In statistics, the delta method is a method of deriving the asymptotic distribution of a random variable. It is applicable when the random variable being considered can be defined as a differentiable function of a random variable which is asymptoticallyGaussian.
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
where θ and σ2 are finite valued constants and denotes
convergence in distribution
, then
for any function g satisfying the property that its first derivative, evaluated at , exists and is non-zero valued.
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
which implies the variance of h(B) is approximately
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
or in univariate terms,
Example: the binomial proportion
Suppose Xn is binomial with parameters and n. Since
we can apply the Delta method with g(θ) = log(θ) to see
Hence, even though for any finite n, the variance of does not actually exist (since Xn can be zero), the asymptotic variance of does exist and is equal to
Note that since p>0, as , so with probability converging to one, is finite for large n.
Moreover, 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 has asymptotic variance equal to
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]
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, , so that the variance of relies on up to the 4th moment of .
The second-order delta method is also useful in conducting a more accurate approximation of 's distribution when sample size is small.
.
For example, when follows the standard normal distribution, can be approximated as the weighted sum of a standard normal and a chi-square with degree-of-freedom of 1.