momentat

Momentat is a term used in statistics and data analysis to denote a parameter-dependent transformation of a distribution's moments. It is not a standard term with a single formal definition, but in common usage it refers to the family of moments obtained when the underlying variable is shifted by a real parameter t.

Formal definition: If X is a random variable with moments m_k = E[X^k] for k = 0,1,2,…, then for a real shift t the momentat of order k is M_k(t) = E[(X+t)^k]. Expanding yields M_k(t) = sum_{j=0}^k binom(k,j) m_j t^{k-j}. Thus the momentat vector (M_1(t),…,M_n(t)) is the binomial transform of the base moments, viewed as a function of t.

Properties: The functions M_k(t) are polynomials in t with coefficients given by the original moments. They describe how the distribution's shape changes under a location shift, and they connect to related concepts such as central moments through identities involving t = -μ.

Applications: Momentat is used in discussing sensitivity of moment-based estimators to baseline drift, in preparing moment-based models for data streams, and in change-detection workflows where higher-order moments provide signals of distributional change.

Status and notes: Momentat is not a canonical term in standard statistical references. When used, it is defined explicitly by the formula above to avoid ambiguity. It serves as a convenient descriptive label for the moment sequence as a function of a shift parameter t.