Delta method

In statistics, the delta method is a result concerning the approximate probability distribution for a function of an asymptotically normal statistical estimator from knowledge of the limiting variance of that estimator.

History

The delta method was derived from propagation of error, and the idea behind was known in the early 19th century.[1] Its statistical application can be traced as far back as 1928 by T. L. Kelley.[2] A formal description of the method was presented by J. L. Doob in 1935.[3] Robert Dorfman also described a version of it in 1938.[4]
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, if there is a sequence of random variables Xn satisfying

\( {{\sqrt {n}}[X_{n}-\theta ]\,{\xrightarrow {D}}\,{\mathcal {N}}(0,\sigma ^{2})}, \)

where θ and σ2 are finite valued constants and \( \xrightarrow{D} \) denotes convergence in distribution, then

\( {\displaystyle {{\sqrt {n}}[g(X_{n})-g(\theta )]\,{\xrightarrow {D}}\,{\mathcal {N}}(0,\sigma ^{2}\cdot [g'(\theta )]^{2})}} \)

for any function g satisfying the property that g′(θ) exists and is non-zero valued.
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 (i.e.: the first order approximation of a Taylor series using Taylor's theorem):

\( g(X_{n})=g(\theta )+g'({\tilde {\theta }})(X_{n}-\theta ), \)

where \( {\tilde {\theta }} \) lies between Xn and θ. Note that since \( X_{n}\,{\xrightarrow {P}}\,\theta \) and \( {\displaystyle |{\tilde {\theta }}-\theta |<|X_{n}-\theta |} \) , it must be that \( {\tilde {\theta }}\,{\xrightarrow {P}}\,\theta \) and since g′(θ) is continuous, applying the continuous mapping theorem yields

\( g'({\tilde {\theta }})\,{\xrightarrow {P}}\,g'(\theta ), \)

where \( {\xrightarrow {P}} \) denotes convergence in probability.

Rearranging the terms and multiplying by \( {\sqrt {n}} \) gives

\( {\sqrt {n}}[g(X_{n})-g(\theta )]=g'\left({\tilde {\theta }}\right){\sqrt {n}}[X_{n}-\theta ]. \)

Since

\( {{\sqrt {n}}[X_{n}-\theta ]{\xrightarrow {D}}{\mathcal {N}}(0,\sigma ^{2})} \)

by assumption, it follows immediately from appeal to Slutsky's theorem that

\( {{\sqrt {n}}[g(X_{n})-g(\theta )]{\xrightarrow {D}}{\mathcal {N}}(0,\sigma ^{2}[g'(\theta )]^{2})}. \)

This concludes the proof.
Proof with an explicit order of approximation

Alternatively, one can add one more step at the end, to obtain the order of approximation:

\( {\begin{aligned}{\sqrt {n}}[g(X_{n})-g(\theta )]&=g'\left({\tilde {\theta }}\right){\sqrt {n}}[X_{n}-\theta ]={\sqrt {n}}[X_{n}-\theta ]\left[g'({\tilde {\theta }})+g'(\theta )-g'(\theta )\right]\\&={\sqrt {n}}[X_{n}-\theta ]\left[g'(\theta )\right]+{\sqrt {n}}[X_{n}-\theta ]\left[g'({\tilde {\theta }})-g'(\theta )\right]\\&={\sqrt {n}}[X_{n}-\theta ]\left[g'(\theta )\right]+O_{p}(1)\cdot o_{p}(1)\\&={\sqrt {n}}[X_{n}-\theta ]\left[g'(\theta )\right]+o_{p}(1)\end{aligned}} \)

This suggests that the error in the approximation converges to 0 in probability.
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:

\( {\sqrt {n}}\left(B-\beta \right)\,{\xrightarrow {D}}\,N\left(0,\Sigma \right), \)

where n is the number of observations and Σ is a (symmetric positive semi-definite) covariance matrix. Suppose we want to estimate the variance of a scalar-valued 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

\( h(B)\approx h(\beta )+\nabla h(\beta )^{T}\cdot (B-\beta ) \)

which implies the variance of h(B) is approximately

\( {\displaystyle {\begin{aligned}\operatorname {Var} \left(h(B)\right)&\approx \operatorname {Var} \left(h(\beta )+\nabla h(\beta )^{T}\cdot (B-\beta )\right)\\&=\operatorname {Var} \left(h(\beta )+\nabla h(\beta )^{T}\cdot B-\nabla h(\beta )^{T}\cdot \beta \right)\\&=\operatorname {Var} \left(\nabla h(\beta )^{T}\cdot B\right)\\&=\nabla h(\beta )^{T}\cdot \operatorname {Cov} (B)\cdot \nabla h(\beta )\\&=\nabla h(\beta )^{T}\cdot (\Sigma )\cdot \nabla h(\beta )\end{aligned}}} \)

One can use the mean value theorem (for real-valued functions of many variables) to see that this does not rely on taking first order approximation.

The delta method therefore implies that

\( {\sqrt {n}}\left(h(B)-h(\beta )\right)\,{\xrightarrow {D}}\,N\left(0,\nabla h(\beta )^{T}\cdot \Sigma \cdot \nabla h(\beta )\right) \)

or in univariate terms,

\( {\sqrt {n}}\left(h(B)-h(\beta )\right)\,{\xrightarrow {D}}\,N\left(0,\sigma ^{2}\cdot \left(h^{\prime }(\beta )\right)^{2}\right). \)

Example: the binomial proportion

Suppose Xn is binomial with parameters \( p \in (0,1] \) and n. Since

\( {{\sqrt {n}}\left[{\frac {X_{n}}{n}}-p\right]\,{\xrightarrow {D}}\,N(0,p(1-p))}, \)

we can apply the Delta method with g(θ) = log(θ) to see

\( {{\sqrt {n}}\left[\log \left({\frac {X_{n}}{n}}\right)-\log(p)\right]\,{\xrightarrow {D}}\,N(0,p(1-p)[1/p]^{2})} \)

Hence, even though for any finite n, the variance of \( {\displaystyle \log \left({\frac {X_{n}}{n}}\right)} \) does not actually exist (since Xn can be zero), the asymptotic variance of \( \log \left({\frac {X_{n}}{n}}\right) \) does exist and is equal to

\( {\displaystyle {\frac {1-p}{np}}.} \)

Note that since \( {\displaystyle \Pr \left({\frac {X_{n}}{n}}>0\right)\rightarrow 1} \) as\(} n \rightarrow \infty \) , so with probability converging to one, \({\displaystyle \log \left({\frac {X_{n}}{n}}\right)} \) is finite for large n.

Moreover, if \( {\hat p} \) and \( \hat q \) are estimates of different group rates from independent samples of sizes n and m respectively, then the logarithm of the estimated relative risk \( {\frac {{\hat p}}{{\hat q}}} \) has asymptotic variance equal to

\( {\displaystyle {\frac {1-p}{p\,n}}+{\frac {1-q}{q\,m}}.} \)

This is useful to construct a hypothesis test or to make a confidence interval for the relative risk.
Alternative form

The delta method is often used in a form that is essentially identical to that above, but without the assumption that Xn 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:[5]

\( {\begin{aligned}\operatorname {Var}\left(h_{r}\right)=&\sum _{i}\left({\frac {\partial h_{r}}{\partial B_{i}}}\right)^{2}\operatorname {Var}\left(B_{i}\right)+\sum _{i}\sum _{{j\neq i}}\left({\frac {\partial h_{r}}{\partial B_{i}}}\right)\left({\frac {\partial h_{r}}{\partial B_{j}}}\right)\operatorname {Cov}\left(B_{i},B_{j}\right)\\\operatorname {Cov}\left(h_{r},h_{s}\right)=&\sum _{i}\left({\frac {\partial h_{r}}{\partial B_{i}}}\right)\left({\frac {\partial h_{s}}{\partial B_{i}}}\right)\operatorname {Var}\left(B_{i}\right)+\sum _{i}\sum _{{j\neq i}}\left({\frac {\partial h_{r}}{\partial B_{i}}}\right)\left({\frac {\partial h_{s}}{\partial B_{j}}}\right)\operatorname {Cov}\left(B_{i},B_{j}\right)\end{aligned}} \)

where hr is the rth element of h(B) and Bi is the ith element of B.
Second-order delta method

When g′(θ) = 0 the delta method cannot be applied. However, if g′′(θ) exists and is not zero, the second-order delta method can be applied. By the Taylor expansion,\( {\displaystyle {\sqrt {n}}[g(X_{n})-g(\theta )]={\frac {1}{2}}{\sqrt {n}}[X_{n}-\theta ]^{2}\left[g''(\theta )\right]+o_{p}(1)} \) , so that the variance of \( {\displaystyle g\left(X_{n}\right)} \) relies on up to the 4th moment of X n {\displaystyle X_{n}} X_{n}.

The second-order delta method is also useful in conducting a more accurate approximation of \( {\displaystyle g\left(X_{n}\right)} \) 's distribution when sample size is small. \( {\displaystyle {\sqrt {n}}[g(X_{n})-g(\theta )]={\sqrt {n}}[X_{n}-\theta ]g'(\theta )+{\frac {1}{2}}{\sqrt {n}}[X_{n}-\theta ]^{2}g''(\theta )+o_{p}(1)} \). For example, when \( X_{n} \) follows the standard normal distribution, \( {\displaystyle g\left(X_{n}\right)} \) can be approximated as the weighted sum of a standard normal and a chi-square with degree-of-freedom of 1.
See also

Taylor expansions for the moments of functions of random variables
Variance-stabilizing transformation

References

Portnoy, Stephen (2013). "Letter to the Editor". The American Statistician. 67 (3): 190–190. doi:10.1080/00031305.2013.820668.
Kelley, Truman L. (1928). Crossroads in the Mind of Man: A Study of Differentiable Mental Abilities. pp. 49–50. ISBN 978-1-4338-0048-1.
Doob, J. L. (1935). "The Limiting Distributions of Certain Statistics". The Annals of Mathematical Statistics. 6: 160–169. doi:10.1214/aoms/1177732594. JSTOR 2957546.
Ver Hoef, J. M. (2012). "Who invented the delta method?". The American Statistician. 66 (2): 124–127. doi:10.1080/00031305.2012.687494. JSTOR 23339471.

Klein, L. R. (1953). A Textbook of Econometrics. p. 258.