Interrelationship between statistical methods for estimating the size of the maximum inclusion in clean steels

Abstract

Two types of approach based on the statistics of extremes have been developed recently to estimate the sizes of large inclusions in clean steel. The first type (termed here “threshold” approaches) includes methods based on the Generalized Pareto distribution (GPD) and the Exponential distribution (EXPGPD). Both of these methods use measurements of the sizes of all inclusions larger than a certain threshold size in a sample. The second type of approach (termed here the “extreme values” type) includes the Statistics of Extreme Values (SEV) method and the Generalized Extreme Values (GEV) method. In these, only the size of the largest inclusion in each of a set of samples is measured. This paper compares the four methods and describes their inter-relationship. The distribution of large sizes depends on a shape parameter ξ. The influence of ξ on confidence intervals for the characteristic size of the maximum inclusion is studied by considering the shape of the likelihood function. The value of ξ is found to have a considerable effect on the precision of estimation. In the methods based on the GPD and GEV the value of ξ is not specified in advance. In such a case the GPD method gives more precise estimation of the characteristic size of the maximum inclusion than the GEV method. On the other hand in the EXPGPD and SEV methods ξ is assumed to be zero. In this case the EXPGPD method gives more precise estimation than the SEV method. The choice of a method of estimation of the characteristic size of the maximum inclusion is discussed in the light of these findings.