On thresholds and environmental curve tensiometers

This paper considers distinctions between lognormal and mixture models. Emphasis is placed on two component mixtures where the lower valued subpopulation has a large mixing parameter. The density of this sort of mixture can be easily mistaken for a lognormal density. In order to compare such a mixture to a lognormal it is demonstrated that Galton's two parameter logmodel and Pearson's five parameternormal mixture are special, or limiting, cases of the same general mixture model. Consideration is given to the lognormal threshold parameter in order to devise a tool that can help distinguish mixtures from lognormals. Based on the threshold parameter, piloted procedures can help measure whether or not a curve is friable, in the sense that a brittle curve is better represented as a mixture than as a skewed lognormal. It is also shown that generalizations of Galton's product risk model can be represented interms of the threshold parameter Based on a tool called a curve tensiometer was designed to be applied as a graphical friability check in the ecological context of Fisher's classic Iris data and in the environmental context of a Santa Monica Bay fish consumption study.