The Delta Method.

Let vN be a sequence of q × 1 random vectors and suppose that

√

N(vN − θ) L

→ N

0, Σ(θ)

.

Suppose that F(·) is a differentiable function taking q vectors into r vectors.

Let dF be the r × q matrix of partial derivatives of F. Then

√

N

F(vN ) − F(θ)

L

→ N

0, dF Σ(θ) dF

.

For technical reasons, it is advantageous to assume that dF and Σ(θ) are also continuous. For mathematical details, see Bishop, Fienberg, and Holland (1975, Section 14.6.3).
Assuming the saturated model for a 2×2 table, use Theorem 10.2.1c and the delta method to find an asymptotic standard error and asymptotic confidence intervals for the odds ratio. Show that the intervals do not change if based on Theorem 10.3.1b. Apply this method to the data of Example 2.1.1 to get a 95% interval. How does this interval compare to the interval given at the end of the subsection on The Odds Ratio in Section 2.1?