deltamethode

The deltamethode, often translated as the delta method, is a technique used in statistics to approximate the variance of a function of a random variable. It is particularly useful when dealing with complex functions for which an exact variance calculation is difficult or impossible. The method relies on a first-order Taylor series expansion of the function around the mean of the random variable.

The core idea of the deltamethode is to linearize a nonlinear function of a random variable. By approximating the function with its tangent line at the expected value of the random variable, the deltamethode essentially treats the function as linear in the vicinity of its mean. This linearization allows for the application of known formulas for the variance of linear combinations of random variables.

The formula for the deltamethode states that if Y = g(X), where X is a random variable with mean $\mu_X$ and variance $\sigma_X^2$, and g is a differentiable function, then the variance of Y can be approximated as Var(Y) $\approx$ [g'($\mu_X$)]^2 Var(X). Here, g'($\mu_X$) represents the derivative of g evaluated at the mean of X.

The deltamethode is widely applied in econometrics, biostatistics, and other fields where statistical inference is performed on parameters that are functions of other estimated parameters. For instance, it is commonly used to derive standard errors for estimated ratios, products, or other nonlinear combinations of regression coefficients. While it provides an approximation, the deltamethode is generally accurate when the function is sufficiently smooth and the variance of the random variable is small relative to its mean.