delmoment

Delmoment is a term used in probability theory and statistics to describe the instantaneous change in the moments of a random variable as a function of a continuous parameter, such as time or a model parameter. It is not a standard, widely adopted concept in mainstream texts, but has appeared in discussions and exploratory papers as a way to study sensitivity of distributional properties.

Formally, if X(θ) denotes a random variable that depends on a parameter θ, and μ_k(θ) = E[(X(θ))^k] is the k-th moment, then the delmoment of order k at θ is defined (when differentiability holds) as D_k(θ) = dμ_k(θ)/dθ. This quantity measures how the k-th moment reacts to small changes in θ. Variations include normalized forms that compare the moment to its baseline, such as derivatives of μ_k relative to μ_1^k, to assess proportional sensitivity.

Delmoment connects to related concepts like the moment generating function and cumulants through derivatives with respect to θ and t, and it can be related to the Jacobian of moment maps in multivariate settings. In practice, calculating delmoments often relies on differentiability assumptions and may employ numerical methods, including finite differences or smoothing techniques when moments are estimated from data.

Applications of delmoment ideas lie in sensitivity analysis, parametric forecasting, and risk assessment, where understanding how distributional features respond to parameter changes can improve model calibration and interpretation. As a developing notion, delmoment remains more a conceptual tool than a standardized metric in current literature.