Momentprovided

Momentprovided is a term used in probability and statistics to denote a function or operator that returns the statistical moments of a random variable or stochastic process at a given time or state. The concept is used in time-dependent modeling to summarize distributional properties as they evolve.

Formally, for a stochastic process {X_t} with information set F_t, the k-th moment provided at time t can be defined as M_k(t) = E[(X_t - μ_t)^k | F_t], where μ_t = E[X_t | F_t] is the conditional mean. The non-centered k-th moment is E[X_t^k | F_t]. Unconditional moments are obtained by dropping the conditioning on F_t.

Computation and estimation: If the distribution is known, the moment is obtained by integration. In data settings, moments are estimated from samples: M_k_hat(t) ≈ (1/n) sum_{i=1}^n (X_t^{(i)} - μ̂_t)^k for centered moments, with μ̂_t the sample conditional mean; or M_k_hat(t) ≈ (1/n) sum_{i=1}^n (X_t^{(i)})^k for non-centered moments. For conditional moments, estimation may use stratification on covariates, regression residuals, or bootstrapped conditioning.

Applications: Momentprovided values are used in time-series analysis, risk assessment, model diagnostics, and feature engineering in machine learning. They help quantify how the distribution of X_t changes over time, beyond the mean.

Relation and notes: The term is related to standard moments and moment-generating concepts in probability theory. It is a descriptive tool rather than a standalone distributional model and is sensitive to sample size and dependence structure.