Exact two‐sided tolerance interval controlling tail proportions for sample variances

In the traditional industrial verification process, when the aim is the compliance to assigned specifications, it is difficult to find an affordable statistical method for the purpose. Most data tables in industrial procedures and standards deal with tolerance limits neglecting the potential needs to verify assigned specification limits. A two‐sided tolerance interval, combined with a bivariate statistical hypothesis test can be used to address this problem. The proposed risk‐based approach leads to the determination of the minimum sample size with preestablished probabilities of Type I and Type II errors, that are essential elements for estimating the safety and reliability risk. A novel method is proposed for determination of the tolerance interval testing factors. This approach calculates the testing factors based on the deviation of the mean and the variance from the null hypothesis when a specified value of Type II error is found. The deviations of the mean and variance are determined in such a way that an assigned proportion of the population falls within the specification limits. Additional studies are provided to assess the robustness of the method for nonnormal environments and to compare it with other methods.

α

and

β

that relate the random tolerance limits with the specification limits.…”

Section: The New Approachmentioning

confidence: 99%

“…A more recent reference by Guo et al 9 proposes an alternative that controls the precision of the proportion in each of the two tails of the distribution, which shall occur with an assigned probability

γ^{'}

close to 1.…”

Section: Introductionmentioning

confidence: 99%

Determining sample size for specification limits verification with tolerance intervals

Vasta

2021

“…This means that the TI is balanced when

m \to \infty ,

in which case

{S^2} \to {\sigma ^2}.

Guo et al. [10] proposed an outer TI, see Guenther [11], for S 2 , which guarantees the probabilities on the tails to be less than

( {1 - \beta } )/2

with probability γ, as a remedy to the imbalance in the TI by Sarmiento et al. [8].…”

Section: Introductionmentioning

confidence: 99%

Exact tolerance interval with specified tail probabilities and a control chart for the sample variance

Alqurashi

Chakraborti

Holcombe

2022

A 𝛽-content tolerance interval (TI) is a statistical interval which contains at least some fraction (proportion) 𝛽 of the population with a given confidence level. When we are interested in the precision of a quality characteristic, a TI for the sample variance is useful. In this paper, we consider an exact two-sided 𝛽-content TI for the sample variance from a normal distribution with a specified ratio of the tail probabilities. The proposed tolerance interval allows the practitioner more control over how the probabilities in the tails are distributed, which may be useful in certain applications. A comparison with an existing two-sided 𝛽-content TI shows that the proposed TI is better on the basis of expected coverage and standard deviation of the coverage. In addition, the proposed TI is shown to require fewer subgroups to achieve a specific accuracy level. Moreover, a phase II control chart with guaranteed performance is obtained from the proposed TI. Finally, a real and a simulated data are used for illustration.

“…For example, researchers have proposed TIs for a normal distribution. [12][13][14][15][16] Simultaneously, many studies on the TI for an exponential distribution also emerged. [17][18][19][20] Besides, TIs have also been studied for the Pareto distribution, 21 hypergeometric distribution or negative hypergeometric distribution, 22 and binomial distribution.…”

Section: Introductionmentioning

confidence: 99%

“…Most of the existing TIs have been constructed based on complete data. For example, researchers have proposed TIs for a normal distribution 12‐16 . Simultaneously, many studies on the TI for an exponential distribution also emerged 17‐20 .…”

Section: Introductionmentioning

confidence: 99%

Constructing exact tolerance intervals for the exponential distribution based on record values

Guo

Zhu

Wang

et al. 2020

Self Cite

Recently, with the wide application of tolerance intervals (TIs), especially in quality management, the construction of TIs has attracted increasing attention. However, TIs applied to record data have not been well established as they have been for complete data. In many industrial stress tests, only record data are stored instead of complete data, which leads to the fact that the developments of methods based on record data are as important as those based on complete data. In this paper, we propose the exact two-sided TIs for the exponential distribution based on record values from the frequentist and Bayesian perspectives. The accuracy of each type of TIs is quantified. The results show that the Bayesian approach is superior to the frequentist approach in terms of the accuracy. A real data example is used to illustrate the constructions and implementations of the proposed TIs.