What is the delta method and how is it used to estimate the standard error of a transformed parameter?
||Explanation of the delta method
Alan H. Feiveson, NASA
December 1999; minor revisions May 2005
The delta method, in its essence, expands a function of a random variable
about its mean, usually with a one-step Taylor approximation, and then takes
the variance. For example, if we want to approximate the variance of G(X)
where X is a random variable with mean mu and G() is differentiable, we can
G(X) = G(mu) + (X-mu)G'(mu) (approximately)
Var(G(X)) = Var(X)*[G'(mu)]2 (approximately)
where G'() = dG/dX. This is a good approximation only if X has a high
probability of being close enough to its mean (mu) so that the Taylor
approximation is still good.
This idea can easily be expanded to vector-valued functions of random vectors,
Var(G(X)) = G'(mu) Var(X) [G'(mu)]T
and that, in fact, is the basis for deriving the asymptotic variance of
maximum-likelihood estimators. In the above, X is a 1 x m
column vector; Var(X) is its m x m
variance–covariance matrix; G() is a vector function returning
a 1 x n column vector; and G'() is its n x m
matrix of first derivatives. T is the transpose operator.
Var(G(X)) is the resulting n x n
variance–covariance matrix of G(X).
Nicholas Cox of Durham University and John Gleason of Syracuse University
provided the references.
- Oehlert, G. W. 1992.
- A note on the delta method.
American Statistician 46: 27–29.
- Rice, John. 1994.
- Mathematical Statistics and Data Analysis. 2nd ed. Duxbury.