Delta method
The Delta method is used to calculate the asymptotic variance of a random variable that is a function of another random variable. The derivative of the function and the mean and asymptotic variance of the second RV are used.
Let Zn be a sequence of random variables such that
√n(Zn−θ)(d)→n→∞N(0,σ2)
where σ2 is the asymptotic variance, and θ∈R. This means that Zn is asymptotically normal.
Given a function g:R→R that is continuously differentiable at θ,
- g(Zn)(P)→n→∞g(θ)
- (g(Zn))n≥1 is also asymptotically normal with asymptotic variance g′(θ)2σ2
- In other words,
√n(g(Zn)−g(θ))(d)→n→∞N(0,g′(θ)2σ2)