momentssuch

Momentssuch is a term used in some speculative discussions to denote a generalized family of moment-like quantities associated with a random variable X. It is not part of standard probability theory and has no formal consensus on its definition, but serves as a conceptual placeholder in thought experiments and informal analyses.

Definition and scope: In this informal sense, a momentssuch M_p(X) is defined as the expectation of a measurable transformation f_p of X, that is M_p(X) = E[f_p(X)], where the family {f_p} is chosen to emphasize particular distributional features such as tails, asymmetry, or higher-order structure. The simplest instantiation with f_p(x) = x^p yields the ordinary p-th moment (when p is a positive integer and the moment exists). Other choices, such as f_p(x) = sign(x)|x|^p or f_p(x) = e^{t x} for a fixed t, produce alternative, nonstandard summaries.

Relation to conventional moments: Momentssuch are intended to generalize the idea of moments but do not fix a unique normalization or transformation. Because the functions f_p are not standardized, different authors may propose different families, which limits cross-study comparability and theoretical development.

Applications and critique: In practice, momentssuch are mainly mentioned in informal discussions about distributional shape, tail behavior, or feature extraction in data analysis. They can, in principle, capture features that ordinary moments miss, but they lack a unified theory and may encounter issues of existence, interpretation, and computation.

See also: Moments, expected value, skewness and kurtosis, L-moments, cumulants.