Statistics 550 Notes 18
Reading: Section 3.3-3.4
I. Finding Minimax Estimators (Section 3.3)
Review: Theorem 3.3.2 is a tool for finding minimax
estimators. It says that if an estimator is a Bayes estimator
and has maximum risk equal to its Bayes risk, th
Stat 550 Notes 20
Reading: Chapter 4.4-4.5, 4.7
I. Confidence Bounds, Intervals and Regions (Section 4.4)
A set estimate of is a set S ( X ) that is estimated to belong to
based on the sample X.
Example 1: For a sample X 1 , X 2 , X 3 , X 4 from a N ( ,1)
Stat 550 Notes 19
Reading: Chapter 4.3
I. Uniformly Most Powerful Tests
When the alternative hypothesis is composite, H1 : 1 , then
the power can be different for different alternatives. For each
particular alternative 1 , a test is the most powerful leve
Statistics 550 Notes 18
Reading: Sections 4.1-4.2
I. Hypothesis Testing Motivation and Framework
Motivating Example: A graphologist claims to be able to
distinguish the writing of a schizophrenic person from a
nonschizophrenic person. The graphologist is
Statistics 550 Notes 17
Reading: Section 3.3
For finding minimax estimators, there is not a constructive
approach like for finding Bayes estimators and it is often
difficult to find the minimax estimators. However, there
are some tools that allow us to fi
Statistics 550 Notes 16
Reading: Sections 3.2
I. Computation of Bayes procedures for complex
problems
For nonconjugate priors, the posterior mean (which is the
Bayes estimator under squared error loss) is not typically
available in closed form.
Example 1:
Statistics 550 Notes 15
Reading: Sections 3.1-3.2
In Chapter 3, we return to the theme of Section 1.3 which is
select among point estimators and decision procedures the
optimal estimator for the given sample size, rather than
select a procedure based on a
Statistics 550 Notes 14
Reading: Sections 2.4.3-2.4.4
I. Newtons Method
Newtons method is a numerical method for approximating
solutions to equations. The method produces a sequence of
values ( 0) , (1) ,K that, under ideal conditions, converges
to the ML
Statistics 550 Notes 13
Reading: Sections 2.4.2
The bisection method is a root finding algorithm which
works by repeatedly dividing an interval in half and then
selecting the subinterval in which the root exists.
The bisection method can be used to solve
Statistics 550 Notes 12
Reading: Section 2.3
I. Invariance Property of the MLE
Consider the model that the data X ~ p ( x | ), . For
an invertible mapping g ( ) , the model can be
reparameterized as X ~ p ( x | ), g () where
= g ( ) .
Theorem 1: The inva
Statistics 550/Biostatistics 622: Mathematical Statistics
Fall 2009
Professor: Dylan Small
E-mail: dsmall@wharton.upenn.edu
Office: 464 Huntsman Hall
Class hours: Tuesday and Thursday, 10:30-11:50.
Office hours: Tuesday, 2:45-4:45; by appointment.
Teachin
Statistics 550 Notes 1
Reading: Section 1.1.
I. Basic definitions and examples of models (Section 1.1.1)
Goal of statistics: Draw useful information from data.
Model based approach to statistics: Treat data as the
outcome of a random experiment that we mo
Statistics 550 Notes 11
Reading: Section 2.2.
Take-home midterm: I will e-mail it to you by Saturday,
October 14th. It will be due Wednesday, October 25th by 5
p.m.
I. Maximum Likelihood
The method of maximum likelihood is an approach for
estimating param
Statistics 550 Notes 1
Reading: Section 1.1.
I. Basic definitions and examples of models (Section 1.1.1)
Goal of statistics: Draw useful information from data.
Model based approach to statistics: Treat data as the
outcome of a random experiment that we mo
Take Home Midterm, Statistics 550, Fall 2006
This is a take home midterm exam and is due Wednesday, October 25th by 5 pm (put in
my mailbox in the statistics department or e-mail to me). You can consult any references
but cannot speak with anyone (except
Homework 5, Statistics 550, Fall 2006
This homework is due Friday, October 13th by 5 p.m. You can put in my mailbox in the
statistics department on the fourth floor of Huntsman Hall if you do not hand it in during
class on Thursday
1. Bickel and Doksum, P
# Simulation study of beta MLE vs. beta Method of Moments
sims=1000;
rhatmom=rep(0,sims);
shatmom=rep(0,sims);
rhatmle=rep(0,sims);
shatmle=rep(0,sims);
derivrfunc=function(r,s,xvec)cfw_
n=length(xvec);
sum(log(xvec)-n*digamma(r)+n*digamma(r+s);
derivs
Statistics 550 Notes 15
Reading: Section 3.3
For finding minimax estimators, there is not a constructive
approach like for finding Bayes estimators and it is often
difficult to find the minimax estimators. However, there
are some tools that allow us to fi
Statistics 550 Notes 14
Reading: Sections 3.2
I. Computation of Bayes procedures for complex
problems
For nonconjugate priors, the posterior mean (which is the
Bayes estimator under squared error loss) is not typically
available in closed form.
Example 1:
Stat 550 Notes 10
Reading: Chapter 3.4.2
We will give an outline of a proof of the asymptotic
optimality of the MLE:
Theorem: If n is the MLE and 0 is any other estimator,
then
%
ARE ( n , n ) 1 .
As a tool for proving this fact, we develop a lower bound
Stat 550 Notes 8
Notes:
1. We have no class this coming Tuesday because its fall
break.
2. The midterm is due at Wednesday by 5. Ill be around on
Monday and Tuesday if you have any questions about it. Ill
hold my usual office hours on Tuesday from 4:45-5:
Stat 550 Notes 11
Reading: Section 3.4.2
I. Asymptotic Relative Efficiency
Suppose that X 1 , K , X n ~ N ( ,1) . The MLE is n = X n , the
sample mean based on the n observations. Another reasonable
estimator of is the sample median 0 . The MLE satisfies
Stat 550 Notes 10
Reading Section 2.2
The likelihood function is defined by LX ( ) = p( X | ) .
The maximum likelihood estimator (the MLE), denoted by MLE
, is the value of that maximizes the likelihood:
MLE = arg max Lx ( ) , or equivalently the value of
Homework 10 (Last Homework Before Take Home Final),
Statistics 550, Fall 2009
This homework is due Friday, December 11th by 5 p.m. You can put in my mailbox in
the statistics department on the fourth floor of Huntsman Hall.
1. Bickel and Doksum, Problem 4
Homework 9, Statistics 550, Fall 2009
This homework is due Friday, December 4th by 5 p.m. You can put in my mailbox in
the statistics department on the fourth floor of Huntsman Hall if you do not hand it in
during class on Thursday
1. Bickel and Doksum, P
Homework 9, Statistics 550, Fall 2009
This homework is due Friday, November 20th by 5 p.m. You can put in my mailbox in
the statistics department on the fourth floor of Huntsman Hall if you do not hand it in
during class on Thursday
1. Bickel and Doksum,
Homework 7, Statistics 550, Fall 2008
This homework is due Friday, November 13th by 5 p.m. You can put in my mailbox in
the statistics department on the fourth floor of Huntsman Hall if you do not hand it in
during class on Thursday
1. Suppose X 1 ,K , X
Homework 6, Statistics 550, Fall 2009
This homework is due Friday, November 6th by 5 p.m. You can put in my mailbox in
the statistics department on the fourth floor of Huntsman Hall if you do not hand it in
during class on Thursday
1. Let X 1 ,K , X n be
Homework 5, Statistics 550, Fall 2009
This homework is due Friday, October 30th by 5 p.m. You can put in my mailbox in the
statistics department on the fourth floor of Huntsman Hall if you do not hand it in during
class on Thursday
1. Bickel and Doksum, 2
Homework 4, Statistics 550, Fall 2009
This homework is due Friday, October 9th by 5 p.m. You can put in my mailbox in the
statistics department on the fourth floor of Huntsman Hall if you do not hand it in during
class on Thursday
1. Bickel and Doksum, 1.