Spring 2016 Statistics 153 (Time Series) : Lecture Thirteen
Aditya Guntuboyina
01 March 2016
1
Last Class: Best Linear Prediction
Let Y and W1 , . . . , Wm represent mean zero random variables which have finite variances. Let cov(Y, Wi ) =
i for i = 1, .
Spring 2016 Statistics 153 (Time Series) : Lecture Three
Aditya Guntuboyina
26 January 2016
1
Last Class
We covered:
1. Sample autocorrelation r1 , r2 , . . . and how they can be used to detect departures from white noise.
2. Deterministic Trend models fo
Spring 2016 Statistics 153 (Time Series) : Lecture Twenty Three
Aditya Guntuboyina
14 April 2016
We started discussing process representation in the last class. We considered simple stationary models
of the form A cos(2t) + B sin(2t) and formed linear com
Spring 2015 Statistics 153 (Time Series) : Lecture Seven
Aditya Guntuboyina
09 February 2016
1
Infinite Order Moving Average
We can extend the definition of moving average processes to even infinite order by taking:
Xt = Z t + 1 Z t
1
+ 2 Z t
2
+ + q Zt
+
Spring 2013 Statistics 153 (Time Series) : Lecture Four
Aditya Guntuboyina
29 January 2015
1
Stationary Time Series
Definition 1.1 (Second-Order or Weakly or Wide-Sense Stationarity). A doubly infinite sequence of
random variables cfw_Xt is weak stationa
Stat 135 Fall 2015: A study guide for the final exam
Chapters covered (in whole, or part) from the text: 6, 7, 8, 9, 10, 11, 12, 13, 14
You will be provided with some basic information that I will post a couple of days before the exam.
Necessary tables
MATH 104-04: INTRODUCTION TO ANALYSIS
SOLUTION 1
(1) For two sets X and Y , show that X Y = X Y if and only if X = Y .
Proof. Suppose that X Y = X Y holds.
Then for any x X, x X Y = X Y Y , so X Y .
On the other hand, for any y Y , y X Y = X Y X, so Y X.
Stat 135 Fall 2013
FINAL EXAM
Name:
December 18, 2013
SID:
Person on right
Person on left
There will be one, double sided, handwritten, 8.5in x 11in page of notes allowed during the exam.
The exam is closed book and will be 2 hours and 50 minutes long.
Spring 2015 Statistics 153 (Time Series) : Lecture Eight
Aditya Guntuboyina
11 February 2016
1
Backshift Notation
A convenient piece of notation avoids the trouble of writing huge expressions in the sequel. Let B denote
the backshift operator defined by
B
Spring 2016 Statistics 153 (Time Series) : Lecture Twenty Two
Aditya Guntuboyina
12 April 2016
1
DFT Recap
Given data x0 , . . . , xn1 , their DFT is given by bj , j = 0, 1, . . . , n 1, where
n1
X
2ijt
xt exp
bj =
for j = 0, . . . , n 1.
n
t=0
P
Two imm
Spring 2016 Statistics 153 (Time Series) : Lecture Twenty Five
Aditya Guntuboyina
21 April 2016
1
Spectral Density
Given a stationary process cfw_Xt with autocovariance function X (h), its spectral density is defined as
f () :=
X
X (h) exp (2ih)
for 1/2
Spring 2016 Statistics 153 (Time Series) : Lecture Eleven
Aditya Guntuboyina
23 February 2016
1
Autocovariance and Autocorrelation Functions of ARMA Processes
Consider the ARMA(p, q) process: (B)Xt = (B)Zt with (z) = 11 z p z p and (z) = 1+1 z+
+ q z q .
Spring 2016 Statistics 153 (Time Series) : Lecture Twenty Six
Aditya Guntuboyina
26 April 2016
1
Nonparametric Estimation of the Spectral Density
Let cfw_Xt be a stationary process with
density that is given by
f () =
P
X
h=
|X (h)| < . We have then seen
Spring 2016 Statistics 153 (Time Series) : Lecture Ten
Aditya Guntuboyina
18 February 2016
1
Autocovariance and Autocorrelation Functions of ARMA Processes
Consider the ARMA(p, q) process: (B)Xt = (B)Zt with (z) = 1 1 z p z p and (z) =
1 + 1 z + + q z q .
Spring 2016 Statistics 153 (Time Series) : Lecture Twelve
Aditya Guntuboyina
25 February 2016
1
Data Analysis via ARMA Models
The ARMA models provide a reasonably versatile collection for modelling stationary time series data.
We shall now study how to ch
Nunez 7
~ . _ _, N ;.I:;2;WW'WWW
W
and no part of n stud-y, the Germ-gin
1 Gmtwl them were eh t3 ' h 1 ' in the US,
on .000 msututlons Of big er eammg I in the 400 munple animal was 95,799?
ommtnsion took a simple random sample of 400 of these. The av
Hank Ibser Statistics 2
. _ Study Guide For_the Midterm
The midterm Will cover the Follow1ng chapters with certain arts omitted:
Ch 1-2: Ideas in these chapters and terminology will be inc uded as
it relates to Ch 12, baSically section 1 oF each chapter.
Hank Ibser Statistics 2 Final Study Guide
See bspace for updates on office hours and instructions. The final
exam will be held on Tuesday Dec 15 from B-llam, in RSF Field House.
Please get to the final a little early as we will seat people by GSI.
The fin
Probability
Stat 2
Hank Ibser
Notation and Definitions
A = an event which may or may not occur (A)
Ac = the complement, or opposite, of event A (A complement)
P (A) = the probability that event A occurs (probability of A)
P (B|A) = the conditional probabi
Statistics 155: Game Theory Fall 2016 - Discussion 4
Definition 1. Let G = (X, Y, A) be a finite game, and let g be a one-to-one transformation
of Y onto itself. The game G is said to be invariant under g if there is a unique g, a
one-to-one transformatio
Statistics 155 Homework Assignment 5 (due Tuesday, October 18, 2016)
1. (Series and parallel games) Karlin and Peres, Exercise 3.1, p72.
(Note: As well as giving the value of the game, specify optimal strategies for the Troll and the Traveler.)
2. (Two-pl
Stat 155 Midterm Spring 2014
Name:
SID:
This exam has 5 problems and a total of 75 points. Attempt all questions and show your
working - solutions without explanation will not receive full credit. One double sided sheets
of notes are permitted. Answer que
Statistics 155: Game Theory Fall 2016 - Discussion 1
Definition 1. Recall from Stat 134 that the expectation of a random variable X is the
mean of the distribution of X, denoted E(X).
If X is a discrete random variable, then
X
E(X) =
x P (X = x).
x
If X
Stat 155 Midterm Practice Solutions
Problems:
Attempt all questions and show your working - solutions without explanation will not receive
full credit. One double sided sheets of notes are permitted.
Q 1 Find the value and optimal strategy of the followin
Statistics 155: Game Theory Fall 2016 - Discussion 2
Definition 1. A combinatorial game in which the legal moves in some positions, ot the sets
of winning positions, differ for the two players, are called partisan.
Example of partisan games: Chess, Hex, t
Midterm 1
Stat155
Game Theory
Lecture 11: Midterm 1 Review
This is an open book exam: you can use any printed or written material,
but you cannot use a laptop, tablet, or phone (or any device that can
communicate). There are three questions, each consisti
Stat 155 Midterm Spring 2014
Name:
SID:
This exam has 5 problems and a total of 75 points. Attempt all questions and show your
working - solutions without explanation will not receive full credit. One double sided sheets
of notes are permitted. Answer que