Czekanowski's Index of Overlap, its L_p-Type Extension, and BIAS Reduction

SYNOPTIC ABSTRACT

Czekanowski's proportional similarity index, which is the area of overlap of two density functions, has played a prominent role in anthropology, ecology, econometrics, engineering, health and other sciences. It has been shown that the corresponding non-parametric kernel-density estimator is consistent. Based on simulation studies several authors have also noted that the estimator exhibits a larger than anticipated bias. In the present paper we examine the situation in detail and learn that the bias is an integral part of the estimator. We then suggest an L_p-extension of Czekanowski's index; the latter being an L₁-type quantity. We show in particular that if p > 4/3, then under an appropriate choice of the bandwidth the kernel-density estimator of the L_p-extension loses its bias.

Key words and phrases:

• Index of overlap
• Czekanowski index
• Bray-Curtis index
• Renkonen index
• Weitzman index
• kernel-density estimation