Springer Texts in Statistics
Advisors:
George Casella
Stephen Fienberg
Ingram Olkin
Springer Texts in Statistics
Alfred: Elements of Statistics for the Life and Social Sciences
Berger: An Introduction to Probability and Stochastic Processes
Bilodeau and B
Assignments for Stat 709:
First Exam Period:
Assigned Problems:
(Due on the first quiz day)
Chapter 1: 5,12,14,24,30,34,35,51,55,56,74,83,85,97
Suggested Problems:
Chapter 1: 2,3,6,23,25,31,36,46,50,53,57,61,65,66,70,73,78,79,81,82,86, 88,91,93,98
Second
Lecture 29: Information inequality
Theorem 3.3 (Cramr-Rao lower bound)
Let X = (X1 , ., Xn ) be a sample from P P = cfw_P : , where
is an open set in R k .
Suppose that T (X ) is an estimator with E [T (X )] = g ( ) being a
differentiable function of ; P
Lecture 25: Asymptotic bias, variance, and mse
Approximate and asymptotic bias
Unbiasedness as a criterion for point estimators is discussed in
2.3.2.
In some cases, however, there is no unbiased estimator.
Furthermore, having a slight" bias in some cases
Chapter 3: Unbiased Estimation
Lecture 26: UMVUE and the method of using a
sufcient and complete statistic
Unbiased estimation
Unbiased or asymptotically unbiased estimation plays an important
role in point estimation theory.
Unbiased estimators can be us
Lecture 24: Asymptotic approach and consistency
Asymptotic approach
In decision theory and inference, a key to the success of nding a
good decision rule or inference procedure is being able to nd
some moments and/or distributions of various statistics.
Th
Lecture 22: Sufciency and Rao-Blackwell theorem,
and two common approaches to derive decision rules
Sufciency
Suppose that we have a sufcient statistic T (X ) for P P .
Intuitively, our decision rule should be a function of T .
This is not true in general
Lecture 23: Statistical inference
Three components in statistical inference
Point estimators (Chapters 3-5)
Hypothesis tests (Chapter 6)
Condence sets (Chapter 7)
Point estimators
Let T (X ) be an estimator of R
Bias: bT (P ) = E [T (X )]
Mean squared er
Lecture 27: UMVUE and the method of conditioning
The 2nd method of deriving a UMVUE when a sufcient and
complete statistic is available
Find an unbiased estimator of , say U (X ).
Conditioning on a sufcient and complete statistic T (X ):
E [U (X )|T ] is
Chapter 2: Fundamentals of Statistics
Lecture 16: Populations, samples, models, and
statistics
Appilcation
One or a series of random experiments is performed.
Some data from the experiment(s) are collected.
Planning experiments and collecting data (not di
Lecture 21: Decision approach
Statistical decision theory: basic elements
X : a sample from a population P P
Decision: an action we take after observing X
A : the set of allowable actions
(A , FA ): the action space
X : the range of X
Decision rule: a mea
Lecture 20: Completeness
Motivation
A statistic V (X ) is ancillary if its distribution does not depend on the
population P
V (X ) is rst-order ancillary if E [V (X )] is independent of P .
A trivial ancillary statistic is the constant statistic V (X ) c
Lecture 19: Minimal sufciency
Maximal reduction without loss of information
There are many sufcient statistics for a given family P .
In fact, X (the whole data set) is sufcient.
If T is a sufcient statistic and T = (S ), where is measurable
and S is anot
Lecture 18: Sufcient statistics and factorization
theorem
Data reduction without loss of information
A statistic T (X ) provides a reduction of the -eld (X )
Does such a reduction results in any loss of information concerning the
unknown population?
If a
Lecture 17: Exponential families and location-scale
families
Two important types of parametric families in statistical applications:
Exponential families and location-scale families
Denition 2.2 (Exponential families)
A parametric family cfw_P : dominate
Lecture 30: U- and V-statistics and their variances
U-statistics
Let X1 , ., Xn be i.i.d. from an unknown population P in a
nonparametric family P .
If the vector of order statistic is sufcient and complete for P P , then
a symmetric unbiased estimator of
Lecture 32: Functions of unbiased estimators and
method of moments
Deriving asymptotically unbiased estimators
An exactly unbiased estimator may not exist, or is hard to obtained.
We often derive asymptotically unbiased estimators.
Functions of unbiased e
STAT 709: MATHEMATICAL STATISTICS I
Fall 2013
Instructor
Oce
Phone
Email
Oce hours
Professor Yazhen Wang
1175 MSC
262-6399
yzwang@stat.wisc.edu
By appointment
Lecture
Discussion
9:55-10:45am MWF (SMI 133)
4:35-5:50pm M (MSC CTR 5295)
Discussions taught by
Statistics 709, Exam 3
Instructor: Dr. Yazhen Wang, November 19, 2012
First Name:
Last Name.:
Be sure to show all relevant work !
P
Notations: stands for convergence in probability, denote by EQ the (conditional or uncondiLp
d
tional) expectation under pr
Statistics 709, Exam 2
Instructor: Dr. Yazhen Wang, October 24, 2012
First Name:
Last Name.:
Be sure to show all relevant work !
Lp
P
d
Notations: stands for convergence in probability, for convergence in Lp , for converW
gence in distribution, for weak c
Statistics 709, Exam 1
Instructor: Dr. Yazhen Wang, September 28, 2012
First Name:
Last Name.:
Be sure to show all relevant work !
1. Suppose that X is a random variable on probability space (, F , P ). Denote by B the Borel
sigma eld on real line R. Let
STAT 709 First Exam
8:25am-9:15am, Sept 28, 2010
Please show all your work for full credits.
1. Let X1 , ., Xk be random variables dened on a probability space.
(a) (6 points) Let X = (X1 , ., Xk ). Show that
(X ) =
k
j =1
( Xj ) .
(b) (2 points) Let Y
Statistics 709, Final
Instructor: Dr. Yazhen Wang, December 20, 2011
First Name:
Last Name.:
Be sure to show all relevant work !
Lp
P
d
Notations: stands for convergence in probability, for convergence in Lp , for converW
gence in distribution, for weak c
Statistics 709, Exam 3
Instructor: Dr. Yazhen Wang, November 23, 2011
First Name:
Last Name.:
Be sure to show all relevant formulas and work !
P
Notations: stands for convergence in probability, denote by EQ the (conditional or unconditional) expectation
Statistics 709, Exam 2
Instructor: Dr. Yazhen Wang, October 26, 2011
First Name:
Last Name.:
Be sure to show all relevant formulas and work !
Lp
P
d
Notations: stands for convergence in probability, for convergence in Lp , for converW
gence in distributio
Lecture 35: Robustness of LSEs
Consider model
X = Z + .
(1)
under assumption A3 (E ( ) = 0 and Var( ) is an unknown matrix).
An interesting question is under what conditions on Var( ) is the LSE
of l with l R (Z ) still the BLUE.
If l is still the BLUE, t
Lecture 36: Asymptotic properties of LSEs and
weighted LSEs
Theorem 3.11 (Consistency)
Consider model
X = Z +
(1)
under assumption A3 (E ( ) = 0 and Var( ) is an unknown matrix).
Consider the LSE l with l R (Z ) for every n.
Suppose that supn + [ Var( )]
Lecture 34: The UMVUE and BLUE in linear models
Theorem 3.7.
Consider model
X = Z +
(1)
with assumption A1 ( is distributed as Nn (0, 2 In ) with an unknown
2 > 0).
(i) The LSE l is the UMVUE of l for any estimable l .
(ii) The UMVUE of 2 is 2 = (n r )1
Lecture 31: The projection method
Since P is nonparametric, the exact distribution of any U-statistic is
hard to derive.
We study asymptotic distributions of U-statistics by using the method
of projection.
Denition 3.3
Let Tn be a given statistic based on
Lecture 33: Linear model, the LSE, and estimability
Linear Models
One of the most useful statistical models
X i = Z i + i ,
i = 1, ., n,
where Xi is the i th observation and is often called the i th response;
is a p -vector of unknown parameters (main pa