Math 541/643: Statistical Theory II, Fall 2011
Instructor: Songfeng (Andy) Zheng
Email: [email protected] Phone: 417-836-3037
Room and Time: Cheek 173, 9:40am 10:30am, MWF
Office and Hours: Cheek 22M, 2:00pm 3:30pm, Monday, Wednesday, and Fr
QUIZ 9
MATH
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502AB
(]v t.i) \ ~
problems .
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~
. of () for which nth e a random sample from
etX1,'"
Xb
ere exists an unbiased
~(xIO)
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= (X log()/(
ower Boun
Solution for weekly review exercises #5
Chen-Yen Lin
Feb. 09, 2011
(a) Let g (s, t) =
s
t
with gradient
g
s
g
t
g=
1
t
s
2t3
=
It is clear that the rst order partial derivative is continuous. By Delta Method, we
have
d
n(g (X, S 2 ) g (, 2 ) N (0, g T g
Solution for weekly review exercises #4
Chen-Yen Lin
Feb. 02, 2011
(a) First realize that F (Xi ) follow a U (0, 1) distribution, then for any > 0
P (|F (X(1) ) 0| > ) = P (F (X(1) ) > ) = [P (F (Xi ) > )]n = (1 )n 0
P (|F (X(n) ) 1| > ) = P (1 F (X(n) )
Solution for weekly review exercises #9
Prepared by Chen-Yen Lin
Mar. 23, 2011
(a) First argue that X(n) is complete sucient for .
X
(
n
i. Since n) Beta(n, 1), it follows that EX(n) = n+1 . Thus,
UMVUE of .
ii. Similarly, E
X(n)
iii. The reciprocal of
(b
Solution for weekly review exercises #14
Prepared by Chen-Yen Lin
Apr. 27, 2011
(a)
i. First derive the CDF of T
n
1c
P (T < t) = 1 P (X1 > c, . . . , Xn > c) = 1
1
, <t<1
Then the density of T is given by
fT (t) = n(1 )n (1 t)n1 , < t < 1
To show MLR pr
Solution for weekly review exercises #3
Chen-Yen Lin
Jan. 26, 2011
(a) Suppose Z N (0, 2 ), the expectation of |Z | is given by
0
z
E |Z | =
=
0
1
2 2
2
2 2
z2
2 2
e
z
dz +
0
1
2 2
z2
e 22 dz
z2
ze 22 dz
2 2
Using the above fact, the expectation of EY1 a
Solution for weekly review exercises #2
Chen-Yen Lin
Jan. 19, 2011
(a)
i
EE (X |)
E
2n
EV ar(X |) + V ar(EX |)
E 1 + V ar()
1 + 4n
EX =
=
=
V ar(X ) =
=
=
ii
MX (t) = E (etX ) = EEetX |
1
= E exp t + t2
2
2
t
1
= e2
et
n1 e 2 d
n
(n)2
0
n
2
t
1
1
, t<
= e
Solution for weekly review exercises #12
Prepared by Chen-Yen Lin
Apr. 13, 2011
(a)
i. Denote T = n=1 log Xi . First derive the MLE of , =
i
the likelihood ratio test rejects the null if and only if
n
n
i=1
log xi
n
0 ( i xi )0 1
<c
n ( i xi )1
n
n
(0 1)(
Solution for weekly review exercises #8
Prepared by Chen-Yen Lin
Mar. 16, 2011
(a) Since the likelihood function is monotone increasing in , the MLE of is X(1) . Next,
by equating X and 1+ , the moment estimator is give by = 2X 1. The bias and
2
variance
Solution for weekly review exercises #10
Prepared by Chen-Yen Lin
Mar. 30, 2011
(a)
i.
(p|x) f (x|p) (p) p1 (1 p) 1 pn (1 p)
n
i=1
xi
= p+n (1 p)
n
i=1
xi + 1
Thus, p|x has Beta(n + , n=1 xi + ) distribution.
i
ii. Using square error loss, the Bayes esti
The Laws of Large Numbers Compared
Tom Verhoeff
July 1993
1 Introduction
Probability Theory includes various theorems known as Laws of Large Numbers;
for instance, see [Fel68, Hea71, Ros89]. Usually two major categories are distinguished: Weak Laws versus
Solution for weekly review exercises #11
Prepared by Chen-Yen Lin
Apr. 06, 2011
(a) Skip
(b)
i. Under the null, the MLE of is given by = mincfw_0 , n1 n=1 Yi2 . Similarly,
i
under the whole parameter space, the MLE of is = n1 n=1 Yi2 . Then the
i
likeliho
ST 522-002: Weekly Review #11
Prepared by Chen-Yen Lin
Apr. 06, 2011
1. Concept Review:
Likelihood Ratio Test
Evaluating Tests
Neyman-Pearson Lemma
2. Exercises
(a) Exercise (10.10)
(b) Suppose Y1 , . . . , Yn forms a random sample from the pdf
f (y |
Solution for weekly review exercises #6
Chen-Yen Lin
Feb. 17, 2011
(5.44)(e)
n(S 2 p(1 p) =
n(S 2 Yn (1 Yn ) + Yn (1 Yn ) p(1 p)
n
=
n
Yn (1 Yn ) Yn (1 Y )n) + n[Yn (1 Yn ) p(1 p)]
n
=
Yn (1 Yn ) + n[Yn (1 Yn ) p(1 p)]
n1
p
n
Since Yn (1 Yn ) p(1 p) and n
Solution for weekly review exercises #7
Chen-Yen Lin
Feb. 23, 2011
(a)
i. First recall that EX1 = + , where is the Euler constant. Thus, by method
of moment, we solve
n
n
1
xi = + ,
M OM = X
i=
Secondly, the log-likelihood function can be written as
n
n
ST 522-002: Weekly Review #14
Prepared by Chen-Yen Lin
Apr. 27, 2011
1. Concept Review:
Interval estimators
UMP test and Karlin-Rubin Theorem (Revisit)
More about hypothesis tets
2. Exercises
iid
(a) Let X1 , . . . , Xn U (, 1) and denote T = mincfw_X1
ST 522-002: Weekly Review #13
Prepared by Chen-Yen Lin
Apr. 20, 2011
1. Concept Review:
Interval estimators
2. Exercises
(a) (Revisit) Let X1 , . . . , Xn be a random sample from U (, 1)
X
(1)
i. Show that 1 is a pivot quantity
ii. Derive a (1 )100% cond
ST522-002 Weekly Review #4
Chen-Yen Lin
Feb. 02, 2011
1. Concept Review:
Convergence
Delta Method
Slutskys Theorem
2. Exercises
(a) Let X1 , . . . , Xn denotes a random sample from common distribution F (x).
i. Show that F (X(n) ) converges in probabil
ST522-002 Weekly Review #3
Chen-Yen Lin
Jan. 26, 2011
1. Concept Review:
Convergence
Consistency of an estimator
2. Exercises
(a) Exercise 5.12: Let X1 , . . . , Xn be a random sample from a N(0,1) population.
Dene
n
n
1
1
Xi , Y2 =
|Xi |
Y1 =
n i=1
n i
ST522-002 Weekly Review #7
Chen-Yen Lin
Feb. 23, 2011
1. Concept Review:
Moment Estimator
Maximum Likelihood Estimator
Assessment of Estimators
2. Exercises
(a) Find the MOME and MLE of the location parameter in the following distributions
iid
i. Extre
ST 522-002: Weekly Review #8
Prepared by Chen-Yen Lin
Mar. 16, 2011
1. Concept Review:
Assessing Estimators
Cramr-Rao Lower Bound
e
Uniformly Minimal Variance Unbiased Estimator
2. Exercises
iid
(a) Let X1 , . . . , Xn U (, 1). Find the maximum likelih
ST 522-002: Weekly Review #10
Prepared by Chen-Yen Lin
Mar. 23, 2011
1. Concept Review:
Bayes Estimator
Asymptotic Evacuation
Likelihood Ratio Test
2. Exercises
(a) Let X1 , . . . , Xn be iid random variable from geometric distribution with success
pro