Convex functions of several variables and on normed spaces

[5/0%]

LaTeX

Show LaTeX

Problem 1.7.13 from Lehmann E.L., Casella G. "Theory of Point Estimation" [LeCa98]More problems in Section 1.7 "Convex Loss Functions"

(a)(b)(c)

(a)

Show that $\phi (\mathbf{x})=e^{\Sigma x_{i}}$ is convex by showing that its Hessian matrix is positive semidefinite.

(b)

Show that the result of Problem 7.4 remains valid if $\phi$ is a convex function defined over an open convex set in $E_{k}$ .

(c)

Use (b) to obtain an alternative proof of the result of part (a).

Problem 1.7.17 from Lehmann E.L., Casella G. "Theory of Point Estimation" [LeCa98]More problems in Section 1.7 "Convex Loss Functions"

Use the convexity of the function $\phi$ of Problem 7.13 to show that the natural parameter space of the exponential family (5.2) is convex.

Problem 1.7.19 from Lehmann E.L., Casella G. "Theory of Point Estimation" [LeCa98]More problems in Section 1.7 "Convex Loss Functions"

Show that $\phi (x, y)=-\sqrt{x y}$ is convex over $x>0, y>0$ .

Problem 1.7.27 from Lehmann E.L., Casella G. "Theory of Point Estimation" [LeCa98]More problems in Section 1.7 "Convex Loss Functions"

Corollary 7.19 states: under the assumptions of Theorem 7.15 (let $\rho$ be a convex function defined on $(-\infty , \infty )$ and $X$ a random variable such that $\phi (a)=\mathbb {E}\left[\rho (X-a)\right]$ is finite for some $a$ ), suppose that $\rho$ is even and $X$ is symmetric about $\mu$ . Then, $\phi (a)$ attains its minimum at $a=\mu$ .

Generalize Corollary 7.19 to the case where $\mathbf{X}$ and $\boldsymbol {\mu }$ are vectors.

Problem C.4 from Pollard D. "A User's Guide to Measure Theoretic Probability" [Pol02]More problems in Appendix C "Convexity"

(i)(ii)

Let $P$ be a probability measure on $\mathbb {R}^{k}$ . Define $M(t) ; = P^{x}\left(e^{x \cdot t}\right)$ for $t \in \mathbb {R}^{k}$ .

(i)

Show that the set $C: = \left\{ t \in \mathbb {R}^{k}: M(t) < \infty \right\}$ is convex.

(ii)

Show that $\log M(t)$ is convex on rel-int $(C)$ .

References

[LeCa98] Lehmann E.L., Casella G. Theory of Point Estimation. 2nd ed., Springer, 1998.

[Pol02] Pollard D. A User's Guide to Measure Theoretic Probability. Cambridge University Press, 2002.