ST 522-002: Statistical Theory II
Solution to Homework Assignment - 1
Prepared by Chen-Yen Lin
Spring, 2010
CB-6.2: Dene g(t|) = expcfw_n(n + 1)/2I(,) (t) and h(x) = expcfw_ n xi . Then it
i=1
follows that p(x|) = g(T (x)|)h(x) and hence by Factorization
ST 522-002: Statistical Theory II
Solution to Homework Assignment - 2
Prepared by Chen-Yen Lin & Na Cai
Spring, 2010
CB-6.10: To prove the minimal sucient statistic T (X) = (X(n) , X(1) ) is not complete, we
want to nd g[T (X)] such that Eg[T (X)] = 0 for
ST 522-002: Statistical Theory II
Solution to Homework Assignment - 4
Prepared by Chen-Yen Lin & Na Cai
Spring, 2010
CB-7.19: (a)
n
1
1
exp( 2 (yi xi )2 )
2
2
2
i=1
1 n 2
(y 2xi yi + 2 x2 )
= (2 2 )n/2 exp( 2
i
2 i=1 i
L(|y) =
= (2 2 )n/2 exp(
By Theorem
ST 522-002: Statistical Theory II
Solution to Homework Assignment - 3
Prepared by Chen-Yen Lin
Spring, 2010
CB-7.7: Since the likelihood function can only take two distinct values, L(1; x) = 2n
and L(0; x) = 1, the MLE of can be expressed as
=
1 if 2n
0 i
ST 522-002: Statistical Theory II
Solution to Homework Assignment - 5
Prepared by Chen-Yen Lin
Spring, 2010
CB-7.26: Since X1 , . . . , Xn conditional on are iid N (, 2 ) and prior distribution () =
e|/a /2a, the joint density of (X, ) is given by
(2 2 )n
ST 522-002: Statistical Theory II
Solution to Homework Assignment - 6
Prepared by Chen-Yen Lin & Na Cai
Spring, 2010
CB-7.62: (a)
For squared error loss, the risk function is
R(, ) = V ar (X) + (E (X) )2
= a2 V ar(X) + (aE(X) + b )2
= a2
2
+ (b (1 a)2 .
n
ST 522-002: Statistical Theory II
Solution to Homework Assignment - 7
Prepared by Chen-Yen Lin and Na Cai
Spring, 2010
1
1
1
CB-10.1: Since EX = 1 x 2 (1 + x)dx = 3 and EX 2 = 1 x2 1 (1 + x)dx = 1 . It follows
2
3
that = 3X is a consistent estimator of fo
ST 522-002: Statistical Theory II
Solution to Homework Assignment - 8
Prepared by Chen-Yen Lin & Na Cai
Spring, 2010
CB-8.2: Let X Poisson(), and we observed X=10. To assess if the accident rate has
dropped, we could calculate
10
P (X 10| = 15) =
e15 15i
ST 522-002: Statistical Theory II
Solution to Homework Assignment - 9
Prepared by Chen-Yen Lin and Na Cai
Spring, 2010
CB-8.10: (a) The posterior kernel of can be found as follows
(|x) =
f (x|)()
+
m(x|)
i
xi 1 (n+1/)
e
By observing the kernel, it can be
ST 522-002: Statistical Theory II
Solution to Homework Assignment - 10
Prepared by Chen-Yen Lin & Na Cai
Spring, 2010
CB-8.20: By the Neyman-Pearson Lemma, the UMP test rejects for large values of f (x|H1 )/f (x|H0 ).
Compute the ratio we can get
x
f (x|H
ST 522-002: Statistical Theory II
Solution to Homework Assignment - 11
Prepared by Chen-Yen Lin and Na Cai
Spring, 2010
CB-10.35: (a) Given that Xi are i.i.d random variables from N (, 2 ), the MLE and informa
tion of are = X and I() = n/ 2 . Set Wn = X,
ST 522-001: Statistical Theory II
Solution to Lab Exercises - 1
Prepared by Chen-Yen Lin
Jan. 19, 2011
5.13: Use the facts that
(n1)S 2
2
2 (n 1) and E
E
E
So, c =
2 (n 1) =
(n/2)
(n1)/2)
2, we have
(n 1)S 2
(n/2)
=
2
2
(n 1)/2)
n 1 (n 1)/2) 2
S
=
2
(n/
ST 522-001: Statistical Theory II
Solution to Lab Exercises - 2
Prepared by Chen-Yen Lin
Jan. 26, 2011
6.9: (a)
n
(2) 2 exp 1
f (x|)
2
=
n
f (y|)
(2) 2 exp 1 (
2
x2 2
i
2
i yi 2
xi + 2
1
= exp
2)
2
i yi +
i
i
2
yi
i
n
i=1
exp(
i
yi . By Theorem
i xi )
ST 522-001: Statistical Theory II
Solution to Lab Exercises - 3
Prepared by Chen-Yen Lin
Feb. 02, 2011
(6.10): From Example 6.2.15, T (X) = cfw_X(1) , X(n) is sucient for . To show T (X) is not
complete, it suces to nd a function g(T ) such that Eg(T ) =
ST 522-001: Statistical Theory II
Solution to Lab Exercises - 4
Prepared by Chen-Yen Lin
Feb. 09, 2011
6.24; (i) When = 0, random variable X degenerates to 0. It follows immediately that
this family is not complete. (ii) When = 1, to show this family is n
3T9; le.
I 6.13 , t=I x, y: =¢ X2 <3, I = e
I v vi #4 _(eV 3
I Y: r . '
Ngf _Q_°"
I = u e 9.
Av: .1.
I A 2: =D¢\| 2'). 2 0! u,_ a M
I - 3" 47" I
2;(3ro<)=o<ee -3?
I 31 dis"- . l
= P. 2 IS re; 7 M_
I e. Hm dis-bbwm .. J. 2 and a I mt ,,. of n
ST 522-001: Statistical Theory II
Solution to Lab Exercises - 5
Prepared by Chen-Yen Lin
Feb. 16, 2011
7.1:
1 if X = 0, 1
=
2 if X = 2,
3 if X = 2, 3, 4
7.6: (a) The joint density is given by
n
x2 n I(0,x(1) ] ()
i
f (x|) =
i=1
By Factorization Theorem,
Hw H gm 91] , 9.28. 9.41 , M}
8.20. B? NF «Loam, m; UMP ted rejwk for («77¢ values «79
+(XIH13/{IKIH9). Cowqu the ra'bo we 05423431.-
2.
l a t; S 6 7
43mm)
{(14140}
6 g 1' 3 l. I 944-
The. WHO is walvj ru X. 3: for
ST 522-002: Statistical Theory II
Lab 7 (March 19, 2014)
1. (Basic Exam 01/2005) Let Xn be a sequence of random variables with Xn having a 2 distribution
with n degrees of freedom.
(a) Using the CLT or MGF to show that
Xn n
1
lim P rcfw_
x =
n
2
2n
x
eu