Optimal stratification and clustering on the line using the 1-norm

A random sample of continuous measurements can be partitioned into g groups or clusters by minimizing the within group dispersion as measured by the 1-norm. The central limit theory associated with such partitions which are universally optimal or locally optimal is derived. A procedure is presented for determining the number of groups represented by the data based on a plot of a sequence of asymptotic nonparametric confidence intervals for the fractional reduction of within group error due to (g + 1)-clustering over g-clustering for g = 1, 2,....