Semivarians

Semivarians are a family of statistical measures that generalize semivariance by quantifying asymmetric moments of a distribution relative to a chosen threshold. Whereas variance treats deviations above and below the mean symmetrically, semivarians focus on one side of the distribution and can be defined for different orders to capture magnitude and frequency of directional deviations.

Formally, for a random variable X, threshold t and order k ≥ 1, a semivarian of order k can be expressed as E[(t − X)^k_+], where (·)_+ denotes the positive part. The second-order case (k = 2) coincides with the common semivariance used to measure downside variability below t. Sample semivarians are estimated by averaging the kth powers of threshold exceedances in empirical data.

Semivarians are nonnegative and depend on the chosen threshold and order; they are not symmetric and therefore convey direction-specific information absent from ordinary variance. They are useful in finance for downside risk assessment, in quality control for identifying below-target variation, and in environmental studies for quantifying exceedances below critical thresholds.

Limitations include sensitivity to threshold selection and potential challenges in optimization contexts due to nonlinearity. Semivarians relate to variance, higher moments, and tail risk measures such as value-at-risk and expected shortfall, and are often used in complementary roles to provide a more nuanced picture of distributional asymmetry.