Biweight

Biweight is a term used in statistics to refer to a family of methods and kernels that downweight outliers in data analysis. In robust statistics, Tukey's biweight denotes two related estimators for location and scale, while the biweight kernel is used in kernel density estimation and nonparametric smoothing.

Biweight location estimator: This robust estimator reduces the influence of outliers on the estimated center. Given data x1, x2, ..., xn, an initial center m (often the median) and a robust scale s (such as MAD) are used. For each observation, compute u_i = (x_i − m)/(c s), where c is a tuning constant (commonly around 6 for location). If |u_i| < 1, assign a weight w_i = (1 − u_i^2)^2; otherwise w_i = 0. The biweight location is T = (Σ x_i w_i) / (Σ w_i). The weighting downweights observations far from the center, providing robustness to outliers.

Biweight midvariance (biweight scale) is a robust measure of dispersion built from the same weighting scheme, using the deviations from the biweight location and the weights to reduce the impact of outliers. Different formulations exist, but the core idea is to form a weighted sum of squared deviations with weights that shrink toward zero for distant points.

Biweight kernel: In kernel methods, the biweight kernel is K(u) = (15/16)(1 − u^2)^2 for |u| ≤ 1 and K(u) = 0 otherwise. It is a smooth, bounded kernel with compact support, used in kernel density estimation and nonparametric regression to obtain robust smoothing.

History and use: The biweight family traces to John Tukey’s work on robust statistics in the 1960s. It is valued for reducing sensitivity to outliers while retaining efficiency under normality, with tuning constants guiding the balance between robustness and efficiency.