Momentcan

Momentcan is a theoretical construct used in probability, statistics, and signal processing to describe a distribution through a compact descriptor of its moments. It refers to a finite tuple that collects the moments of a random variable into a single vector, enabling straightforward comparison and integration with downstream methods. In its common form, a momentcan of order four is the vector consisting of the mean, the standard deviation, the skewness, and the excess kurtosis of a variable X: M = (μ, σ, γ1, γ2), where μ = E[X], σ = sqrt(E[(X−μ)^2]), γ1 = E[(X−μ)^3]/σ^3, and γ2 = E[(X−μ)^4]/σ^4 − 3. More generally, one can define a momentcan M_k = (μ, σ, γ1, γ2, ..., γ_{k−2}) using standardized moments up to order k.

Momentcan captures distributional shape and scale while remaining invariant to location and linear rescaling for the standardized moments. This makes momentcan suitable for tasks such as distribution comparison, feature extraction in machine learning, quality control, and pattern analysis in time series.

Estimation is typically done from samples, via sample moments or robust alternatives; high-order estimates require large samples and are sensitive to outliers. Limitations include instability for heavy-tailed distributions, sensitivity to outliers, and potential information loss for complex distributions not well described by a small moment set. In practice, momentcan is most useful as a concise descriptor when accompanied by diagnostic checks and, where appropriate, supplementary features such as cumulants or moment invariants.