delmomentet

Delmomentet, or the partial moment, is a statistical concept used to quantify how much a random variable deviates from a chosen threshold. Unlike traditional moments that summarize the whole distribution, delmomentet focuses on deviations on one side of a reference point, making it useful for assessing downside risk or concentration around a target.

Definition and notation: For a random variable X, a threshold t, and a nonnegative order r, the lower partial moment is LPM_r(t) = E[(t − X)+^r], and the upper partial moment is UPM_r(t) = E[(X − t)+^r], where (a)+ = max(a, 0). These quantities depend on the chosen threshold t and the order r. For r = 2, LPM_2(t) measures the dispersion of values below t, while UPM_2(t) measures dispersion above t.

Properties and interpretation: Delmomentet focuses on deviations relative to t, so it captures downside or upside risk separately. Higher r intensifies penalties for larger deviations. For symmetric distributions around t, LPM_r and UPM_r convey complementary information, but they are generally distinct from conventional central moments like variance or skewness.

Applications: In finance, partial moments are used to define downside risk measures and related performance metrics, such as the Sortino ratio, which relies on downside deviation derived from LPM_2. In actuarial science and reliability engineering, partial moments assess losses or failures relative to a critical threshold. They also appear in portfolio optimization and decision theory as objective functions that penalize adverse outcomes more heavily than favorable ones.

Etymology and usage: In Scandinavian languages, "delmomentet" literally means the partial moment. The term is commonly used in textbooks and articles on risk management, decision theory, and statistics to describe these threshold-relative moments. Example values for a discrete distribution can illustrate how LPM_r(t) and UPM_r(t) summarize tail behavior on each side of t.

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