Momenttiteoreemalla

Momenttiteoreemalla is a theoretical concept in probability theory describing a formal link between the moments of a random variable and the underlying distribution within a parametric family, under mild regularity conditions. The term blends word roots for moment and theorem, underscoring its role as a result about moments rather than a specific distribution.

Informally, momenttiteoreemalla states that, if a random variable X has finite moments up to a certain order and the map from distribution parameters to the first several moments is injective, then the first few population moments uniquely determine the parameters of the distribution. In practice, this means that one can identify parameters by equating theoretical moments to their observed counterparts.

A standard implication is the method of moments: parameters θ are estimated by solving equations Eθ[X^j] = m̂_j for j = 1, ..., r, where m̂_j are sample moments. Under regularity conditions and by the law of large numbers, these estimators tend to the true parameter values as sample size increases. The theorem also implies identifiability: distinct parameter values should yield distinct moment vectors.

Applications of momenttiteoreemalla include parameter estimation in situations where likelihood methods are difficult to apply or when moments provide a robust alternative. It is commonly discussed in the context of normal, gamma, and Poisson families, as well as more general distributions where moments offer a convenient characterization. Limitations include potential non-identifiability, poor efficiency relative to maximum likelihood in small samples, and sensitivity to outliers when higher-order moments are used.

See also: method of moments, cumulants, moment generating function.