varExp

varExp is the name given to an exponential variance function commonly used in statistical modeling to describe heteroscedasticity (non-constant variance). It is most widely known from the nlme package in the R programming environment, where it is implemented as a varFunc-type object and can be supplied to generalized least squares (gls) and linear mixed-effects (lme) models through the weights argument.

The varExp formulation models the variance of residuals as a positive exponential function of a covariate or the fitted values. In its usual form, Var(e_i) = σ^2 * exp(2 δ v_i), where σ^2 is a scale parameter, δ is an estimated coefficient governing how rapidly variance changes, and v_i is a covariate (often the fitted value or a predictor). When δ = 0 the model reduces to homoscedastic errors.

Typical usage in R looks like weights = varExp(form = ~ covariate) or weights = varExp(form = ~ fitted(.)), which instructs the fitting algorithm to estimate δ jointly with the model parameters using maximum likelihood or restricted maximum likelihood. The exponential form ensures strictly positive variances and can accommodate increasing or decreasing variance patterns depending on the sign of δ.

varExp is related to other variance functions such as varPower (power-of-mean) and varIdent (heterogeneous variances by group). Choice among these depends on diagnostic evidence and the expected relationship between variance and predictors.