Math Problems Vault

Problem 2.6.1

For any random variables $\left(\psi_{1}, \ldots , \psi_{s}\right)$ , show that the matrices $\left\| E \psi_{i} \psi_{j}\right\|$ and $C= \left\| \operatorname {cov}\left(\psi_{i}, \psi_{j}\right)\right\|$ are positive semidefinite.

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Problem 2.6.2

In this problem, we establish some facts about eigenvalues and eigenvectors of square matrices. (For a more general treatment, see, for example, Marshall and Olkin 1979, Chapter 20.) We use the facts that a scalar $\lambda >0$ is an eigenvalue of the $n \times n$ symmetric matrix $A$ if there exists an $n \times 1$ vector $p$ , the corresponding eigenvector, satisfying $A p=\lambda p$ . If A is nonsingular, there are $n$ eigenvalues with corresponding linearly independent eigenvectors.

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(a)

Show that $A=P^{\prime } D_{\lambda } P$ , where $D_{\lambda }$ is a diagonal matrix of eigenvalues of $A$ and $P$ is and $n \times n$ matrix whose rows are the corresponding eigenvalues that satisfies $P^{\prime } P=P P^{\prime }=I$ , the identity matrix.

(b)

Show that $\max_{x} \frac{x^{\prime } A x}{x^{\prime } x}=$ largest eigenvalue of $A$ .

(c)

If $B$ is a nonsingular symmetric matrix with eigenvector-eigenvalue representation $B=Q^{\prime } D_{\beta } Q$ , then $\max_{x} \frac{x^{\prime } A x}{x^{\prime } B x}=$ largest eigenvalue of $A^{*}$ , where $A^{*}= D_{\beta }^{-1 / 2} Q A Q^{\prime } D_{\beta }^{-1 / 2}$ and $D_{\beta }^{-1 / 2}$ is a diagonal matrix whose elements are the reciprocals of the square roots of the eigenvalues of $B$ .

(d)

For any square matrices $C$ and $D$ , show that the eigenvalues of the matrix $C D$ are the same as the eigenvalues of the matrix $D C$ , and hence that $\max_{x} \frac{x^{\prime } A x}{x^{\prime } B x}=$ largest eigenvalue of $A B^{-1}$ .

(e)

If $A=a a^{\prime }$ , where $a$ is a $n \times 1$ vector ( $A$ is thus a rank-one matrix), then $\max_{x} \frac{x^{\prime } a a^{\prime } x}{x^{\prime } B x}= a^{\prime } B^{-1} a$ .

Problem 2.6.3

An alternate proof of Theorem 6.1 uses the method of Lagrange (or undetermined) multipliers. Show that, for fixed $\gamma$ , the maximum value of $a^{\prime } \gamma$ , subject to the constraint that $a^{\prime } C a=1$ , is obtained by the solutions to

\frac{\partial }{\partial a_{i}}\left\{ a^{\prime } \gamma -\frac{1}{2} \lambda \left[a^{\prime } C a-1\right]\right\} =0,

where $\lambda$ is the undetermined multiplier. (The solution is $a= \pm C^{-1} \gamma / \sqrt{\gamma^{\prime } C^{-1} \gamma }$ .)

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Problem 2.6.4

Prove (6.11) under the assumptions of the text.

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Problem 2.6.5

(a)(b)

Verify

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(a)

the information matrices of Table 6.1 and

(b)

Equations (6.15) and (6.16).

Problem 2.6.6

If $p(x)=(1-\varepsilon ) \phi (x-\xi )+(\varepsilon / \tau ) \phi [(x-\xi ) / \tau ]$ where $\phi$ is the standard normal density, find $I(\varepsilon , \xi , \tau )$ .

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Problem 2.6.7

Verify the expressions (6.20) and (6.21).

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Problem 2.6.8

Let $A=\left(\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right)$ be a partitioned matrix with $A_{22}$ square and nonsingular, and let

B=\left(\begin{array}{cc} I-A_{12} A_{22}^{-1} 0 & I \end{array}\right).

Show that $\left|A\right|=\left|A_{11}-A_{12} A_{22}^{-1} A_{21}\right| \cdot \left|A_{22}\right|$ .

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Problem 2.6.9

(a)(b)

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(a)

Let

A=\left(\begin{array}{ll} a & b^{\prime } \\ b & C \\ \end{array}\right)

where $a$ is a scalar and $b$ a column matrix, and suppose that $A$ is positive definite. Show that $\left|A\right| \leq a\left|C\right|$ with equality holding if and only if $b=0$ .

(b)

More generally, if the matrix $A$ of Problem 6.8 is positive definite, show that $\left|A\right| \leq \left|A_{11}\right| \cdot \left|A_{22}\right|$ with equality holding if and only if $A_{12}=0$ .

Problem 2.6.10

(a)(b)

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(a)

Show that if the matrix $A$ is nonsingular, then for any vector $x,\left(x^{\prime } A x\right)\left(x^{\prime } A^{-1} x\right)> \left(x^{\prime } x\right)^{2}$ .

(b)

Show that, in the notation of Theorem 6.6 and the following discussion,

\frac{\left[\frac{\partial }{\partial \theta _{i}} \mathbb {E}_{\theta }\left[\delta \right]\right]^{2}}{I_{i i}(\theta )}=\frac{\left(\varepsilon _{i}^{\prime } \alpha \right)^{2}}{\varepsilon _{i}^{\prime } I(\theta ) \varepsilon _{i}},

and if $\alpha =\left(0, \ldots , 0, \alpha_{i}, 0, \ldots 0\right), \alpha^{\prime } I(\theta )^{-1} \alpha =\left(\varepsilon_{i}^{\prime } \alpha \right)^{2} \varepsilon_{i}^{\prime } I(\theta )^{-1} \varepsilon_{i}$ , and hence establish (6.25).

Problem 2.6.11

Prove that (6.26) is necessary for equality in (6.25).

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Problem 2.6.12

Prove the Bhattacharyya inequality (6.29) and show that the condition of equality is as stated.

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