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@HaomingJiang 2017-09-06T16:32:39.000000Z 字数 3745 阅读 1152

ISYE 6412 HW #2

Name: Haoming Jiang


Problem #1

(a)

,
with the fact that ,

(b)

, is the density function of . So

(c)

Since , the derivative is positive if .

(d)

When . The risk is minimized when the derivative is 0. leads to . So

(e)

In the case . So . Which means

Problem #2

(a)




(b)




(c)

(I) are admissible
(II)



is Bayes.
(III)



and are Bayes.

(d)

Assume for some prior distribution, it is also Bayes. Assume that prior distribution is . Then,




And is the smallest, if and only if

Problem #3


.
For convenience, is denoted as
For a certain , if , , if ,

In order to minimize .
, when
, when
, when

Problem #4

(a)

This follows at once from our discussion in class that a procedure is Bayes relative to if and only if, for every ; it assigns a decision which minimizes (over )
or equivalently, to minimize

(b)

When , we have , which is minimized at

(c)

When , we have

, which is monotonically increasing and reaches 0 when . In orther words, is minimized when is the median of the posterior distribution.

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