A method for computing tolerance intervals for a location-scale family of distributions
- PDF / 400,994 Bytes
- 28 Pages / 439.37 x 666.142 pts Page_size
- 104 Downloads / 173 Views
A method for computing tolerance intervals for a location-scale family of distributions Ngan Hoang-Nguyen-Thuy1 · K. Krishnamoorthy1 Received: 2 May 2019 / Accepted: 27 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The problems of computing two-sided tolerance intervals (TIs) and equal-tailed TIs for a location-scale family of distributions are considered. The TIs are constructed using one-sided tolerance limits with the Bonferroni adjustments and then adjusting the confidence levels so that the coverage probabilities of the TIs are equal to the specified nominal confidence level. The methods are simple, exact and can be used to find TIs for all location-scale families of distributions including log-location-scale families. The computational methods are illustrated for the normal, Weibull, twoparameter Rayleigh and two-parameter exponential distributions. The computational method is applicable to find TIs based on a type II censored sample. Factors for computing two-sided TIs and equal-tailed TIs are tabulated and R functions to find tolerance factors are provided in a supplementary file. The methods are illustrated using a few practical examples. Keywords Asymmetric location-scale · Bisection method · Bonferroni · Content · Coverage level · Equivariant estimators · type II censored
1 Introduction In many practical applications, such as medical, environmental and engineering, it is desired to find an interval estimate based on a sample that would capture at least a proportion p of the sampled population with confidence γ . Such a statistical interval is referred to as the tolerance interval (TI). A tolerance interval based on a random sample is constructed so that it would include at least a proportion p of the sampled population with confidence level γ . This type of interval estimate is referred to as a p content—γ
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s00180020-01031-w) contains supplementary material, which is available to authorized users.
B 1
K. Krishnamoorthy [email protected] Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USA
123
N. Hoang-Nguyen-Thuy, K. Krishnamoorthy
coverage tolerance interval (TI) or simply ( p, γ ) TI. Another type of TI (L e , Ue ) is constructed so that at most a proportion (1 − p)/2 of the population is less than the lower endpoint L e and at most a proportion (1 − p)/2 of the population is greater than the upper endpoint Ue (Owen 1964). This type of TI controls the percentages in both tails and is referred to as the ( p, γ ) “equal-tailed” TI. A ( p, γ ) one-sided lower tolerance limit (TL) is constructed so that at least a proportion p of the population falls above the limit with confidence γ while a ( p, γ ) one-sided upper TL is constructed so that at least a proportion p of the population falls below the limit with confidence γ . For earlier work and numerous applications of TIs, see the book by Guttman (1970), and the book by Krishnamo
Data Loading...
