Math 5p85
Lecture 8
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 24, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Sucient statistic and factorization theorem
Def 6.2.1
A statistic T (X) is a sucient stati
Math 5p85
Lecture 12
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 6, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
LRT
Def 8.2.1
The LRT stat for testing H0 : 0 versus H1 : c is
0
(x) =
sup0 L(|x)
sup L(|
Math 5p85
Lecture 9
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 28, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Likelihood principle
Def 6.3.1
Given that X = x is observed, the function of dened by
L(|x
Math 5p85
Lecture 11
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 6, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Cramr-Rao inequality
e
Thm 7.3.9 (general version)
X has joint pdf f (x|). W (X) an estim
Math 5p85
Lecture 10
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 6, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Methods of evaluating estimators.
Def 7.3.1
Mean squared error (MSE) of estimator W for p
Math 5p85
Lecture 20
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 17, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Asymptotic evaluations: IE
Use Thm 10.1.12 and Slutskys Theorem: approximate ML
interval is
Math 5p85
Lecture 13
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 23, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
LRT and suciency
In Example 8.2.3, the LRT stat is a function of X(1) . So
observing the
Math 5p85
Lecture 14
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 23, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Method of evaluating tests
Type I error: rejecting H0 when H0 is true
Type II error: acc
Math 5p85
Lecture 19
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 17, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Bootstrap standard errors
A method to estimate the standard error of an estimator.
Example
Math 5p85
Lecture 17
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 10, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Asymptotic evaluations
How to evaluate estimators and tests based on statistics that
do not
Math 5p85
Lecture 18
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 17, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Asymptotic relative eciency (ARE)
Def 10.1.16
Suppose
and
2
n[Wn ()] n[0, W ]
2
n[Vn ()] n[
Math 5p85
Lecture 16
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
March 7, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Pivotal quantities
Useful to choose our condence set based on r.v.s which is
independent of
Math 5p85
Lecture 15
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
February 27, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Monotone likelihood ratio
Def 8.3.16
pdf/pmf of a univariate r.v. g (t|), real-valued pa
Math 5p85
Lecture 2
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 6, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Exponential family
A family of distributions is called an exponential family if
k
f (x|) =
Math 5p85
Lecture 5
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 6, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Order Statistics
Def 5.4.1: Order statistics of a random samples denoted by
X(1) , ., X(n)
Math 5p85
Lecture 3
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 6, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Statistics
Def 5.1.1: Random sample from population f (x) if X1 , .Xn
are iid. (Key: Joint
Math 5p85
Lecture 4
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 11, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Delta Method
CLT - useful to approximate standardized rvs.
What if we are interested in a
Math 5p85
Lecture 1
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 6, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Foundations
Def 1.2.4
Sample space S, -algebra B, a function P : B [0, 1] is a
probability
Math 5p85
Lecture 6
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 21, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
A transformation proof of Thm 5.4.4
In general, by the n-d transformation, the joint pdf
f
Math 5p85
Lecture 7
Wai Kong (John) Yuen
wyuen@brocku.ca
Department of Mathematics, Brock University
January 24, 2014
Wai Kong (John) Yuen wyuen@brocku.ca
Math 5p85
Thm 6.2.2
If p(x|) is the joint pdf of X and q(t|) is the pdf of T (X),
then T (X) is a su