(Quadratic Approximation) Confirm the quadratic approximation in (17.21) using the following steps.

(a) Write out a second-order Taylor series expansion for $E_\theta[L(\theta_B)]$ around $\theta = \theta_0$.

(b) Use the argument in Section 15.3.2 to show that the Hessian in the expansion converges in probability to the information matrix $I(\theta_0)$.

(c) Use a relationship such as (15.4) to show that $\sqrt{N}E_\theta[L(\theta_A)]$ converges in distribution.

(d) Finally, combine these results with $\sqrt{N}(\hat{\theta}_A - \theta_0) \xrightarrow{d} \mathcal{N}(0, V_A)$ and $\sqrt{N}(\hat{\theta}_B - \theta_0) \xrightarrow{d}$

$\mathcal{N}(0, V_B)$ to obtain (17.21).