Abstract
We develop a clustering framework for observations from a population with a smooth probability distribution function and derive its asymptotic properties. A clustering criterion based on a linear combination of order statistics is proposed. The asymptotic behavior of the point at which the observations are split into two clusters is examined. The results obtained can then be utilized to construct an interval estimate of the point which splits the data and develop tests for bimodality and presence of clusters.
Highlights
We develop a general framework for univariate clustering based on the ideas in Hartigan (1978) for the case of observations from a population with smooth and invertible distribution function
Contrary to Hartigan’s approach, which was based on a quadratic function of the observed data, our clustering criterion function possesses the advantage of being a linear combination of order statistics—it is a combination of trimmed sums and sample quantiles
We deviate from the Hartigan’s framework and concentrate our attention on a function of the derivative of his split function. This approach permits us to obviate the existence of a finite fourth moment assumption imposed by Hartigan in the asymptotic investigation of his criterion function—a second moment assumption at the cost of an additional smoothness condition on our criterion function suffices
Summary
We develop a general framework for univariate clustering based on the ideas in Hartigan (1978) for the case of observations from a population with smooth and invertible distribution function. One important example is modeling in continuoustime mathematical finance, wherein observations are typically increments from a continuoustime stochastic process, and have smooth distributions because of presence of Ito integral components Keeping this in mind, we deviate from the Hartigan’s framework and concentrate our attention on a function of the derivative of his split function. Holzmann and Vollmer (2008) proposed a parametric test for bimodality based on the likelihood principle by using two-component mixtures Their method was applied to investigate the modal structure of the cross-sectional distribution of per-capita log GDP across EU regions.
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