zweighted

Zweighted is a weighting scheme used in statistics and data analysis in which each observation is assigned a weight determined by the observation's z-score, the standardized deviation from a central tendency. The z-score is computed as z_i = (x_i − μ)/σ, where μ and σ can be the sample mean and standard deviation or robust estimates of location and scale. In a zweighted approach, the weight w_i is a function of z_i, w_i = w(z_i).

Rationale and common forms: The method aims to modulate the influence of observations according to how far they lie from the center, reducing the impact of outliers or extreme values. Typical weight functions include Gaussian downweighting w(z) = exp(−λ z^2), reciprocal forms w(z) = 1/(1+|z|^p), or robust alternatives such as Tukey's biweight and Huber-type weights. The specific choice of w(z) reflects a trade-off between efficiency and robustness.

Applications: Zweighted weighting is used in robust regression, robust principal component analysis, and data preprocessing to stabilize estimates against outliers. It is commonly implemented in iterative procedures where μ and σ (or their robust equivalents) are themselves estimated from the data and updated as part of the analysis.

Notes: The approach assumes a sensible scale and center; its behavior depends on the distribution of z-scores and on the chosen weight function. As with other weighting schemes, incorrect specification can lead to biased results or loss of efficiency, particularly when data are not approximately normal. Zweighted is a descriptive label used in some modern statistical literature and software to denote z-score-based weighting schemes.