Shortest Confidence Intervals of Weibull Modulus for Small Samples in Materials Reliability Analysis

Abstract

The Weibull distribution has been widely used to model strength properties of brittle materials. Estimation of confidence intervals for Weibull shape parameter has been an important concern, since small sample sizes in materials science experiments bring about large intervals. Many methods have been proposed in the literature for constructing shorter intervals; the methods of maximum likelihood, least square, and Menon are among the most extensively studied methods. However, they all use an equal-tails approach. The pivotal quantities used for constructing confidence intervals have right-skewed and unimodal distributions, thus, they clearly do not produce the shortest intervals for a given confidence level in equal tail form. This study constructs the shortest confidence intervals for the three aforementioned methods and compares their performances by their equal-tails counterparts. To this end, a comprehensive simulation study has been conducted for the shape parameter values between 1 to 80 and the sample sizes between 3 to 20. The comparison criterion is chosen as the expected interval length. The results show that the shortest confidence intervals in each of three methods have yielded considerably narrower intervals. Further, the unknown parameter values are more centered in these intervals.