Shiftsmoments

Shiftsmoments is a term used in probability theory to describe the set of moments of a real-valued random variable after an additive shift. It focuses on how the distribution’s moments change when the variable is translated by a constant.

For a random variable X with ordinary moments μ_k = E[X^k] for integers k ≥ 0, the k-th shifted moment with shift a ∈ R is ν_k(a) = E[(X + a)^k]. The collection of shifted moments up to a chosen order n, ν(a) = (ν_0(a), ν_1(a), ..., ν_n(a)), captures the impact of the translation on the distribution’s moments.

The shifted moments can be computed from the original moments using the binomial theorem: ν_k(a) = sum_{j=0}^k binomial(k, j) a^{k-j} μ_j, with μ_0 defined as 1. This makes shifted moments polynomial functions of the shift parameter a, where each coefficient is determined by the original moments.

A simple example: if X has first two moments μ_1 and μ_2, then ν_1(a) = μ_1 + a and ν_2(a) = μ_2 + 2a μ_1 + a^2. These relations illustrate how translation affects both mean and dispersion, and how higher-order moments incorporate mixed terms of the shift and the original moments.

Applications of shiftsmoments include parameter estimation under affine transformations, data alignment and preprocessing in statistics, and moment-based distribution fitting where translations are common. The concept is often discussed alongside related ideas such as moment-generating functions and cumulants, which describe how moments and their relationships transform under shifts. See also: moments, moment-generating function, cumulants, translation in probability.