Analysis of survival data by a Weibull–Bessel distribution

In survival analysis and reliability studies, problems with random sample size arise quite frequently. More specifically, in cancer studies, the number of clonogens is unknown and the time to relapse of the cancer is defined by the minimum of the incubation times of the various clonogenic cells. In this article, we have proposed a new model where the distribution of the incubation time is taken as Weibull and the distribution of the random sample size as Bessel, giving rise to a Weibull–Bessel distribution. The maximum likelihood estimation of the model parameters is studied and a score test is developed to compare it with its special submodel, namely, exponential–Bessel distribution. To illustrate the model, two real datasets are examined, and it is shown that the proposed model, presented here, fits better than several other existing models in the literature. Extensive simulation studies are also carried out to examine the performance of the estimates.     

A HEWMA-CUSUM control chart for the Weibull distribution

The Weibull distribution is one of the most popular distributions for lifetime modeling. However, there has not been much research on control charts for a Weibull distribution. Shewhart control is known to be inefficient to detect a small shift in the process, while exponentially weighted moving average (EWMA) and cumulative sum control chart (CUSUM) charts have the ability to detect small changes in the process. To enhance the performance of a control chart for a Weibull distribution, we introduce a new control chart based on hybrid EWMA and CUSUM statistic, called the HEWMA-CUSUM chart. The performance of the proposed chart is compared with the existing chart in terms of the average run length (ARL). The proposed chart is found to be more sensitive than the existing chart in ARL. A simulation study is provided for illustration purposes. A real data is also applied to the proposed chart for practical use.     

A control chart for monitoring process variation using multiple dependent state sampling

A control chart for monitoring process variation by using multiple dependent state (MDS) sampling is constructed in the present article. The operational formulas for in-control and out-of-control average run lengths (ARLs) are derived. Control constants are established by considering the target in-control ARL at a normal process. The extensive ARL tables are reported for various parameters and shifted values of process parameters. The performance of the proposed control chart has been evaluated with several existing charts in regard of ARLs, which empowered the presented chart and proved far better for timely detection of assignable causes. The application of the proposed concept is illustrated with a real-life industrial example and a simulation-based study to elaborate strength of the proposed chart over the existing concepts.     

A double acceptance sampling plan for generalized log-logistic distributions with known shape parameters

A double acceptance sampling plan for the truncated life test is developed assuming that the lifetime of a product follows a generalized log-logistic distribution with known shape parameters. The zero and one failure scheme is mainly considered, where the lot is accepted if no failures are observed from the first sample and it is rejected if two or more failures occur. When there is one failure from the first sample, the second sample is drawn and tested for the same duration as the first sample. The minimum sample sizes of the first and second samples are determined to ensure that the true median life is longer than the given life at the specified consumer’s confidence level. The operating characteristics are analyzed according to various ratios of the true median life to the specified life. The minimum such ratios are also obtained so as to lower the producer’s risk at the specified level. The results are explained with examples.     

Optimized estimation for population mean using conventional and non-conventional measures under the joint influence of measurement error and non-response

Most of the research work in the theory of survey sampling only deals with the sampling errors under the assumptions: (i) there is a complete response and (ii) recorded information from individuals is correct but in practice it is not always true. Non-sampling errors like non-response and measurement errors (MEs) mostly creep into the survey and become more influential for estimators than sampling errors. Considering this practical situation of non-response and MEs jointly, we proposed an optimum class of estimators for population mean under simple random sampling using conventional and non-conventional measures. Bias and mean square error of the proposed estimators are derived up to first degree of approximation. Moreover, a simulation study is conducted to assess the performance of new estimators which proves that proposed estimators are more efficient than the traditional Hansen and Hurwitz estimator and other competing estimators.     

Bayesian inference for the randomly censored Burr-type XII distribution

The article presents the Bayesian inference for the parameters of randomly censored Burr-type XII distribution with proportional hazards. The joint conjugate prior of the proposed model parameters does not exist; we consider two different systems of priors for Bayesian estimation. The explicit forms of the Bayes estimators are not possible; we use Lindley's method to obtain the Bayes estimates. However, it is not possible to obtain the Bayesian credible intervals with Lindley's method; we suggest the Gibbs sampling procedure for this purpose. Numerical experiments are performed to check the properties of the different estimators. The proposed methodology is applied to a real-life data for illustrative purposes. The Bayes estimators are compared with the Maximum likelihood estimators via numerical experiments and real data analysis. The model is validated using posterior predictive simulation in order to ascertain its appropriateness.     

Multiple dependent state repetitive sampling plans with or without auxiliary variable

In this article, a new variable acceptance sampling plan has been developed using the multiple dependent state repetitive sampling scheme for the normal distribution. The plan parameters have been determined so as to minimize the average sample number while satisfying the producer's and the consumer's risks under the operating characteristic function. The multiple dependent state repetitive sampling scheme has also been extended to the case of utilizing an auxiliary variable. The proposed sampling plans are compared with the existing sampling plans.     

Progressive Variance Control Charts for Monitoring Process Dispersion

In a process, the deviation from location or scale parameters affects the quality of the process and waste resources. So it is essential to monitor such processes for possible changes due to any assignable causes. Control charts are the most famous tool used to meet this intention. It is useless to monitor process location until the assurance that process dispersion is in-control. This study proposes some new two-sided memory control charts named as progressive variance (PV) control charts which are based on sample variance to monitor changes in process dispersion assuming normality of quality characteristic to be monitored. Simulation studies are made, and an example is discussed to evaluate the performance of the proposed charts. The comparison of the proposed chart is made with exponentially weighted moving average- and cumulative sum-type charts for process dispersion. The study shows that performance of the proposed charts are uniformly better than its competitors for detecting positive shifts while for detecting negative shift in the variance their performance is better for small shifts and reasonably good for moderated shifts.     

On the estimation of parameters,survival functions,and hazard rates based on fuzzy life time data

Life time data analysis is regarded as one of the significant out-shoots of statistics. Classical statistical techniques reckon life time observations as precise numbers and solely cover variation among the observations. In fact, there are two types of uncertainty in data: variation among the observations and the fuzziness. To this effect, the analysis techniques, which do not consider fuzziness and are only based on precise life time observations, use incomplete information; hence lead to pseudo results. This study aimed at generalizing parameters estimation, survival functions, and hazard rates for fuzzy life time data.     

A Note On Beta Inverse-Weibull Distribution

The inverse Weibull distribution is one of the widely applied distribution for problems in reliability theory. In this article, we introduce a generalization—referred to as the Beta Inverse-Weibull distribution—generated from the logit of a beta random variable. We provide a comprehensive treatment of the mathematical properties of the Beta Inverse-Weibull distribution. The shapes of the corresponding probability density function and the hazard rate function have been obtained and graphical illustrations have been given. The distribution is found to be unimodal. Results for the non central moments are obtained. The relationship between the parameters and the mean, variance, skewness, and kurtosis are provided. The method of maximum likelihood is proposed for estimating the model parameters. We hope that this generalization will attract wider applicability to the problems in reliability theory and mechanical engineering.