fourmoment

Fourmoment is a term used to denote the fourth central moment of a real-valued random variable. If X has mean μ and finite variance σ^2, the fourth central moment is μ4 = E[(X − μ)^4]. It measures the heaviness of tails and the peakedness of the distribution, providing information beyond the mean and variance. The quantity is closely related to kurtosis: the standardized fourth moment κ = μ4 / σ^4, and the excess kurtosis κe = κ − 3, with κ = 3 corresponding to the normal distribution.

In practice, μ4 and σ^2 are estimated from data via sample moments. The sample second moment is m2 = (1/n) Σ (xi − x̄)^2 and the sample fourth moment is m4 = (1/n) Σ (xi − x̄)^4. The sample kurtosis is often reported as κ̂ = m4 / m2^2, with excess kurtosis given by κ̂ − 3. These statistics are sensitive to outliers and sample size, and unbiased variants exist for small samples.

Interpretation and use: A larger μ4 relative to σ^4 indicates heavier tails or a sharper peak than the normal distribution. Kurtosis and related measures are used to assess normality, compare distributions, and inform models in fields such as finance, physics, and engineering. In finance, for example, higher kurtosis is associated with greater tail risk, influencing risk management and pricing of extreme events.

History and terminology: The concept originates in the method of moments in probability theory. While “fourmoment” is not a standard technical term, it is sometimes used informally to refer to the fourth central moment and its associated kurtosis measure.