Smallest Extreme Value Distribution Research Articles

Model Selection Approaches for Predicting Future Order Statistics from Type II Censored Data

This paper studies a discriminant problem of location-scale family in case of prediction from type II censored samples. Three model selection approaches and two types of predictors are, respectively, proposed to predict the future order statistics from censored data when the best underlying distribution is not clear with several candidates. Two members in the location-scale family, the normal distribution and smallest extreme value distribution, are used as candidates to illustrate the best model competition for the underlying distribution via using the proposed prediction methods. The performance of correct and incorrect selections under correct specification and misspecification is evaluated via using Monte Carlo simulations. Simulation results show that model misspecification has impact on the prediction precision and the proposed three model selection approaches perform well when more than one candidate distributions are competing for the best underlying distribution. Finally, the proposed approaches are applied to three data sets.

Application of Mixture Models for Analyzing Reliability Data: A Case Study

Whenever purchasing durable goods, customers expect it to perform properly at least for a reasonable period of time. Over the last 30 years there has been a heightened interest in improving quality, productivity and reliability of manufactured products. As discussed by Murthy, Xie and Jiang, [1] analyzed “Aircraft windshield failure data” and fitted 2-fold Weibull mixture model for this data set. They estimated the model parameters from WPP plot. In this study, a set of competitive 2-fold mixture models (including Weibull, Exponential, Normal, Lognormal, Smallest extreme value distribution) are applied to find out the suitable statistical models. The data consist of both failure and censored lifetimes of the windshield. In the existing literature, there are many uses of mixture models for the complete data, but very limited literature available about it uses for the censored data case. Maximum likelihood estimation method is used to estimate the model parameters and the Akaike Information Criterion (AIC), Anderson-Darling (AD) and Adjusted Anderson Darling (AD*) test statistics are applied to select the suitable models among a set of competitive models. Various characteristics of the mixture models, such as the cumulative distribution function, reliability function, mean time to failure, B10 life, etc. are estimated to assess the reliability of the component.

Monitoring the Weibull Shape Parameter with Type II Censored Data

In this article, we introduce a method for monitoring the Weibull shape parameter β with type II (failure) censored data. The control limits depend on the sample size, the number of censored observations, the target average run length, and the stable value of β. The method assumes that the scale parameter α is constant during each sampling period, which is true under rational subgrouping. The proposed method utilizes the relationship between Weibull and smallest extreme value distribution. We propose an unbiased estimator of σ = 1/β as the monitoring statistic. We derive the control limits for one-sided and two-sided charts for several stable process average run lengths. We discuss two schemes, namely, the control-limits-only scheme and the control-limits-with-warning-lines scheme. The stable process average run length performance of the proposed charts is studied and compared with those of other charts for monitoring β under similar assumptions. Copyright © 2014 John Wiley & Sons, Ltd.

Moving range charts for monitoring the Weibull shape parameter with single observation samples

This paper proposes an approach to monitor shifts in the Weibull shape parameter bfβ via control charts based on the moving range of single-point samples from a smallest extreme value distribution. The average run length (ARL) of the proposed charts are computed using Fredholm integral equations of the second kind. The derived control limits for one-sided and two-sided control charts are unbiased in the sense that the ARL when β has shifted is shorter than the desired stable-process ARL. These control limits depend only on the desired stable-process ARL and the stable value of β. The paper also discusses the sample size requirements for Phase I so that the run length distributions are similar under standards-given scenario (β is given) and retrospective scenario (β is estimated from past data). The proposed methods are then applied to data on the breaking strengths of carbon fibers. The results suggest that one-sided control charts can detect small shifts in β sooner than two-sided charts. Copyright © 2011 John Wiley & Sons, Ltd.

Monitoring the Weibull shape parameter by control charts for the sample range

AbstractIn this paper, we propose control charts for monitoring changes in the Weibull shape parameter β. These charts are based on the range of a random sample from the smallest extreme value distribution. The control chart limits depend only on the sample size, the desired stable average run length (ARL), and the stable value of β. We derive control limits for both one‐ and two‐sided control charts. They are unbiased with respect to the ARL. We discuss sample size requirements if the stable value of βis estimated from past data. The proposed method is applied to data on the breaking strengths of carbon fibers. We recommend one‐sided charts for detecting specific changes in βbecause they are expected to signal out‐of‐control sooner than the two‐sided charts. Copyright © 2010 John Wiley & Sons, Ltd.

Total-Probability Method in Reliability Design of Gear Strength

Total probability method is based on stress-strength interference theory. There are four acceptable distribution types for gear (strength, stress), which are normal distribution, lognormal distribution, Weibull distribution and the smallest extreme value distribution. Based on the four distribution types, 16 different stress-strength distributions were obtained through permutation respectively. The results show that they summarize the mathematical model of reliability calculation for the gear actual failure mode and get the exact solutions of gear reliability.

Theory for Optimum Accelerated Censored Life Tests for Weibull and Extreme Value Distributions

This paper presents maximum likelihood theory for large-sample optimum accelerated life test plans. The plans are used to estimate a simple linear relationship between (transformed) stress and product life, which has a Weibull or smallest extreme value distribution. Censored data are to be analyzed before all test units fail. The plans show that all test units need not run to failure and that more units should be tested at low test stresses than at high ones. The plans are illustrated with a voltage-accelerated life test of an electrical insulating fluid.

Optimum Accelerated Life-Tests for the Weibull and Extreme Value Distributions

This paper presents charts for optimum accelerated life-test plans for estimating a simple linear relationship between an accelerating stress and product life, which has a Weibull or smallest extreme value distribution, when the data are to be analyzed before all tests units fail. The plans show that one need not run all test units to failure and that more units ought to be tested at low test stresses than at high ones. The plans are illustrated with a voltage-accelerated life test of an electrical insulating fluid.