Thresholdsmoments

Thresholdsmoments is a term used in statistics to describe a set of conditional moments of a random variable relative to a specified threshold. The concept arises in contexts where interest focuses on behavior when a variable crosses a boundary, such as extreme events or regime changes.

Formally, let X be a real-valued random variable and t a threshold. The upper-threshold k-th moment can be defined as m_k+(t) = E[X^k | X > t]. Another common variant is the threshold excess (or exceedance) moment e_k(t) = E[(X - t)^k | X > t]. Analogous lower-threshold moments m_k-(t) = E[X^k | X < t] may be used when attention is on values below the threshold. These are piecewise or conditional moments that describe the distribution in the region determined by the threshold.

Estimation from data uses sample analogs: compute sample means of X^k for observations above t; for exceedance moments, compute mean of (X_i - t)^k for X_i > t. Threshold selection affects bias and variance; common approaches include fixed thresholds, percentile-based thresholds (e.g., choose t as a high quantile), or data-driven methods. In finance and engineering, threshold moments are used to study tail behavior and regime-specific dynamics.

Relation to related concepts: threshold models (threshold autoregressive), tail risk measures like Value at Risk and Expected Shortfall are connected to threshold moments via threshold exceedances. They complement full unconditional moments by focusing on non-central regions of the distribution.

See also: conditional expectation, moments, tail risk, threshold model.