Test Termination Time Research Articles

Order Statistics and ML Estimators of a 3-Component Mixture of Power Distribution: A Simulation Study

The probability density functions of , and order statistics are explored in this study. Also, the moments, their mean and variance of , and order statistics are derived. The Maximum Likelihood (ML) method is employed to estimate the parameters, and a system of non-linear equations is solved to derive the limiting expressions of the ML estimators and their variances, utilizing Fisher’s information matrix. A detailed numerical analysis of the ML estimators’ performance is carried out through a Monte Carlo simulation for various sample sizes, test termination times, and parameter values. Additionally, the practical applicability of the 3-component mixture of Power distributions is demonstrated by estimating the ML parameters using real-world data. The study's results indicated that maximum likelihood estimators based on complete (uncensored) data perform significantly better than those based on censored data. Additionally, the estimators derived from uncensored data are more dependable and precise.

The 3-component mixture of power distributions under Bayesian paradigm with application of life span of fatigue fracture

Mixture distributions are naturally extra attractive to model the heterogeneous environment of processes in reliability analysis than simple probability models. This focus of the study is to develop and Bayesian inference on the 3-component mixture of power distributions. Under symmetric and asymmetric loss functions, the Bayes estimators and posterior risk using priors are derived. The presentation of Bayes estimators for various sample sizes and test termination time (a fact of time after that test is terminated) is examined in this article. To assess the performance of Bayes estimators in terms of posterior risks, a Monte Carlo simulation along with real data study is presented.

Bayesian Analysis of 3-Component Unit Lindley Mixture Model with Application to Extreme Observations

Bayesian inference of the 3-component unit Lindley right censored mixture is presented in this paper. The posterior distributions of the parameters are derived assuming informative (gamma) as well as noninformative (uniform and Jeffreys) priors. For the gamma (informative) prior, hyperparameters are elicited using prior predictive distribution. The Bayesian estimation has been carried out considering both symmetric and asymmetric loss functions (squared error, quadratic, weighted, and precautionary loss functions). Simulation studies for various sample sizes and different threshold values (test termination times) are considered to evaluate the performances of the Bayes estimators w.r.t their posterior risks under the said loss functions. Real life flood data from Naser Lake is also analyzed as a 3-component mixture for the sake of illustrative purpose. The simulation study and data analysis reveals that the estimates under informative prior perform better than the noninformative priors. Also, it is observed that the increase in sample size and the threshold value (test termination time) are inversely proportional to the posterior risks. Among the loss functions considered, the loss functions performance from the best to the least, w.r.t the posterior risk, is as follows: precautionary loss function < weighted loss function < quadratic loss function < squared error loss function. Posterior risks are directly proportional to the sample size and threshold value (test termination time).

Analytical approach for designing accelerated degradation tests under an exponential dispersion model

To provide timely lifetime information of manufactured products to potential customers, designing an efficient accelerated degradation test (ADT) is an important task for reliability analysts. In the literature, several papers have addressed a general k-level ADT design problem (including the determinations of the number of stress levels (k), testing stresses, allocation of test units and termination times) when the underlying degradation path follows an exponential dispersion model. The results are practical and interesting. However, most of those studies only addressed the case of k≤4. Generally, if a direct proof (such as using the Karush–Kuhn–Tucker conditions) for the case of k>4 is intractable, we can consider an indirect proof via the general equivalence theorem.In this paper, we first propose a conjecture optimal design for a k-level ADT design problem and then apply the general equivalence theorem to show that this conjecture design turns out to be the global V-optimal design. In addition, an example is used to illustrate the proposed procedure. The main contribution of this work is that this analytical approach can provide reliability analysts with better insight for designing an efficient ADT plan.

On Estimation of Three-Component Mixture of Distributions via Bayesian and Classical Approaches

In this study, we model a heterogeneous population assuming the three-component mixture of the Pareto distributions assuming type I censored data. In particular, we study some statistical properties (such as various entropies, different inequality indices, and order statistics) of the three-component mixture distribution. The ML estimation and the Bayesian estimation of the mixture parameters have been performed in this study. For the ML estimation, we used the Newton Raphson method. To derive the posterior distributions, different noninformative priors are assumed to derive the Bayes estimators. Furthermore, we also discussed the Bayesian predictive intervals. We presented a detailed simulation study to compare the ML estimates and Bayes estimates. Moreover, we evaluated the performance of different estimates assuming various sample sizes, mixing weights and test termination times (a fixed point of time after which all other tests are dismissed). The real-life data application is also a part of this study.

Reliability Test Plan for an Extended Birnbaum-Saunders Distribution

Birnbaum-Saunders distribution has been widely studied in statistical literature because this distribution accommodates several interesting properties. The purpose of this paper is to introduce a new parametric distribution based on the Birnbaum-Saunders model and develop a new acceptance sampling plans for derived extended Birnbaum-Saunders distribution when the mean lifetime test is truncated at a predetermined time. For various acceptance numbers, confidence levels and values of the ratio of the fixed experimental time to the specified mean life, the minimum sample size necessary to assure a specified mean lifetime worked out. The results are illustrated by a numerical example. The operating characteristic functions of the sampling plans and producer’s risk and the ratio of true mean life to a specified mean life that ensures acceptance with a pre-assigned probability are tabulated. This paper presents relevant characteristics of the new distribution and a new acceptance sampling plans when the lifetime of a product adopts an extended Birnbaum-Saunders distribution. Based on this study, the optimal number of testers demanded is decreases as test termination time increases. Moreover, the operating characteristic values increases as the mean life ratio increases, which indicate that items with increased mean life will be accepted with higher probability compared with items with lower mean life ratio.

Bayesian Modeling of 3-Component Mixture of Exponentiated Inverted Weibull Distribution under Noninformative Prior

Bayesian study of 3-component mixture modeling of exponentiated inverted Weibull distribution under right type I censoring technique is conducted in this research work. The posterior distribution of the parameters is obtained assuming the noninformative (Jeffreys and uniform) priors. The different loss functions (squared error, quadratic, precautionary, and DeGroot loss function) are used to obtain the Bayes estimators and posterior risks. The performance of the Bayes estimators through posterior risks under the said loss functions is investigated through simulation process. Real data analysis of tensile strength of carbon fiber is also applied for 3 components to conclude the presentation of Bayes estimators. The limiting expressions are also elaborated for Bayes estimators and posterior risks in this study. The impact of some test termination times and sample sizes is reported on Bayes estimators.

Bayesian analysis of 3-components Kumaraswamy mixture model: Quadrature method vs. Importance sampling

This paper presents the Bayesian inference of the 3-component Kumaraswamy mixture distribution. The paper formulates the 3-component model, provides Bayesian estimates and their respective posterior risks assuming non-informative prior under type-I censoring. Since the Bayes estimates are not in closed form, the paper provides a comparative analysis of the estimates and their respective risks derived under quadrature method and importance sampling using different loss functions, assuming different sample sizes, test termination times and mixture probabilities. Finally, the proposed 3 component mixture model is then applied to the real life data to assess its validity. The convergence of the estimates under importance sampling is rapidly observed as compared to quadrature methods.

Bayesian analysis for 3-component mixture of exponentiated Weibull distribution assuming non-informative priors

ABSTRACTBayesian analysis of 3-component mixture model, of exponentiated Weibull distribution under type-I right censoring scheme, is explored in this study. With the help of non-informative (uniform and Jeffreys) priors and loss functions (e.g. squared error loss function, quadratic loss function, precautionary loss function and DeGroot loss function), Bayes estimators and posterior risks are derived. The Bayes estimators and posterior risks are observed as a function of the test termination time. Simulation study as well as a practical example is given in this study.

Bayesian estimation of the mixture of Burr Type-XII distributions using doubly censored data

This study discusses the Bayesian and maximum likelihood estimation methods for analyzing the data from a 3-component mixture of Burr Type-XII probability distributions. The maximum likelihood estimators with their variances cannot be obtained in an explicit form and thus an iterative procedure is used to calculate them numerically. Contrary to this, elegant closed form algebraic expressions of Bayes estimators and their posterior risks are derived. Using the informative and noninformative priors, the posterior predictive distributions along with predictive intervals are also discussed. A method of eliciting hyperparameters using prior predictive distribution is also a part of this study. Some interesting properties (including, posterior risks and Bayesian predictive intervals) of Bayes estimators and their behavior across different sample sizes, left and right test termination times, informative prior versus Jeffreys noninformative prior, are provided via a detailed Monte Carlo simulation study. To assess the suitability and application of the proposed model, a real life data example is also discussed in this article. Based on the simulated results and real data application, it is concluded that the IP paired with DLF (SELF) is a more suitable for estimating mixing component.

A Mathematical Model for the Effect of Parathyroid Hormone using Fuzzy Log-Normal Distribution

The theoretical study for the effect of parathyroid hormone secretion in chronic renal failure was determined. Formulae for fuzzy mean and variance of log-normal distribution and its alpha cuts were presented. Using the fuzzy log-normal distribution model, we showed that the effect of parathyroid hormone secretion is reasonably higher when the test termination time increases.

Acceptance sampling plan for truncated life test having generalized Pareto distribution

In this study, repetitive acceptance sampling plan for truncated life test is developed for that situations in which the lifetime of the product follows the generalized Pareto distribution. The parameters of the proposed plan such as sample size, acceptance numbers and probability of lot acceptance are determined when the producer’s risk, consumer’s risk and test termination time are specified. The mean lifetime of the test unit is taken as the quality parameter. Single acceptance plan is also developed for the given distribution and compared with the proposed plan in terms of the sample size required to make a decision. The proposed plan requires less sample size than the single acceptance plan. Numerical cases are presented to exemplify the applications of acceptance sampling plans.

THE ODD GENERALIZED EXPONENTIAL LOG-LOGISTIC DISTRIBUTION GROUP ACCEPTANCE SAMPLING PLAN

Abstract In this manuscript, a group acceptance sampling plan (GASP) is developed when the lifetime of the items follows odd generalized exponential log-logistic distribution (OGELLD), the multiple number of items as a group can be tested simultaneously in a tester. The design parameters such as the minimum group size and the acceptance number are derived when the consumer’s risk and the test termination time are specified. The operating characteristic (OC) function values are calculated (intended) according to various quality levels and the minimum ratios of the true average life to the specified average life at the specified producer’s risk are derived. The methodology is illustrated through real data.

Bayesian Estimation of 3-Component Mixture of the Inverse Weibull Distributions

This article focuses on the study of a 3-component mixture of the inverse Weibull distributions under Bayesian perspective. The censored sampling scheme is used because it is popular in reliability theory and survival analysis. To achieve this objective, the Bayes estimates of the parameter of the mixture model along with their posterior risks using informative and non-informative priors are attained. These estimates have been acquired under two cases: (a) when the shape parameter is known and (b) when all parameters are unknown. For the case (a), Bayes estimates are gained under three loss functions while for the case (b) only the squared error loss function is used. To study numerically, the performance of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times.

Weighted Erlang and Lomax Distributions

Two new families of weighted Erlang and Lomax probability distributions are developed. Various properties including the moments and the bias in the estimated mean of a weighted distribution are derived. As applications of these distributions, time truncated acceptance sampling plans are designed to obtain the minimum sample size and the acceptance number for un-weighted and weighted distributions, where various pre-assigned producer’s and consumer’s risks, various ratios of true mean to the specified mean life, and various test termination time multipliers are considered. A real data set is analyzed to illustrate the proposed time-truncated sampling plan.

Bayesian estimation of 3-component mixture of Gumbel type-II distributions under non-informative and informative priors

This paper deals with 3-component mixture of the Gumbel type-II distributions when the scale parameter is known under Bayesian view point. The type-I right censored sampling scheme is considered due to its extensive use in reliability theory and survival analysis, taking different non-informative and informative priors. Bayes estimates of the parameters of the mixture model along with their posterior risks are derived under different loss functions. In case where no or little prior information is available, elicitation of hyperparameters is given. In order to numerically study the execution of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times. The comparisons among the estimators have been made in terms of the corresponding posterior risks. A real life data example is also given to illustrate the study.

Group acceptance sampling plan for resubmitted lots based on life tests for odds exponential log logistic distribution

PurposeThe purpose of this paper is to develop a group acceptance sampling plan (GASP) for a resubmitted lot when the lifetime of a product follows odds exponential log logistic distribution introduced by Rao and Rao (2014). The parameters of the proposed plan such as minimum group size and acceptance number are determined for a pre-specified consumer’s risk, number of testers and the test termination time. The authors compare the proposed plan with the ordinary GASP, and the results are illustrated with live data example.Design/methodology/approachThe parameters of the proposed plan such as minimum group size and acceptance number are determined for a pre-specified consumer’s risk, number of testers and the test termination time.FindingsThe authors determined the group size and acceptance number.Research limitations/implicationsNo specific limitations.Practical implicationsThis methodology can be applicable in industry to study quality control.Social implicationsThis methodology can be applicable in health study.Originality/valueThe parameters of the proposed plan such as minimum group size and acceptance number are determined for a pre-specified consumer’s risk, number of testers and the test termination time.

Bayesian Analysis of the Mixture of Frechet Distribution under Different Loss Functions

This paper has to do with 3-component mixture of the Frechet dis- tributions when the shape parameter is known under Bayesian view point. The type-I right censored sampling scheme is considered due to its extensive use in reliability theory and survival analysis. Taking dif- ferent non-informative and informative priors, Bayes estimates of the parameter of the mixture model along with their posterior risks are derived under squared error loss function, precautionary loss function and DeGroot loss function. In case, no or little prior information is available, elicitation of hyper parameters is given. In order to study numerically, the execution of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times. A real life data example is also given to illustrate the study.

Optimal design of constant-stress accelerated degradation tests using the M-optimality criterion

In this paper, we propose the M-optimality criterion for designing constant-stress accelerated degradation tests (ADTs). The newly proposed criterion concentrates on the degradation mechanism equivalence rather than evaluation precision or prediction accuracy which is usually considered in traditional optimization criteria. Subject to the constraints of total sample number, test termination time as well as the stress region, an optimum constant-stress ADT plan is derived by determining the combination of stress levels and the number of samples allocated to each stress level, when the degradation path comes from inverse Gaussian (IG) process model with covariates and random effects. A numerical example is presented to verify the robustness of our proposed optimum plan and compare its efficiency with other test plans. Results show that, with a slightly relaxed requirement of evaluation precision and prediction accuracy, our proposed optimum plan reduces the dispersion of the estimated acceleration factor between the usage stress level and a higher accelerated stress level, which makes an important contribution to reliability demonstration and assessment tests.

Bayesian Analysis of a 3-Component Mixture of Rayleigh Distributions under Type-I Right Censoring Scheme

Since the last few decades, constructing flexible parametric classes of probability distributions has been the most popular approach in the Bayesian analysis. As compared to simple probability models, a mixture model of some suitable lifetime distributions may be more capable of capturing the heterogeneity of the nature. In this study, a 3-component mixture of Rayleigh distributions is investigated by considering type-I right censoring scheme to obtain data from a heterogeneous population. The closed form expressions for the Bayes estimators and posterior risks assuming the non-informative (uniform and Jeffreys’) priors under squared error loss function, precautionary loss function and DeGroot loss function are derived. The performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. The posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. In addition, the limiting expressions for the Bayes estimators and posterior risks are derived. Simulated data sets are used for the different comparisons and the model is finally illustrated using the real data.