I'm trying to work on #1 from Chapter 15 of Greene's Econometric Analysis and I'm confused about how to use the Delta method. The problem statement is:
For the normal distribution $μ_{2k} = \frac{\sigma^{2k}(2k)!}{k!2^k}$ and $μ_{2k+1} = 0, k = 0, 1,\ldots$. Use this result to analyze the two estimators $b_1= \frac{m_3}{m_2^{3/2}}$ and $b_2=\frac{m_4}{m_2^2}$. where $$ m_k = \frac{1}{n} \sum_{i=1}^n(x_i-\bar{x})^k $$ The following result will be useful: $$ \text{Asy.Cov}[\sqrt{n}m_j,\sqrt{n}m_k]=\mu{j+k}−\mu_j\mu_k+ jk\mu_2\mu_{j−1}\mu_{k−1} − jmu_{j−1}\mu_{k+1} −kmu_{k−1}\mu_{j+1}. $$ Use the delta method to obtain the asymptotic variances and covariance of these two functions, assuming the data are drawn from a normal distribution with mean $\mu$ and variance $\sigma^2$. (Hint: Under the assumptions, the sample mean is a consistent estimator of μ, so for purposes of deriving asymptotic results, the difference between x and μ may be ignored. As such, no generality is lost by assuming the mean is zero, and proceeding from there.) Obtain V, the 3 × 3 covariance matrix for the three moments, then use the delta method to show that the covariance matrix for the two estimators is:
$$ \mathbf{JVJ'}= \begin{bmatrix} 6/n & 0 \\ 0 & 24/n \end{bmatrix} $$
My question is how to properly apply the Delta method. The statement from the book is: If $\mathbf{z_n}$ is a $K\times 1$ sequence of vector-valued random variables such that $\sqrt{n}(\mathbf{z_n}-\boldsymbol{\mu}) \xrightarrow{d} N(\mathbf{0},\boldsymbol\Sigma)$ and if $\mathbf{c}(\mathbf{z_n})$ is a set of $J$ continuous functions of $\mathbf{z_n}$ not involving $n$, then $$\sqrt{n}(\mathbf{c}(\mathbf{z_n})-\mathbf{c}(\boldsymbol{\mu}) \xrightarrow{d} N(\mathbf{0},\mathbf{C}(\boldsymbol\mu)\boldsymbol\Sigma\mathbf{C}(\boldsymbol\mu)') $$, where $\mathbf{C}(\boldsymbol\mu)$ is the $J\times K$ matrix of partial derivatives.
Here, are my $\mathbf{z_n}$ the moments $m_2$, $m_3$, and $m_4$?
