceptually simple and easy to use. The discussion is limited to one-sided prediction limits. Finally, we provide practical numerical examples where the proposed analytical methodology is illustrated in terms of the one-parameter (or two-parameter) exponential distribution. Applications to other log-location-scale distributions can follow directly.

Estimation of Large River Design Floods Using the Peaks-Over-Threshold (POT) Method

This research analyzes the peaks-over-threshold (POT) method for designed flood estimation needed to plan river levees, spillways and water facilities. In this study, a one-parameter exponential probability distribution has been modified by including the coefficient of λ, which represents an average number of floods and enables return period calculation within the specified period of time. The study also compares results using the Log-Pearson Type III distribution of maximum annual flows and a standard exponential distribution of the selected peaks over the threshold level. The aforementioned approach represents the standard mathematical tools for river flood design, while the proposed modification of the exponential distribution highlights the estimation of flood quantiles with longer return periods (e.g., 100, 1000 and 10,000 years). Moreover, the sensitivity analysis of the threshold selection is proposed to assist in the flood design flow estimation alongside the proposed modification of the exponential probability distribution. The study was carried out at the Danube River, and the Novi Sad hydrological station (Republic of Serbia) was used for the long-term recorded period from 1876 to 2015. The results suggest that the POT method derives more reliable estimates of design floods than the traditional statistical tools for flood estimation. The results suggest the theoretical values of the water level of the 10,000 years return period is equal to 867 cm, while the Log-Pearson Type III distribution of annual maximum flows underestimated this value for 14 cm.

Modeling and inference for multivariate time series of counts based on the INGARCH scheme

Modeling multivariate time series of counts using the integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) scheme is proposed. The key idea is to model each component of the time series with a univariate INGARCH model, where the conditional distribution is modeled with a one-parameter exponential family distribution, and to use a (nonlinear) parametric function of all components to recursively produce the conditional means. It is shown that the proposed multivariate INGARCH (MINGARCH) model is strictly stationary and ergodic. For inference, the quasi-maximum likelihood estimator (QMLE) and the minimum density power divergence estimator (MDPDE) for robust estimation are adopted, and their consistency and asymptotic normality are verified. As an application, the change point test based on the QMLE and MDPDE is illustrated. The Monte Carlo simulation study and real data analysis using the number of weekly syphilis cases in the United States are conducted to confirm the validity of the proposed method.

Interval Estimation of the Stress-Strength Reliability for Lower Record Data Based on One-parameter Exponential Distribution Using Median Ranked Set Sampling and Record Ranked Set Sampling

This paper develops interval estimation of stress-strength reliability when data are lower record values from a one-parameter exponential distribution using resampling schemes such as median ranked set sampling (MRSS) and record ranked set sampling (RRSS). For illustrative aims, we inspect the performance of the proposed interval estimation methods, bootstrap, and generalized confidence intervals (CIs) by using a simulation study and real examples. The result indicates that the proposed interval based on a bootstrap percentile (BP) CI and a generalized CI (GC) using MRSS is suitable for the application.

Sampling design for the lifetime performance index of exponential lifetime distribution under progressive type I interval censoring

The lifetime performance index was used for the evaluation on the process performance to promote the quality and productivity especially with products following a one-parameter exponential distribution. Based on the hypothesis testing procedure using the maximum likelihood estimator as testing statistic, the sampling design is determined under different situations. For given power of hypothesis testing, the minimum sample size is determined and tabulated to reach the given power. When the termination time is fixed and the number of inspection intervals is not fixed, the required number of inspection intervals and sample size with minimum total cost are determined and tabulated. When the termination time is not fixed, the required number of inspection intervals, sample size and equal interval length to reach the minimum total cost are determined and tabulated. At last, one numerical example is given to illustrate the use of this sampling design to collect data and then implement the testing procedure to determine whether the process is capable.

An overview of Bayesian prediction of future record statistics using upper record ranked set sampling scheme

Two sample prediction is considered for a one-parameter exponential distribution‎. ‎In practical experiments using sampling methods based on different schemes‎ ‎is crucial‎. This paper addresses the problem of Bayesian prediction of record values from a future sequence‎, ‎based on an upper record ranked set sampling scheme‎. First‎, ‎under an upper record ranked set sample (RRSS) and different values of hyperparameters‎, ‎point predictions have been studied with respect to both symmetric and asymmetric loss functions‎. ‎These predictors are compared in the sense of their mean squared prediction errors‎. ‎Next‎, we have derived two prediction intervals for future record values‎. ‎Prediction intervals are compared in terms of coverage probability and expected length‎. ‎Finally‎, a simulation study is performed to compare the performances of the predictors‎. ‎The real data set is also analyzed for an illustration of the findings.‎

Phase-II monitoring of exponentially distributed process based on Type-II censored data for a possible shift in location–scale

A large number of researches addressed the problem of monitoring statistical processes using complete data. Nevertheless, in engineering applications, especially reliability engineering and lifetime test, we often observe the incomplete or censored sample. In this paper, we introduce three EWMA schemes for monitoring exponentially distributed processes based on type-II censored data. Our proposed procedure is capable of detecting a shift in both the origin (location) and scale parameters of a shifted (two-parameter) exponential distribution. As one-parameter exponential distribution is a particular case of the two-parameter exponential with the origin equals zero, the present paper will be useful to detect a shift in the location (origin) or scale or both from traditional one-parameter exponential processes. Also, one can achieve a complete sample situation as a particular case. Therefore, the current article is very general and inclusive. We provide the design parameters, such as control limits of the three schemes and evaluate the performance in terms of average run length characteristics. We investigate the effects of sample size and censoring design on the performance of the proposed schemes. We discuss some realistic examples to illustrate the implementation strategies of the proposed schemes.

Bayesian sampling plan for the exponential distribution with generalized Type - II hybrid censoring scheme

In this paper, a decision-theoretic approach is used to obtain the Bayesian sampling plan (BSP) for the generalized Type-II hybrid censoring scheme when lifetimes of sampled units follow a one-parameter exponential distribution. An efficient loss function is used to decide whether to accept or reject the batch. The BSP is obtained by constructing the closed form of the Bayes decision function. It is observed that the closed form of the Bayes decision function cannot be obtained analytically for any arbitrary loss function. A numerical approach is proposed to determine the optimum BSP for an arbitrary loss function. To illustrate this, we consider a higher degree polynomial loss function and obtain the optimum BSP numerically. Some numerical studies have been conducted to check the performance of the optimum BSP in different cases.

Minimum risk point estimation (MRPE) of the mean in an exponential distribution under powered absolute error loss (PAEL) due to estimation plus cost of sampling

We begin with a review of asymptotic properties of a purely sequential minimum risk point estimation (MRPE) methodology for an unknown mean in a one-parameter exponential distribution under a class of generalized loss functions. This class of powered absolute error loss (PAEL) includes both squared error loss (SEL) and absolute error loss (AEL) plus cost of sampling. We prove the asymptotic second-order efficiency property and asymptotic first-order risk efficiency property associated with the purely sequential MRPE problem. For operational convenience, we then move to implement an accelerated sequential MRPE methodology and prove the analogous asymptotic second-order efficiency property and asymptotic first-order risk efficiency property. We follow up with extensive data analysis from simulations and provide illustrations using cancer data.

A confidence interval for the population mean of a one-parameter exponential distribution based on the Wilson-Hilferty transformation

In this paper, an approximate confidence interval (CI) is proposed for the population mean of a one-parameter exponential distribution. The Wilson-Hilferty approximation is used to transform the exponential random variable to a normal rando

Inference for a simple step-stress model based on ordered ranked set sampling

In this paper, we considered the inference problem on simple step-stress accelerated life test data from one-parameter exponential distribution under type-I censored ordered ranked set sample with cumulative exposure model. The Bayesian estimators and credible intervals for the model parameters are developed and compared with the corresponding estimators based on simple random sampling. Two real data sets and numerical simulation evaluations are presented to illustrate all the results developed here. The simulation study indicated that the proposed Bayes estimators and credible intervals based on ordered ranked set sampling performed better than their counterparts using simple random sampling.

The Logarithmic Burr-Hatke Exponential Distribution for Modeling Reliability and Medical Data

In this work, we introduced a new one-parameter exponential distribution. Some of its structural properties are derived% \textbf{.} The maximum likelihood method is used to estimate the model parameters by means of numerical Monte Carlo simulation study. The justification for the practicality of the new lifetime model is based on the wider use of the exponential model. The new model can be viewed as a mixtureof the exponentiated exponential distribution. It can also be considered as a suitable model for fitting right skewed data.\textbf{\ }We prove empirically the importance and flexibility of the new model in modelingcancer patients data, the new model provides adequate fits as compared to other related models with small values for $W^{\ast }$\ \ and $A^{\ast }$. The new model is much better than the Modified beta-Weibull, Weibull, exponentiated transmuted generalized Rayleig, the transmuted modified-Weibull, and transmuted additive Weibull models in modeling cancer patients data. We are also motivated to introduce this new model because it has only one parameter and we can generate some new families based on it such as the the odd Burr-Hatke exponential-G family of distributions, the logarithmic\textbf{\ }Burr-Hatke exponential-G family of distributions and the generalized\textbf{\ }Burr-Hatke exponential-G family of distributions, among others.

On the Construction of Preliminary Test Estimators of the Reliability Characteristics for the Exponential Distribution Based on Records

SYNOPTIC ABSTRACTThe one-parameter exponential distribution plays an important role in reliability theory. Two measures of reliability for exponential distribution are considered, R(t) = P(X > t) and P = P(X > Y). Sometimes, due to past knowledge or experience, the experimenter may be in a position to make an initial guess on some of the parameters of interest. In such cases, we can provide an improved estimator by incorporating the prior information on the parameters. Preliminary test estimators (PTES) have been developed in the literature for the parameters of various distributions. To the best of the knowledge of the authors, PTES are not available for reliability functions R(t) and P. For record values from exponential distribution, we define PTES based on uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE), and empirical Bayes estimator (EBE) for the powers of the parameter, R(t) and P. Bias and mean square error (MSE) expressions for the proposed estimators are derived to examine their efficiency. A comparative study of different methods of estimation is done through simulations, and it is established that PTES perform better than ordinary UMVUES, MLES, and EBES.

Lifetime performance index based on ranked set sampling

ABSTRACTIn this article, we assume that the lifetimes of products follow a one-parameter exponential distribution and use both conjugate and Jeffreys prior distributions for finding a Bayes estimator based on the RSS scheme for lifetime performance index CL. We also obtained maximum likelihood and adhac estimators for CL. Monte Carlo simulation study is done for comparing the obtained estimators in two sampling schemes SRS and RSS. Fisher information has been utilized to obtain lower confidence bound of CL for both schemes in asymptotically state and use this bounds for hypothesis testing of CL.

A new method for generating distributions with an application to exponential distribution

ABSTRACTA new method has been proposed to introduce an extra parameter to a family of distributions for more flexibility. A special case has been considered in detail, namely one-parameter exponential distribution. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, mode, moment-generating function, mean residual lifetime, stochastic orders, order statistics, and expression of the entropies, are derived. The maximum likelihood estimators of unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non linear equations only. Further, we consider an extension of the two-parameter exponential distribution also, mainly for data analysis purposes. Two datasets have been analyzed to show how the proposed models work in practice.

Computational testing algorithmic procedure of assessment for lifetime performance index of products with one-parameter exponential distribution under progressive type I interval censoring

Process capability indices had been widely used to evaluate the process performance to the continuous improvement of quality and productivity. When the lifetime of products possesses a one-parameter exponential distribution, the larger-the-better lifetime performance index is considered. The maximum likelihood estimator is used to estimate the lifetime performance index based on the progressive type I interval censored sample. The asymptotic distribution of this estimator is also investigated. We use this estimator to develop the new hypothesis testing algorithmic procedure in the condition of known lower specification limit. Finally, two practical examples are given to illustrate the use of this testing algorithmic procedure to determine whether the process is capable.

Estimation of stress-strength reliability using record ranked set sampling scheme from the exponential distribution

In this paper, point and interval estimation of stress-strength reliability based on upper record ranked set sampling (RRSS) from one-parameter exponential distribution are considered. Maximum likelihood estimator (MLE) as well as the uniformly minimum variance unbiased estimator (UMVUE) of stress-strength parameter are derived and their performance are studied. Also, some confidence intervals for stress-strength parameter based on upper RRSS are constructed and then compared on the basis of a simulation study. Finally, a data set has been analyzed for illustrative purposes.

Effect of the Seasonal Rainfall Distribution on Storm-Water Quality Capture Volume Estimation

Thewater quality capturevolume (WQCV) is a design parameter that is used for sizing urban storm-water treatment facilities. The storage, treatment, overflow, and runoff model (STORM) in NetSTORM was used to translate historical hourly rainfall records into storm event runoff data and to compute the WQCV (set at the 85th percentile capture of the annual runoff volume) of eight cities located in various climatological areas of South Korea. For each city, one-parameter and two-parameter exponential distributions were fitted to the set of storm event runoff volumes that were computed by STORM, and the 85th percentile runoff volume that was determined from the STORM was compared to the 85th percentile runoff volumes that were obtained from the two exponential distributions. The results demonstrate that the more easily applied distribution approach is adequate in South Korea for determining the WQCV. In Seoul, which is a city located in the northwestern part of South Korea, the two-parameter exponential distribution yielded the best estimate of the WQCV computed by STORM. For Chuncheon, Wonju, and Daejeon, which are cities located in the north-central and central parts of South Korea, the WQCVs were accurately estimated using both the one-parameter and two-parameter exponential distributions. Finally, in the cities of Gangneung, Gwangju, Daegu, and Busan, which are located in the northeastern and southern parts of South Korea, the WQCVs were best estimated using the one- parameter exponential distribution. It was found that seasonal variations in rainfall and rainfall event depth are important factors in assessing the WQCV. This study also found that the one-parameter exponential distribution approach, which is the most common approach used in the United States, may not be adequate for WQCVestimation if the standard deviation for average monthly rainfall is significant. DOI: 10.1061/ (ASCE)WR.1943-5452.0000204. © 2013 American Society of Civil Engineers. CE Database subject headings: Nonpoint pollution; Parameters; Rainfall; Seasonal variations; South Korea; Stormwater management; Water quality.