A Note on the Frequency Polygon Based on the Weighted Sums of Binned Data

Abstract

We revisit the generalized midpoint frequency polygons of Scott (1985), and the edge frequency polygons of Jones et al. (1998) and Dong and Zheng (2001). Their estimators are linear interpolants of the appropriate values above the bin centers or edges, those values being weighted averages of the heights of r, r ∈ N, neighboring histogram bins. We propose a simple kernel evaluation method to generate weights for binned values. The proposed kernel method can provide near-optimal weights in the sense of minimizing asymptotic mean integrated square error. In addition, we prove that the discrete uniform weights minimize the variance of the generalized frequency polygon under some mild conditions. Analogous results are obtained for the generalized frequency polygon based on linearly prebinned data. Finally, we use two examples and a simulation study to compare the generalized midpoint and edge frequency polygons.

Keywords:

• Edge frequency polygon
• Kernel-based weights
• Midpoint frequency polygon
• Minimum variance weights
• Mixture weights
• Uniform weights

Mathematics Subject Classification:

• Primary 62G07
• Secondary 62G20