Design And Applications Of The Exact Tolerance Intervals For The Skewed Populations:the Frequentist And The Bayesian Cases

The tolerance interval can be used to judge whether the production process is on control,and improve the efficiency of detecting abnormal conditions in the process,and achieve the purpose of pre-prevention.Tolerance intervals are powerful tools and widely used in various fields,including manufacturing,environmental monitoring,biometry,medicine,pharmacy,biochemistry,reliability analysis and quality assessment of a product.Therefore,it is an inevitable trend that tolerance interval is widely used in process control.At present,the study of the tolerance interval is mostly based on the assumption which the data coms from a normal population or an approximate normal population.But in reality,the normality assumption often cannot be met.Therefore,this thesis mainly focuses on the applications of tolerance intervals in skewness populations.We calculate exact two-sided control-the-center tolerance intervals,control-both-tails tolerance intervals and -expectation tolerance intervals for the exponential distribution based on record values,the Maxwell distribution and sample variances,respectively.In order to provide essential improvements in performance and save cost,the Bayes theorem is applied to the construction of the tolerance intervals.The Bayesian tolerance interval allows an analyst to incorporate prior knowledge into a data analysis problem to supplement limited data.And we use different accuracy criteria to measure the performances of the frequentist tolerance intervals and the Bayesian tolerance interval.The minimum sample sizes needed to achieve the desired accuracy are also provided.The results show that the Bayesian approach is superior to the traditional perspective,indicating a higher accuracy level.As a consequence,the Bayesian paradigm just needs a smaller sample size in order to achieve a specified level of accuracy.The main content of this thesis is as follows:(1)We construct the control-the-center tolerance interval,the control-both-tails tolerance interval and the -expectation tolerance interval for the exponential distribution based on record values,the Maxwell distribution and sample variances,respectively.(2)Based on the Bayesian viewpoint,the Bayesian control-the-center tolerance intervals,the Bayesian control-both-tails tolerance intervals and the Bayesian -expectation tolerance intervals are constructed by using Jeffrey Prior and the conjugate prior.(3)We provide a comparison of the frequentist and the Bayesian approaches in terms of the accuracy,and investigate the minimum sample sizes to attain a specified accuracy.The higher the accuracy is,the better the tolerance interval can meet the demand of the producer.