Frequency Polygons and Average Shifted Histograms in Higher Dimensions

摘要

Scott has recently studied two simple variations on the ordinary histogram, namely the frequency polygon and the average shifted histogram, and found that they are able to compete with for example kernel density estimators in performance while retaining the advantage of being conceptually and computationally simple. The present paper proposes a way of generalizing frequency polygons to d-dimensional space that performs better than Scott's generalization. Expressions for integrated mean squared error and for integrated mean absolute deviation plus integrated absolute bias are obtained for generalized frequency polygons, for average shifted histograms, and for generalized frequency polygons of average shifted histograms. These expressions are used to give guidelines for window sizes. Keywords: Frequency polygons; Average shifted historgrams; Multidimensional; Integrated mean squared error; Integrated mean absolute deviation; Integrated absolute bias.