Valuation of primary diamond deposits by extreme value statistics

The valuation of diamondiferous kimberlites and lamproites depends on the stone size distribution, stone value distribution, and stone density distribution. Diamonds are uniformly distributed in a homogeneous kimberlite facies and simple Poisson distributions for the number of stones can be determined when samples are taken from it. The key variates in predicting the economic potential are the individual sizes (in carats) of diamonds and their associated values (in U.S.$). Usually, a few large stones take up a dominant proportion of the total value of a sample. The empirical tail nature and stochastic behavior of the extreme values of these variates is therefore critical and of crucial importance in any statistical analysis. They must be modeled with care to make any reasonably accurate prediction of the average contained value of the ore, expressed as U.S.$/metric ton (t). Size and value distributions are often regarded as lognormal and Sichel's t estimator can be used as an efficient estimator of the mean. The lognormal model is mainly applicable to alluvial and marine diamond deposits. In primary deposits, however, many cases have been noted where the size and value distributions of the diamonds display a significant deviation from lognormality. New generalized mixing or compounding distributions are needed for modeling mixing of many different diamond populations, sampled in the upper mantle by the ascending kimberlite magma. Although different models belonging to the compound lognormal family, like the loghyperbolic distribution, have been fitted to real data, a more general theoretical framework for analyzing such distributions has yet to be presented. This paper introduces the extreme value theory as a general framework for analyzing the extreme behavior of variables such as sizes and values which are of strategic importance in the tail region. Rather than using statistical techniques based on the central limit theorem, techniques are developed starting from extreme value limit laws. The stochastic behavior of the long tail of the empirical distribution is modeled using extreme value statistics. In conjunction with this theoretical framework, a practical approach is presented based on graphical diagnostic techniques such as quantile plots. A double bootstrap method is used to estimate mean stone size, value and carat price, and associated confidence in case studies where the loghyperbolic-type distributions apply.