Math Problems Vault

Problem 6.5.1

(a)(b)

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(a)

If a vector $\mathbf{Y}_{n}$ in $E_{s}$ converges in probability to a constant vector $\mathbf{a}$ , and if $h$ is a continuous function defined over $E_{s}$ , show that $h\left(\mathbf{Y}_{n}\right) \rightarrow h(\mathbf{a})$ in probability.

(b)

Use (a) to show that the elements of $\left\| A_{j k n}\right\|^{-1}$ tend in probability to the elements of $B$ as claimed in the proof of Lemma 5.2.

Problem 6.5.2

(a)(b)(c)

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(a)

Show that (5.26) with the remainder term neglected has the same form as (5.15) and identify the $A_{j k n}$ .

(b)

Show that the resulting $a_{j k}$ of Lemma 5.2 are the same as those of (5.23).

(c)

Show that the remainder term in (5.26) can be neglected in the proof of Theorem 5.3.

Problem 6.5.3

(a)(b)

Let $X_{1}, \ldots , X_{n}$ be iid according to $N\left(\xi , \sigma^{2}\right)$ .

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(a)

Show that the likelihood equations have a unique root.

(b)

Show directly (i.e., without recourse to Theorem 5.1) that the MLEs $\hat{\xi }$ and $\hat{\sigma }$ are asymptotically efficient.

Problem 6.5.4

(a)(b)

Let $\left(X_{0}, \ldots , X_{s}\right)$ have the multinomial distribution $M\left(p_{0}, \ldots , p_{s} ; n\right)$ .

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(a)

Show that the likelihood equations have a unique root.

(b)

Show directly that the MLEs $\hat{p}_{i}$ are asymptotically efficient.

Problem 6.5.5

Theorem 5.3 states: suppose that the assumptions of Theorem 5.1 hold (the standard multiparameter likelihood-regularity and consistency conditions for an iid sample from a regular parametric family with parameter $\theta =(\theta_1,\ldots ,\theta_s)$ ), and that $\tilde\theta_{jn}$ is a $\sqrt{n}$ -consistent estimator of $\theta_j$ for $j=1,\ldots ,s$ . Let $\left\{ \delta_{kn}, k=1,\ldots ,s\right\}$ be the solution of the equations

\sum _{k=1}^{s}\left(\delta _{k n}-\tilde{\theta }_{k n}\right) l_{j k}''\left(\tilde{\theta }_{n}\right)=-l_{j}'\left(\tilde{\theta }_{n}\right), \qquad j=1,\ldots ,s,

where $l(\theta )$ is the log-likelihood, $l_j' = \partial l/\partial \theta_j$ , and $l_{jk}'' = \partial^2 l /\partial \theta_j\partial \theta_k$ . Then, $\delta_n=(\delta_{1n},\ldots ,\delta_{sn})$ is asymptotically efficient.

Corollary 5.4 states: suppose that the assumptions of Theorem 5.3 hold and that the elements $I_{jk}(\theta )$ of the information matrix of the $X$ 's are continuous. Then, the solutions $\delta_{kn}'$ of the equations

n\left(\delta _{k n}'-\tilde{\theta }_{k n}\right) I_{j k}\left(\tilde{\theta }_{n}\right)=l_{j}'\left(\tilde{\theta }_{n}\right)

are asymptotically efficient.

Prove Corollary 5.4.

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Problem 6.5.6

Show that there exists a function $f$ of two variables for which the equations $\partial f(x, y) / \partial x=$ 0 and $\partial f(x, y) / \partial y=0$ have a unique solution, and this solution is a local but not a global maximum of $f$ .

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