Determination of Sample Sizes for Setting Tolerance Limits

Safety margins confidence estimation for a passive residual heat removal system

“…Thus, tin this work the statistical analyses of the model evaluations for obtaining confidence intervals for safety parameters estimates rely on the use of Order Statistics (OS), along a non-parametric approach initially explored by [Wilks, 1941;Wilks, 1942]; this brings the advantage that the number of code calculations needed for safety margins evaluation is independent of the number of uncertain input parameters and provides a given confidence on the reliability of the calculated point-estimate obtained with a limited number of code runs. Figure 1 shows a schematic sketch of the non-parametric procedure here adopted [Secchi et al, 2008]; for ease of illustration, a single safety parameter y is considered.…”

“…In this case, the confidence in the estimates becomes crucial for decision making and must, thus, be quantified [Wilks, 1941;Wilks, 1942;Guba et al, 2003;Nutt et al, 2004].…”

“…These can be used to estimate a given percentile of the safety margin probability distribution. To obtain the desired confidence in the safety margin percentile, the number N of code runs is defined on the basis of the Order Statistics (OS) methodology, in its nonparametric formulation which applies independently from the type of probability distribution of the output data under study (in this case unknown) [Wilks, 1941;Wilks, 1942]. As we shall see, this amounts to ordering the elements of the random sample by increasing value, the element in the r th place being the statistic of order r, and using the order statistics for estimating the percentiles of the distribution (Section 2.4) with the desired confidence.…”

Small nonparametric tolerance regions for directional data

“…Starting from the early forties many publications have appeared in the literature on tolerance intervals and regions (e.g., Wilks (1941), Wald (1943), Tukey (1947Tukey ( , 1948, Ackermann (1983)). They dealt with both parametric and nonparametric cases and considered two types of tolerance regions (guaranteed coverage and mean coverage in the terminology of Aitchison and Dunsmore (1975) or ,B-content and ,B-expectation in the terminology of Guttman (1970)).…”

“…They dealt with both parametric and nonparametric cases and considered two types of tolerance regions (guaranteed coverage and mean coverage in the terminology of Aitchison and Dunsmore (1975) or ,B-content and ,B-expectation in the terminology of Guttman (1970)). Classical tolerance intervals introduced by Wilks (1941) are intervals with order statistics as endpoints. Since the classical procedure is based on order statistics it was troublesome to extend it to higher dimensions.…”