Discussion

Suppose {X₁, X₂, ......., X_n} is a random sample from \(N(\mu, \sigma^{2})\) distribution. Then the approximate dispersion matrix of

\[ \sqrt{n}\big((\bar{X}_{n},S_{n}^{2})'-(\mu,\sigma^{2})'\big) \]

is :

A.\(\begin{pmatrix}\sigma^{2}&0\\0&\sigma^{4}\end{pmatrix}\)
B.\(\begin{pmatrix}\sigma^{2}&0\\0&2\sigma^{4}\end{pmatrix}\)
C.\(\begin{pmatrix}\frac{1}{\sigma^{2}}&0\\0&\frac{1}{\sigma^{4}}\end{pmatrix}\)
D.\(\begin{pmatrix}\frac{1}{\sigma^{2}}&0\\0&\frac{1}{2\sigma^{4}}\end{pmatrix}\)

Answer: B

The asymptotic dispersion matrix is diag(σ², 2σ⁴).

For a normal sample, \(\bar{X}_{n}\) and \(S_{n}^{2}\) are independent, so the off-diagonal terms are 0. Next,

\[ \begin{aligned} Var(\sqrt{n}\,\bar{X}_{n}) &= \sigma^{2} \\ Var(\sqrt{n}\,S_{n}^{2}) &\to 2\sigma^{4} \end{aligned} \]

the second because \((n - 1)S_{n}^{2}/\sigma^{2}\) is \(\chi^{2}_{n-1}\) with variance \(2(n - 1)\), giving \(Var(S_{n}^{2}) = 2\sigma^{4}/(n - 1)\).

\(\sigma^{4}\) misses the factor 2, and the matrices of reciprocals are information-type quantities, not the dispersion of the estimators.

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