Economics 520, Fall 2011
Lecture Note 10: Convergence and Limit Laws, CB 5.1-5.3, 5.5
1. Introduction: A Simulation Experiment
This note focuses on limiting behavior of sequences of random variables, and particularly on the A sequence of RVs
is just a col
Economics 520, Fall 2011
Lecture Note 11: Limit Laws for Random Vectors
(Based on Van der Vaart, Asymptotic Statistics, Cambridge University Press.)
Most of the results discussed in LN10 extend fairly easily to random vectors in Rk . As before, if
X is a
Economics 520, Fall 2011
Lecture Note 12: Point Estimation: Method of Moments and Maximum Likelihood Estimation
(CB 7.1, 7.2.1, 7.2.2)
1. Point Estimation Problem
In statistical inference, we consider a set of possible probability models, and try to use o
Economics 520, Fall 2011
Lecture Note 13: Bayesian Point Estimation (CB 7.2.3)
We have already discussed two general approaches to constructing point estimators of parameters, the method of moments and the maximum likelihood method. A third important clas
Economics 520, Fall 2011
Lecture Note 14: Bayesian Analysis Part 2
Previously, we considered an example where we observed X given the unknown mean , and
had a normal prior distribution. The posterior distribution turned out to be normal as well.
This was
Economics 520, Fall 2011
Lecture Note 15: Evaluating Estimators Part 1 (CB 7.3, CB 6.1-6.2)
Suppose you do a Bayesian analysis, and you feel comfortable with the subjectivist interpretation
that the distribution over parameters represents your beliefs abo
Economics 520, Fall 2011
Lecture Note 16: Cramr-Rao Bound
Another important result that helps us in the search for a minimum variance unbiased estimator
is the CramrRao bound:
Result 1 (C RAMRR AO BOUND )
Let X be a random variable with pdf/pmf f X (x ; )
Economics 520, Fall 2011
Lecture Note 17: Large Sample Properties of Maximum Likelihood Estimators
CB 10.1.1-10.1.3
Previously, we showed that if there is an Minimum Variance Unbiased Estimator with variance equal
to the CramerRao bound, then the MVUE is
Economics 520, Fall 2011
Lecture Note 18: Hypothesis Testing and Condence Intervals
CB 8.1, 8.3.1, 9.1, 9.2.1
Introductory Example
We now turn to a different type of statistical problem. Suppose we have some population of
individuals, and want to see whet
Economics 520, Fall 2011
Lecture Note 19: Hypothesis Testing via the Neyman-Pearson Lemma
CB 8.1, 8.3.1-8.3.3
Uniformly Most Powerful Tests and the Neyman-Pearson Lemma
Lets return to the hypothesis testing problem within the Neyman-Pearson framework. Rec
Economics 520, Fall 2011
Lecture Note 20: Large Sample Tests and Condence Intervals, CB 10.3, 10.4
Inference for the mean based on asymptotic normality
In the previous lecture we discussed an example where we had a random sample from a normal
distribution
Economics 520, Fall 2011
Midterm Review Questions
Note: I will not provide solutions to these questions. Most of these questions are drawn from previous
years exams.
1. Events M and N are said to be mutually exclusive provided that M N = .
Suppose that P
Economics 520: Introduction to R, Part 1
and Problem Set 1, Due Monday, 8/29/11
Keisuke Hirano
August 17, 2011
1 Installing R
The main web page for the R system is http:/www.r-project.org. From the main R page, you can
download a copy of R by clicking on
Economics 520: Introduction to R, Part 2
and R Exercises for HW2
Keisuke Hirano
August 28, 2011
1 Vectors
In R, a variable can hold a vector of values. An easy way to construct a vector is the c command,
which combines or concatenates numbers into a vecto
Economics 520: Introduction to R, Part 3
Keisuke Hirano
September 19, 2011
1 Random sampling with R
R has extensive facilities for working with random numbers and probability distributions. The
sample command draws samples from a given vector of possible
Final Review Questions Note: Many of these questions are drawn from previous years nals. I will not provide solutions to these questions. The nal will cover material from the entire semester, but with more weight on the second half of the course. 1. Suppo
Econ 520, Fall 2007 Midterm Review Questions Note: I will not provide solutions to these questions. Some of these questions are drawn from previous years exams. 1. Events M and N are said to be mutually exclusive provided that M N = . Suppose that P (A) >
Economics 520, Fall 2011
Lecture Note 9: Introduction to Stochastic Processes
These notes are based on S. Ross, Introduction to Probability Models, Academic Press, and J.
Hamilton, Time Series Analysis, Princeton University Press.
Denition 1 A stochastic
Economics 520, Fall 2011
Lecture Note 7: Joint Distributions, Conditional Distributions, and Independence of Random
Variables (CB 4.1-4.2, 4.6)
Denition 1 A N dimensional random vector is a function from the sample space to RN .
Thus a random vector can b
Economics 520, Fall 2011
Lecture Note 6: Special Distributions continued (CB 3.3-3.4)
1. Gamma Distribution The Gamma distribution with parameters > 0 and > 0 is
x
x 1 e
f X (x ) =
,
()
for x > 0 and 0 elsewhere. Recall that the gamma function is dened a
Econ 520, Final Review Questions
Note: Many of these questions are drawn from previous years nals. I will not provide solutions
to these questions. The nal will cover material from the entire semester, but with more weight
on the second half of the course
Economics 520, Fall 2011
Homework 2
Due Wednesday, September 7
Notes:
Due to Labor Day, this exercise is due on a Wednesday. Future homeworks will generally
be due on Mondays.
CB refers to exercises in Casella and Berger.
You may work in groups, but wr
Economics 520, Fall 2011
Homework 3
Due Monday, September 19
Exercises:
1. CB 1.55
2. Suppose X is a continuous random variable with pdf f (x ) and CDF F (x ). For a xed number x 0 such that F (x 0 ) < 1, dene
g (x ) =
f (x )/(1 F (x 0 ),
x x0
0,
x < x0
(
Economics 520, Fall 2011
Homework 4
Due Monday, September 26
Exercises:
1. Let X be a discrete random variable with P ( X = 1) = 1/8, P ( X = 0) = 6/8 and P ( X = 1) =
1/8. Calculate the bound on P (| X X | k X ) for k = 2 using Chebyshevs inequality.
Com
Economics 520, Fall 2011
Homework 5
Due Monday, October 3
Exercises:
1. CB 3.25
2. CB 3.26, part (a)
3. Show that each of the following distributions is an exponential family:
(a) Gamma distribution.
(b) Beta distribution.
(c) Chisquared distribution.
4.
Economics 520, Fall 2011
Homework 6
Due Monday, October 24
Exercises:
1. Let Y1 and Y2 be two random variables, where the marginal distribution of Y1 is N (1 , 2 )
1
and the conditional distribution of Y2 given Y1 = y 1 is N (+ y 1 , 2 ). Prove that the r
Economics 520, Fall 2011
Homework 7 [minor correction 10/25]
Due Monday, October 31
Exercises:
1. Consider the following sequence of random variables W1 , W2 , . . . with
Wn =
1
n
with probability .5
0
with probability .5
p
Using the denition of convergen
Economics 520, Fall 2011
Homework 8
Due Monday, November 7
Exercises:
1. Let X 1 , X 2 , . . . be a sequence of i.i.d. random variables with nite mean (not equal to zero)
and nite variance, and let X n =
1
n
n
i =1 X i .
Let Zn = 1/( X n ). Find the limit
Economics 520, Fall 2011
Homework 9
Due Monday, November 14
Exercises:
1. Let X 1 , X 2 , . . ., X N represent a random sample from each of the distributions given below.
In each case nd the maximum likelihood estimator of .
(a) f (x ; ) = x exp( )/x !, f