STOR 556 - Homework 2 Solutions
THEORY PART
1.1
a)
E(Y c)2 = E(Y 2 ) 2cE(Y ) + c2
Take derivative with respect to c and set to 0,
2c 2E(Y ) = 0
c = E(Y ) =
b) Since we are given X, E[f (X)|X] = f (X). From the linearity of conditional
expectation, we ge

STOR 556
Statistical Methods II
Lecture 19
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
March 4, 2016
Review
Inference for the Process Mean (2.4)
Kai Zhang (UNC Chapel Hill)
March 4, 2016
2 / 15
Outline
Nonnegative D

STOR 556 Lecture 18 Supplementary Math
1
n
Asymptotic Variance of X
n] =
V ar[X
n
n
1 XX
Cov(Xi , Xj )
n2
i=1 j=1
=
n
ni
1 X X
Cov(Xi , Xi+h )
n2
i=1 h=1i
=
=
=
For large n, we have
becomes
n|h|
n
1
n2
1
n2
1
n
i=min(n,nh)
n1
X
X
X (h)
h=n+1 i=max(1h,1)
n

STOR 556
Statistical Methods II
Lecture 18
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
March 2, 2016
Announcement
Homework 6 is now posted and is due March 9th.
If you are still looking for team members, please send

STOR 556
Statistical Methods II
Lecture 17
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
February 29, 2016
Review
Transforming the Data (Not in Textbook)
Tests of Randomness (1.6)
Kai Zhang (UNC Chapel Hill)
February

STOR 556
Statistical Methods II
Lecture 16
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
February 26, 2016
Review
Classical Estimation of Both Trend and Seasonality (1.5.2)
Trend and Seasonality Elimination by Differe

STOR 556
Statistical Methods II
Lecture 15
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
February 24, 2016
Outline
Classical Estimation of Both Trend and Seasonality (1.5.2)
Trend and Seasonality Elimination by Differ

STOR 556
Statistical Methods II
Lecture 14
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
February 22, 2016
Thanks for your feedback!
Too many questions on concepts
I
I
I
I emphasize deep understanding of concepts.
Rea

STOR 556
Statistical Methods II
Lecture 13
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
February 19, 2016
Outline
Group Project
Kai Zhang (UNC Chapel Hill)
February 19, 2016
2 / 13
STOR 556 Group Project
Purpose: To

STOR 556
Statistical Methods II
Lecture 11
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
February 10, 2016
Review
Lake Huron Data
Smoothing with a Filter
Kai Zhang (UNC Chapel Hill)
February 10, 2016
2 / 44
Outline
Sm

STOR 556
Statistical Methods II
Lecture 10
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
February 8, 2016
Outline
Sample ACVF and ACF
Models with Trend: US Population Data
Kai Zhang (UNC Chapel Hill)
February 8, 2016

STOR 556
Statistical Methods II
Lecture 9
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
February 5, 2016
Announcements
For the computing part of HW4, you would need to include the plots
of simulated time series, the m

STOR 556 Lecture 19 Supplementary Math
1
Nonnegativeness of and the ACF for MA(1)
If | 21 , think about the ACF for MA(1). We solve the equation 1+
2 = and verify that this has a real root in
| < 1: therefore, the process is MA(1) with this . (To have rea

STOR 556
Statistical Methods II
Lecture 20
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
March 7, 2016
Bonus!
Suppose we are given that the process is MA(1) with 2 = 9, = 0.5, and
that xn = 0.51 based on n = 100 obser

STOR 556 Lecture 20 Supplementary Math
1
Invertibility of MA(1)
For a MA(1) process Xt = Zt + Zt1 with Zt W N (0, 2 ) and | < 1, we can write
Zt =
X
j Xtj
(1)
j=0
Proof: We write
Zt = Xt Zt1
= Xt Xt1 + 2 Zt2
.
.
= Xt Xt1 + 2 Xt2 . + ()M XtM + ()M +1 ZtM 1

STOR 556 Lecture 27 Supplementary Math
1
Causality of AR(p)
The general AR(p) process is defined by the equation (B)Xt = Zt , where
(z) = 1 1 z 2 z 2 . p z p
is a polynomial of order p. By the Fundamental Theorem of Algebra, has p roots, i.e. solutions 1

STOR 556
Statistical Methods II
Lecture 27
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
April 6, 2016
Announcement:
Check Your Final Exam Schedule
Any student needing to be excused from the regularly scheduled Final

STOR 556
Statistical Methods II
Lecture 26
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
April 1, 2016
Announcements
Homework 7 posted on Sakai and is due April 6th.
Group project due tomorrow. Make sure to check the

STOR 556 Lecture 25 Supplementary Math
1
The Optimality of Pt Xt+h
Suppose we are trying to predict a variable Y based on observations W1 , ., Wn , where E(Y ) = 0 , E(Wi ) = i .
we compute the matrix = (ij ) where ij = Cov(Wi , Wj ), and the vector = (i

STOR 556
Statistical Methods II
Lecture 25
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
March 30, 2016
Thanks again for your feedback!
More practice problems before examsMore extra credits
I
There will be an extra cr

STOR 556
Statistical Methods II
Lecture 24
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
March 28, 2016
Summary
The midterm consists 20% of your course grade.
Here are some summary statistics of your midterm:
Mean
15.

STOR 556
Statistical Methods II
Lecture 23
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
March 21, 2016
Announcements
Midterm on Wednesday March 23. Instructions posted on Sakai.
I will hold an additional office hour

STOR 556 Lecture 22 Supplementary Math
1
The Optimality of Pt Xt+h
Define
Pt Xt+1 =
X
j Xt+1j
(1)
j=1
and recursively for h > 1,
Pt Xt+h =
h1
X
j Pt Xt+hj
j=1
X
j Xt+hj .
(2)
j=h
Theorem. Pt Xt+h is the best linear predictor of Xt+h given Xs , s t, and

STOR 556
Statistical Methods II
Lecture 22
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
March 11, 2016
Announcements
Midterm 2 will be on Wednesday March 23, 2016. The class on
March 21 will be a review for the midte

STOR 556 Lecture 21 Supplementary Math
1
ACVF of ARMA(1,1)
The ACVF for Xt ARM A(1, 1) processes are
2
h=0
2 (1 + (+)
2 )
1
X (h) =
(+)2
h1 2
h1
+ + 12
Proof: By the equation (0.2),
X (h) =
X
2
j jh .
j=
For h = 0,
X
j jh =
j=
X
j2
j=0
= 1+
X
( + )2 2(

STOR 556
Statistical Methods II
Lecture 21
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
March 9, 2016
Alpha Go Beats a World Champion!
Go: A game of 4000 years history.
I
I
I
I
A day to learn the rules; a life to
mas

STOR 556
Statistical Methods II
Lecture 8
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
February 3, 2016
Announcements
For the final course grade, I will drop your assignments with the two
lowest grades.
If reading is

STOR 556
Statistical Methods II
Lecture 7
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
February 1, 2016
Review
Stationary Models, ACVF, and ACF
Kai Zhang (UNC Chapel Hill)
February 1, 2016
2 / 37
Stationarity
cfw_Xt

STOR 556
Statistical Methods II
Lecture 6
Kai Zhang
Department of Statistics and Operations Research
UNC Chapel Hill
STOR 556
January 29, 2016
Announcement
Homework 3 is posted and is due Feb 3. It serves as a review of
covariance calculations (since we h

STOR 556 Homework 5 (Due Mar. 3rd)
Theory Part:
1. The following variant of textbook problem 1.15(a): If Xt = 2 + t + st + Yt , where st is a seasonal component
with period 12, and Yt = Zt 0.5Zt1 with Zt W N (0, 4). Show that 12 Xt = (1 B)(1 B 12 )Xt is
s

STOR 556 Homework 4 (Due Feb. 10th)
Theory Part:
Textbook problems 1.6 and 1.14. For 1.14, you only need to show that the filter passes third-degree polynomials
without distortion.
Computing Part:
1. Simulate 200 observations from IID(0, 4). Set the rando

STOR 556 Homework 2 (Due Jan. 27th)
Theory Part:
Textbook problems 1.1.
Computing Part:
This problem requires using ITSM package. Having installed ITSM package and with the tutorial (Appendix D)
in hand, do the following. Produce time plots of the time se

STOR 556 Homework 1 (Due Jan. 20th)
Theory Part:
1. Let X be the outcome of throwing a fair dice. Find E[(X 2)2 ].
2. Let X be the outcome of throwing a fair dice. Find V ar[(X 2)2 ].
3. If 4 balls are randomly selected from an urn containing 9 balls of w