The anchor position of histograms and frequency polygons: quantitative and qualitative smoothing

Abstract

Histogram and frequency polygon estimators suffer from the drawback of the need to choose an anchor position for the estimator; different choices can lead to estimates with different appearances, particularly for small samples. In this paper, Monte Carlo simulations are used to investigate the effect of anchor position on both the integrated squared error accuracy and mode resolution of histogram and frequency polygon estimators. Although the effect can be made unimportant with respect to squared error (quantitative smoothing)), it has a stronger effect on mode resolution of the estimators (qualitative smoothing). Further, even estimators based on the best choice of anchor position are noticeably inferior with respect to qualitative smoothing than available kernel estimators. These results imply two important messages for data analysts interested in density estimation: (1) Effective or consistent performance with respect to functional norms such as squared error does not necessarily correspond to effective or consistent performance with respect to the exploratory goal of mode resolution. (2) Histograms and frequency polygons are apparently not very good at mode resolution, and smoother density estimators, such as kernel estimators, should be used instead.

Keywords:

• integrated squared error
• mode resolution
• Monte Carlo simulations