Anyvaries

Anyvaries is a hypothetical construct used in statistics to describe a class of random variables whose variability depends on an index variable. It serves as a pedagogical tool to discuss how dispersion can change across conditions, without committing to a single mathematical specification. In a simple anyvaries model, a response Y_i is assumed to have a distribution whose variance is a function of a covariate x_i: Var(Y_i) = g(x_i), where g is nonnegative. More generally, not only the variance but other aspects of the distribution (skewness, tails) may vary with x, yielding a distribution that adapts to the covariate.

Examples: In econometrics, household income variance might rise with age or income bracket; in environmental science, measurement error variance may depend on sensor quality and time. In finance, time-varying volatility models resemble anyvaries in spirit when the index is time.

Relation to established models: anyvaries is conceptually related to heteroscedastic models, variance-function modeling, and location-scale or GAMLSS (generalized additive models for location, scale, and shape). It emphasizes modeling the dispersion as an explicit function of covariates, not just the mean.

Status: The term is not standard in the statistical literature and is mainly used in teaching or thought experiments to illustrate variable dispersion. Readers should refer to established variance-function or heteroscedastic modeling frameworks for rigorous methods.