Czekanowski’s index of overlap, its \(L_p\)-type extension, and bias reduction. (English) Zbl 1188.62147
Summary: J. Czekanowski’s proportional similarity index [Korrespondenzbl. Dtsche. Gesellschaft Anthropol. 40, 44–47 (1909); Anthrop. Anzeiger 9, 227–249 (1932); Current Anthropol. 3, 481–494 (1962)] 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 nonparametric kernel-density estimator is consistent. Based on simulation studies several authors have also noted that the estimator exhibits a larger than anticipated bias. 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 \(L_1\)-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.
MSC:
| 62G07 | Density estimation |
| 62G20 | Asymptotic properties of nonparametric inference |
| 65C60 | Computational problems in statistics (MSC2010) |
| 62P99 | Applications of statistics |
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