4.29 [Delta method (continued)]. In Example 4.4 we introduced the delta method for distributional approximations. The method can also be used for moment approximations. Let T1, . . . , Tk be random variables whose means and variances exist. Let g(t1, . . . , tk) be a differentiable function. Then, by the Taylor expansion, we can write g(T1, . . . , Tk) ≈ g(μ1, . . . , μk) +

k i=1

∂g

∂ti

(Ti − μi ), where μi = E(Ti ), 1 ≤ i ≤ k, and ∂g/∂ti is evaluated as (μ1, . . . , μk). This leads to the following approximations:

E{g(T1, . . . , Tk)} ≈ g(μ1, . . . , μk), var{g(T1, . . . , Tk)} ≈

k i=1



∂g

∂ti

2 var(Ti )

+2



i



∂g

∂ti



∂g

∂tj



cov(Ti, Tj ).

(i) Suppose that T ∼ Gamma(α, β) with the pdf given in Exercise 4.24(ii).

Use the above delta method to approximate the mean and variance of T

−1.

(ii) Note that the exact mean and variance of T

−1 can be obtained in this case, given a suitable range of α. What is the range of α so that E(T

−1)

exists? What is the range of α so that E(T

−2) exists?

(iii) Obtain the exact mean and variance of T −1 given the suitable range of α and compare the results with the above delta-method approximations.
How do the values of α and β affect the accuracy of the approximations?