momentmatching

Momentmatching, in statistics and related fields, refers to the practice of selecting or adjusting a model so that its moments align with target or observed moments. Moments are expectations of powers of a variable (for example, the mean is the first moment, the variance relates to the second central moment, and higher moments capture skewness and kurtosis). The goal is to make the model reproduce key characteristics of the data or a reference distribution by matching these moments.

The standard approach, often called the method of moments, involves solving equations that set the model’s moments equal to the desired moments. If the model has parameters arranged so that p moments can be matched, one obtains p equations in p unknowns. In practice, the system may be overdetermined or infeasible, leading to optimization formulations that minimize discrepancies across chosen moments, typically in a least-squares sense. When moments are weakly informative or moments do not exist, momentmatching can be supplemented with regularization or alternative summaries.

Momentmatching is used in several contexts. In distribution approximation, a complex distribution may be approximated by a simpler one by matching a few moments (for example, approximating a fat-tailed distribution by a normal with the same mean and variance). In Bayesian inference, momentmatching appears in approximate methods that align moments of a tractable proxy posterior with those of a target distribution. In systems theory and model reduction, moment matching ensures that reduced models reproduce selected moments of a system’s transfer function, preserving behavior near specified expansion points.

Limitations include non-uniqueness, sensitivity to moment choice, and potential instability with higher moments or heavy-tailed data.