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ECO 2400F1
Second Half of Fall 2010
Problem Set 1
Suggested Solutions
1.
(a) We observe X1 , . . . , Xn , where for each i cfw_1, . . . , n, Xi = + i . 1 , . . . , n are independent N (.1, 2 ) random variables, where is known.
As such, the sample X = (X1
ECO 2400F1
Second Half of Fall 2010
Problem Set 4
Suggested Solutions
1.
(a)
n
i. Let T (x) x . If = 0, then T (X) N (0, 1). A level- test of H0 : 0
against H1 : > 0 would reject if T (x) z1 , where z1 is the (1 )-quantile
of the standard normal distribut
ECO 2400F1
Second Half of Fall 2010
Problem Set 3
Suggested Solutions
1.
(a)
i. Let h(Z) be an arbitrary function of Z. By iterated expectations,
E [h(Z)] = E [E [ h(Z)| Z]
= E [h(Z)E [ Y (Z)| Z]
= 0,
since E [ Y (Z)| Z] = (Z) (Z) = 0 with probability one
ECO 2400F1
Second Half of Fall 2010
Problem Set 2
Suggested Solutions
1.
(a)
i. When p = q = .1, we have the risk points given in Table 1 and plotted in Figure 1.
Table 1 and Figure 1 appear at the end of this handout.
ii. When p = 1 q = .1, we have the r
ECO 2400F1
Second Half of Fall 2010
Problem Set 6
Suggested Solutions
1.
(a)
i. The likelihood function is given by
cfw_
n
2
p (x, ) = (2)
n
1
2
(xi )
exp 2
2 i=1
,
cfw_(
)
which is maximized over 0 , 2 : 0 , > 0 at = 0 and 2 = 2
n
2
1
i=1 (xi x) .
n
ECO 2400F1
Proposed Solutions to the Final Examination
17 December 2010
1. Suppose X1 , . . . , Xn are iid N (, 1). Consider the problem of estimating
the population mean , where the quality of the estimate is measured with
respect to squared error loss.
ECO 2400F1
Second Half of Fall 2010
Problem Set 5
Suggested Solutions
1.
(a) Let 1 , which allows the Weibull likelihood to be written as
( n )c1
(
)
n
n n
1
c
p(x, ) = c
xi
exp
xi .
i=1
i=1
For 1 < 2 the LR statistic L (x, 1 , 2 ) has the form
p (x, 2 )