Kurtosiswith

Kurtosiswith is a term used in statistics to denote a family of measures that extend the conventional concept of kurtosis by incorporating additional weighting or conditioning. It is not a standard, universally defined statistic, but appears in some software documentation and methodological discussions as a flexible framework for capturing tail heaviness and peakedness in weighted or conditioned data.

A common formulation treats kurtosiswith as a weighted fourth central moment normalized by the square of the weighted variance. Let x1, x2, ..., xn be observations with weights w1, w2, ..., wn. The weighted mean is mu_w = sum(w_i x_i) / sum(w_i), and the weighted variance is sigma_w^2 = sum(w_i (x_i - mu_w)^2) / sum(w_i). The weighted fourth moment is m4_w = sum(w_i (x_i - mu_w)^4) / sum(w_i). Then kurtosiswith is defined as k_w = m4_w / (sigma_w^2)^2. When all weights are equal, this reduces to the ordinary kurtosis. Subtracting 3 yields the excess kurtosis, a common reference in practice.

Variants of kurtosiswith may use conditioning on a covariate, alternative weighting schemes, or robust weights to mitigate outlier influence. The interpretation follows standard kurtosis logic: higher values indicate fatter tails or sharper peaks relative to the weighted reference distribution, while lower values suggest lighter tails.

Notes and limitations: the value of kurtosiswith depends on the chosen weights or conditioning, so comparisons require consistent weighting schemes. In small samples, bias can be substantial; resampling methods like bootstrapping are often employed to assess uncertainty.

See also: kurtosis, weighted moments, excess kurtosis, Pearson's kurtosis.