Optimal Test Plan of Step Stress Partially Accelerated Life Testing for Alpha Power Inverse Weibull Distribution under Adaptive Progressive Hybrid Censored Data and Different Loss Functions

Accelerated life tests are used to explore the lifetime of extremely reliable items by subjecting them to elevated stress levels from stressors to cause early failures, such as temperature, voltage, pressure, and so on. The alpha power inverse Weibull (APIW) distribution is of great significance and practical applications due to its appealing characteristics, such as its flexibilities in the probability density function and the hazard rate function. We analyze the step stress partially accelerated life testing model with samples from the APIW distribution under adaptive type II progressively hybrid censoring. We first obtain the maximum likelihood estimates and two types of approximate confidence intervals of the distributional parameters and then derive Bayes estimates of the unknownparameters under different loss functions. Furthermore, we analyze three probable optimum test techniques for identifying the best censoring under different optimality criteria methods. We conduct simulation studies to assess the finite sample performance of the proposed methodology. Finally, we provide a real data example to further demonstrate the proposed technique

Classical and Bayesian Inference on Finite Mixture of Exponentiated Kumaraswamy Gompertz and Exponentiated Kumaraswamy Fréchet Distributions under Progressive Type II Censoring with Applications

A finite mixture of exponentiated Kumaraswamy Gompertz and exponentiated Kumaraswamy Fréchet is developed and discussed as a novel probability model. We study some useful structural properties of the proposed model. To estimate the model parameters under the classical method, we use the maximum likelihood estimation using a progressive type II censoring scheme. Under the Bayesian paradigm the estimation is carried out with gamma priors under a progressive type II censored samples with squared error loss function. To demonstrate the efficiency of the proposed model based on progressively type II censoring, a simulation study is carried out. Three actual data sets are used as an example, demonstrating that the suggested model in the new class fits better than the existing finite mixture models available in the literature

Optimal Design for a Bivariate Step-Stress Accelerated Life Test with Alpha Power Exponential Distribution Based on Type-I Progressive Censored Samples

We consider an optimization design for the alpha power exponential (APE) distribution as asymmetrical probability distributions under progressive type-I censoring for a step-stress accelerated life test. In this study, two stress variables are taken into account. To save the time and cost of lifetime testing, progressive censoring and accelerated life testing are utilized. The test units’ lifespans are supposed to follow an APE distribution. A cumulative exposure model is used to study the impact of varying stress levels. A log-linear relationship between the APE distribution’s scale parameter and stress is postulated. The maximum likelihood estimators, Bayesian estimators of the model parameters based on the symmetric loss function, approximate confidence intervals (CIs) and credible intervals are provided. Under normal operating conditions, an ideal test plan is designed by minimizing the asymptotic variance of the percentile life

A New Exponential Distribution to Model Concrete Compressive Strength Data

Concrete mixtures can be developed to deliver a broad spectrum of mechanical and durability properties to satisfy the configuration conditions of construction. One technique for evaluating the compressive strength of concrete is to suppose that it pursues a probabilistic model from which it is reliability estimated. In this paper, a new technique to generate probability distributions is considered and a new three-parameter exponential distribution as a new member of the new family is presented in detail. The proposed distribution is able to model the compressive strength of high-performance concrete rather than some other competitive models. The new distribution delivers decreasing, increasing, upside-down bathtub and bathtub-shaped hazard rates. The maximum likelihood estimation approach is used to estimate model parameters as well as the reliability function. The approximate confidence intervals of these quantities are also obtained. To assess the performance of the point and interval estimations, a simulation study was conducted. We demonstrate the performance of the offered new distribution by investigating one high-performance concrete compressive strength dataset. The numerical outcomes showed that the maximum likelihood method provides consistent and asymptotically unbiased estimators. The estimates of the unknown parameters as well as the reliability function perform well as sample size increases in terms of minimum mean square error. The confidence interval of the reliability function has an appropriate length utilizing the delta method. Moreover, the real data analysis indicated that the new distribution is more suitable when compared to some well-known and some recently proposed distributions to evaluate the reliability of concrete mixtures

Explained variation for survival and recurrent event data

PhD ThesisExplained variation measures are used to quantify the amount of information in a model and especially how useful the model might be when predicting future observations. Such measures are useful in guiding model choice for all types of predictive regression models, including linear and generalized linear models and survival analysis. The rst part of this thesis considers explained variation for survival data and we investigate how individual observations in a data set can in uence the value of various proposed statistics . In uence of a subject is a measure of the e ect on estimates of deleting him/her from the data set. In uence on regression coe cients has had much attention but there has not been work in in uence for explained variation for survival data analysis or other measures of predictive accuracy. Generally in reasonable size data sets the deletion of a single subject has no e ect on conclusions. However, examination of distance between measures with and without the subject can be useful in distinguishing abnormal observations. In the second part of the thesis we investigate how measures of explained variation for survival data can be extended to recurrent event data. We describe an existing rank-based measure and we investigate a new statistic based on observed and expected event count processes. Both methods can be used for all models. Adjustments for missing data are proposed for the count measure and demonstrated through simulation to be e ective. We compare the population values of the two statistics and illustrate their use in comparing an array of non-nested models for data on recurrent episodes of infant diarrhea. There is evidence that the rank-based method is robust to ignored random e ects and also to the presence of unusual observations. The count-based method more directly compares observed and expected intensities. We assess in uence of individual observation on these measures

Complexity Analysis of E-Bayesian Estimation under Type-II Censoring with Application to Organ Transplant Blood Data

The E-Bayesian estimation approach has been presented for estimating the parameter and/or reliability characteristics of various models. Several investigations in the literature have considered this method under the assumption that just one parameter is unknown. So, based on Type-II censoring, this study proposes for the first time an effort to use the E-Bayesian estimation approach to estimate the full model parameters as well as certain related functions such as the reliability and hazard rate functions. To illustrate this purpose, we apply the proposed technique to the two-parameter generalized inverted exponential distribution which can be considered to be one of the most flexible asymmetrical probability distributions. Moreover, the E-Bayesian method, maximum likelihood, and Bayesian estimation approaches are also considered for comparison purposes. Under the assumption of independent gamma priors, the Bayes and E-Bayes estimators are developed using the symmetrical squared error loss function. Due to the complex form of the joint posterior density, two approximation techniques, namely the Lindley and Markov chain Monte Carlo methods, are considered to carry out the Bayes and E-Bayes estimates and also to construct the associate credible intervals. Monte Carlo simulations are performed to assess the performance of the proposed estimators. To demonstrate the applicability of the proposed methods in real phenomenon, one real data set is analyzed and it shows that the proposed method is effective and easy to operate in a real-life scenario

Bayesian and Frequentist Analytical Approaches Using Log-Normal and Gamma Frailty Parametric Models for Breast Cancer Mortality

One of the major causes of death among females in Saudi Arabia is breast cancer. Newly diagnosed cases of breast cancer among the female population in Saudi Arabia is 19.5%. With this high incidence, it is crucial that we explore the determinants associated with breast cancer among the Saudi Arabia populace—the focus of this current study. The total sample size for this study is 8312 (8172 females and about 140 representing 1.68% males) patients that were diagnosed with advanced breast cancer. These are facility-based cross-sectional data collected over a 9-year period (2004 to 2013) from a routine health information system database. The data were obtained from the Saudi Cancer Registry (SCR). Both descriptive and inferential (Cox with log-normal and gamma frailties) statistics were conducted. The deviance information criterion (DIC), Watanabe–Akaike information criterion (WAIC), Bayesian information criterion (BIC), and Akaike information criterion were used to evaluate or discriminate between models. For all the six models fitted, the models which combined the fixed and random effects performed better than those with only the fixed effects. This is so because those models had smaller AIC and BIC values. The analyses were done using R and the INLA statistical software. There are evident disparities by regions with Riyadh, Makkah, and Eastern Province having the highest number of cancer patients at 28%, 26%, and 20% respectively. Grade II (46%) and Grade III (45%) are the most common cancer grades. Left paired site laterality (51%) and regional extent (52%) were also most common characteristics. Overall marital status, grade, and cancer extent increased the risk of a cancer patient dying. Those that were married had a hazard ratio of 1.36 (95% CI: 1.03–1.80) while widowed had a hazard ratio of 1.57 (95% CI: 1.14–2.18). Both the married and widowed were at higher risk of dying with cancer relative to respondents who had divorced. For grade, the risk was higher for all the levels, that is, Grade I (Well diff) (HR = 7.11, 95% CI: 3.32–15.23), Grade II (Mod diff) (HR = 7.89, 95% CI: 3.88–16.06), Grade III (Poor diff) (HR = 5.90, 95% CI (2.91–11.96), and Grade IV (Undiff) (HR = 5.44, 95% (2.48–11.9), relative to B-cell. These findings provide empirical evidence that information about individual patients and their region of residence is an important contributor in understanding the inequalities in cancer mortalities and that the application of robust statistical methodologies is also needed to better understand these issues well

Analysis of the new complementary unit Weibull model from adaptive progressively type-II hybrid

In this study, we look at some estimation issues for complementary unit Weibull distributions in the context of adaptive progressive type-II hybrid censoring. The point and interval estimations of the model parameters, as well as a number of its reliability indices, are explored. The likelihood frequentist approach is used as a classical strategy to obtain the point and approximate confidence ranges. The median parameter of the distribution is produced in a closed form as a function of the shape parameter, while the shape parameter can be obtained iteratively. The squared error loss function and gamma and beta prior distributions are used for evaluating Bayes estimates. The Markov chain Monte Carlo method is used to solve the difficult posterior distribution expression in order to provide Bayes estimates and the highest posterior density credible ranges. A simulation study is done to evaluate the efficacy of various estimating methodologies making use of different circumstances for sample sizes and progressive censoring strategies. Finally, three real-world datasets from veterinary, industrial, and physical applications are examined to highlight the practical importance of the provided methodologies

Inferences for Nadarajah–Haghighi Parameters via Type-II Adaptive Progressive Hybrid Censoring with Applications

This study aims to investigate the estimation problems when the parent distribution of the population under consideration is the Nadarajah–Haghighi distribution in the presence of an adaptive progressive Type-II hybrid censoring scheme. Two approaches are considered in this regard, namely, the maximum likelihood and Bayesian estimation methods. From the classical point of view, the maximum likelihood estimates of the unknown parameters, reliability, and hazard rate functions are obtained as well as the associated approximate confidence intervals. On the other hand, the Bayes estimates are obtained based on symmetric and asymmetric loss functions. The Bayes point estimates and the highest posterior density Bayes credible intervals are computed using the Monte Carlo Markov Chain technique. A comprehensive simulation study is implemented by proposing different scenarios for sample sizes and progressive censoring schemes. Moreover, two applications are considered by analyzing two real data sets. The outcomes of the numerical investigations show that the Bayes estimates using the general entropy loss function are preferred over the other methods

Estimation of Reliability Indices for Alpha Power Exponential Distribution Based on Progressively Censored Competing Risks Data

In reliability analysis and life testing studies, the experimenter is frequently interested in studying a specific risk factor in the presence of other factors. In this paper, the estimation of the unknown parameters, reliability and hazard functions of alpha power exponential distribution is considered based on progressively Type-II censored competing risks data. We assume that the latent cause of failures has independent alpha power exponential distributions with different scale and shape parameters. The maximum likelihood method is considered to estimate the model parameters as well as the reliability and hazard rate functions. The approximate and two parametric bootstrap confidence intervals of the different estimators are constructed. Moreover, the Bayesian estimation method of the unknown parameters, reliability and hazard rate functions are obtained based on the squared error loss function using independent gamma priors. To get the Bayesian estimates as well as the highest posterior credible intervals, the Markov Chain Monte Carlo procedure is implemented. A comprehensive simulation experiment is conducted to compare the performance of the proposed procedures. Finally, a real dataset for the relapse of multiple myeloma with transplant-related mortality is analyzed