Math 5p85
Lecture 14
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
February 23, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Method of evaluating tests
Type I error: rejecting H0 when H0 is true
Type II error: acc
Math 5p85
Lecture 13
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
February 23, 2014
Wai Kong (John) Yuen [email protected]
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 20
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
March 17, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Asymptotic evaluations: IE
Use Thm 10.1.12 and Slutskys Theorem: approximate ML
interval is
Math 5p85
Lecture 10
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
February 6, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Methods of evaluating estimators.
Def 7.3.1
Mean squared error (MSE) of estimator W for p
Math 5p85
Lecture 11
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
February 6, 2014
Wai Kong (John) Yuen [email protected]
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 9
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
January 28, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Likelihood principle
Def 6.3.1
Given that X = x is observed, the function of dened by
L(|x
Math 5p85
Lecture 12
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
February 6, 2014
Wai Kong (John) Yuen [email protected]
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 8
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
January 24, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Sucient statistic and factorization theorem
Def 6.2.1
A statistic T (X) is a sucient stati
BROCK UNIVERSITY
MATH5P85 Mathematical Statistical Inference
Homework 1
Due Date: September 30, 2016
You must quote the theorem numbers from the textbook in your calculations to claim full credit.
1. Show that the beta family (p106) is an exponential fami
Math 5p85
Lecture 4
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 14, 2016
Wai Kong (John) Yuen [email protected]
Math 5p85
Principles of data reduction
Unknown parameter in a distribution
Obtain a sample X = (X1
Math 5p85
Lecture 5
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 21, 2016
Wai Kong (John) Yuen [email protected]
Math 5p85
Sufficient statistic and factorization theorem
Def 6.2.1
A statistic T (X) is a sufficie
Math 5p85
Lecture 3
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 14, 2016
Wai Kong (John) Yuen [email protected]
Math 5p85
Transformation for joint pdf
X = (X1 , .Xn ) r.v., joint pdf fX (x1 , ., xn )
pdf with s
Math 5p85
Lecture 1
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 8, 2016
Wai Kong (John) Yuen [email protected]
Math 5p85
Why Mathematical Statistics?
BREXIT.
Hong Kong Legislative Council election.
US election.
Math 5p85
Lecture 6
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
September 23, 2016
Wai Kong (John) Yuen [email protected]
Math 5p85
Why sufficiency?
For the discrete case: The focus is on
P(X = x|T (X) = T (x), which doe
Math 5p85
Lecture 19
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
March 17, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Bootstrap standard errors
A method to estimate the standard error of an estimator.
Example
Math 5p85
Lecture 17
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
March 10, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Asymptotic evaluations
How to evaluate estimators and tests based on statistics that
do not
Math 5p85
Lecture 7
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
January 24, 2014
Wai Kong (John) Yuen [email protected]
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
Math 5p85
Lecture 6
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
January 21, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
A transformation proof of Thm 5.4.4
In general, by the n-d transformation, the joint pdf
f
Math 5p85
Lecture 1
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
January 6, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Foundations
Def 1.2.4
Sample space S, -algebra B, a function P : B [0, 1] is a
probability
Math 5p85
Lecture 4
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
January 11, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Delta Method
CLT - useful to approximate standardized rvs.
What if we are interested in a
Math 5p85
Lecture 3
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
January 6, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Statistics
Def 5.1.1: Random sample from population f (x) if X1 , .Xn
are iid. (Key: Joint
Math 5p85
Lecture 5
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
January 6, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Order Statistics
Def 5.4.1: Order statistics of a random samples denoted by
X(1) , ., X(n)
Math 5p85
Lecture 2
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
January 6, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Exponential family
A family of distributions is called an exponential family if
k
f (x|) =
Math 5p85
Lecture 15
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
February 27, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Monotone likelihood ratio
Def 8.3.16
pdf/pmf of a univariate r.v. g (t|), real-valued pa
Math 5p85
Lecture 16
Wai Kong (John) Yuen
[email protected]
Department of Mathematics, Brock University
March 7, 2014
Wai Kong (John) Yuen [email protected]
Math 5p85
Pivotal quantities
Useful to choose our condence set based on r.v.s which is
independent of