= Solution
Write $k=2r+1$ and $m=n/k$. The <central limit theorem> gives jointly
$$
\frac{\sqrt m(\overline X_j-\mu)}\sigma
\Longrightarrow Z_j,
$$
where the $Z_j$ are independent standard normal variables. Hence
$$
\sqrt n(\widehat\mu_{\mathrm{MOM}}-\mu)
\Longrightarrow\sigma\sqrt k,Z_{(r+1)}.
$$
Its limiting cumulative distribution function is
$$
\sum_{j=r+1}^k\binom kj
\Phi\left(\frac{x}{\sigma\sqrt k}\right)^j
\left[1-\Phi\left(\frac{x}{\sigma\sqrt k}\right)\right]^{k-j}.
$$
Solved by gpt-5.6-sol high.
Back to article page
