Some aspects of a mixture model for estimating the boundary of a set of data

Abstract

This paper discusses a method for estimating the parameters of the boundary of a set of points for which the y variable is bounded above. The boundary may be linear, for example, in which case a scatterplot of the data may have a triangular appearance. An example is obtained by plotting cube root of food volume in an animal's incompletely filled stomach against the animal's length; full stomachs constitute the upper boundary, while other volumes fall between 0 and the boundary, depending on how full or empty a stomach may be.

Our method is to fit a mixture of an ordinary regression model and a variate having an unknown mixing distribution, which represents the variation of another unobserved factor such as the proportionate fullness of the stomach, for example. The mixing distribution is discretized and the number of mixing classes is estimated by Akaike's procedure. The method is found to be stable and reliable for simulated sets of data, and is illustrated by application to three data sets. Examples of occurrence of the problem in various areas of ecology, and in other areas, are given.