 Delta method

In statistics, the delta method is a method for deriving an approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator. More broadly, the delta method may be considered a fairly general central limit theorem.
Contents
Univariate delta method
While the delta method generalizes easily to a multivariate setting, careful motivation of the technique is more easily demonstrated in univariate terms. Roughly, for some sequence of random variables X_{n} satisfying
where θ and σ^{2} are finite valued constants and denotes convergence in distribution, it is the case that
for any function g satisfying the property that g′(θ) exists and is nonzero valued. (The final restriction is really only needed for purposes of clarity in argument and application. Should the first derivative evaluate to zero at θ, then the delta method may be extended via use of a second or higher order Taylor series expansion.)
Proof in the univariate case
Demonstration of this result is fairly straightforward under the assumption that g′(θ) is continuous. To begin, we use the Mean value theorem:
where lies between X_{n} and θ. Note that since implies and since g′(θ) is continuous, applying the continuous mapping theorem yields
where denotes convergence in probability.
Rearranging the terms and multiplying by gives
Since
by assumption, it follows immediately from appeal to Slutsky's Theorem that
This concludes the proof.
Motivation of multivariate delta method
By definition, a consistent estimator B converges in probability to its true value β, and often a central limit theorem can be applied to obtain asymptotic normality:
where n is the number of observations and Σ is a (symmetric positive semidefinite) covariance matrix. Suppose we want to estimate the variance of a function h of the estimator B. Keeping only the first two terms of the Taylor series, and using vector notation for the gradient, we can estimate h(B) as
which implies the variance of h(B) is approximately
One can use the mean value theorem (for realvalued functions of many variables) to see that this does not rely on taking first order approximation.
The delta method therefore implies that
or in univariate terms,
Example
Suppose X_{n} is Binomial with parameters p and n. Since
we can apply the Delta method with g(θ) = log(θ) to see
Hence, the variance of is approximately
Moreoever, if and are estimates of different group rates from independent samples of sizes n and m respectively, then the logarithm of the estimated relative risk is approximately normally distributed with variance that can be estimated by . This is useful to construct a hypothesis test or to make a confidence interval for the relative risk.
Note
The delta method is often used in a form that is essentially identical to that above, but without the assumption that X_{n} or B is asymptotically normal. Often the only context is that the variance is "small". The results then just give approximations to the means and covariances of the transformed quantities. For example, the formulae presented in Klein (1953, p. 258) are:
where h_{r} is the rth element of h(B) and B_{i}is the ith element of B. The only difference is that Klein stated these as identities, whereas they are actually approximations.
See also
 Taylor expansions for the moments of functions of random variables
 Many special cases and examples of Delta Method
References
 Casella, G. and Berger, R. L. (2002), Statistical Inference, 2nd ed.
 Cramér, H. (1946), Mathematical Models of Statistics, p. 353.
 Davison, A. C. (2003), Statistical Models, pp. 3335.
 Greene, W. H. (2003), Econometric Analysis, 5th ed., pp. 913f.
 Klein, L. R. (1953), A Textbook of Econometrics, p. 258.
 Oehlert, G. W. (1992), A Note on the Delta Method, The American Statistician, Vol. 46, No. 1, p. 2729.
 Lecture notes
 More lecture notes
 Explanation from Stata software corporation
Categories: Econometrics
 Statistical approximations
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