The Delta Method gives a technique for doing this and is based on using a Taylor series approxi-mation. Tr(x) = (x a)k: k!

Example (Delta method for sample variance) For X i i.i.d. with Var(X i) = ˙2 and E[X4 i] <1, let S2 n:= 1 n Xn i=1 (X i X n)2 = 1 n Xn i=1 X2 i X 2 n: Then for ˚(x;y) = y x2 we have S2 n = ˚(X n;X2 n), and p n(S2 n ˙ 2)!Nd 0;E[X4] E[X2]2 dist = N 0;Var(X2): Delta method 2{7

Delta Method: Approximating Moments Delta Method: Approximating Distributions Consistency: A Stronger Definition Definition: qˆ n is a uniformly consistent estimator of q(θ), if for every E> 0, lim n→∞ sup[P θ(|q n(X 1,..., X n) − q(θ)| >E)] = 0. θ∈Θ. Example: Consider the sample mean qˆ. n = X. n. for which E [X. n | θ] = θ ...

Delta Method: Suppose that, p n(X n ) ˙!d N(0;1); and that gis a continuously di erentiable function such that g0( ) 6= 0. Then, p n(g(X n) g( )) ˙!d N(0;[g0( )]2): Proof: The basic idea is simply to use Taylor’s approximation. We know that, g(X n) ˇg( ) + g0( )(X n ); so that, p n(g(X n) g( )) ˙ ˇg0( ) p n(X n ) ˙!d N(0;[g0( )]2):

The Delta method is a theorem that can be used to derive the distribution of a function of an asymptotically normal variable. It is often used to derive standard errors and confidence intervals for functions of parameters whose estimators are asymptotically normal.

The Delta Method (DM) states that we can approximate the asymptotic behaviour of functions over a random variable, if the random variable is itself asymptotically normal. In practice, this theorem tells us that even if we do not know the expected value and variance of the function \(g(X)\) we can still approximate it reasonably.

The delta method allows a normal approx-imation (a normal central limit type or result, that is convergence in distribution to a normal distribution) for a continuous and differentiable function of a sequence of r.v.s that already has a normal limit in distribution. Example : Method of Moments for Exponential Distribution.

17.2 The delta method We would like to be able to quantify our uncertainty about g(^ ) using what we know about the uncertainty of ^ itself. When nis large, this may be done using a rst-order Taylor approximation of g, formalized as the delta method: Theorem 17.3 (Delta method). If a function g: R !R is di erentiable at 0 with g0( 0) 6= 0, and if p

Statisticians use a procedure commonly called the delta method to obtain an estimator of the variance when the estimator is not a simple sum of observations. The basic idea is to use a method from calculus called a Taylor series expansion to

The classical, well known delta method states the following: If n−−√ (Xn − θ) law N(0,σ2). Then the following holds: n−−√ (g(Xn) − g(θ)) law N(0,σ2(g′(θ))2) for any function g satisfying the property that g′(θ) exists and is non-zero valued.