STA3000 - Advanced Theory of Statistics
Announcements
May 10
Here are the solutions to the final exam.
I will email each student their marked exam and final grade in the course
shortly.
April 21
In question 4(f) the notation $ J ( x \rightarrow ||x||, D(x) ) $
is the Jacobian factor you need to go from the
density for x to the joint density for ||x||, D(x). I used a different
notation in the notes but this notation is quite common too.
April 20
Take-home exam is here. Please
write your solutions on the exam paper, scan the completed exam
and send to me by Friday, April 24 at midinight.
If there are ambiguities or questions just email me and I will post the answers here.
April 17 - Final Exam
The exam will be posted here around mignight Sunday, April 19 and will be
due midnight Friday, April 24.
April 14 - Returned Assignments
Send me an email and I'll scan in and send you the marked assignments.
April 12 - Solutions to Assignments 3 and 4 posted
All the solutions for Exercises 4 are now in Solutions under Exercises.
April 2 Typo in Assignment 4
There is a typo in #6(c) as the last sentence should read
"Show that G leaves the decision problem invariant."
So the word "function" is replaced by"problem".
March 31 Hints for #6
So for #6 answer the following questions:
- what is the group
- how does it act on the sample space
- what is a transformation variable and the maximal invariant and what
is their joint distribution (they are independent here so you really only
need the distribution of the transformation variable to get the Pitman
estimator as you don't need to condition on the maximal invariant)
- how does the group act on the parameters space
- what is the parameter of interest and is it equivariant, if so what is the induced action
- is the loss function invariant
- what does it mean for an estimator to be equivariant for this problem
- what is the Pitman estimate (namely what minimizes the relevant integral
- this follows from least squares)
March 31 - Assignment 4 and the material on invariance
- In response to some email I propose the following modifications.
For Assignment 4 you only need to hand in Exercise #6. This is a
fairly easy extension of the location model on page 138.
The only thing you need to add to this is to show that the
Pitman estimate is the sample mean and this is a simplification
of the location-scale example on page 146. The point of this
example is that we will have proved that the sample mean is
optimal unbiased and optimal invariant but not admnissible!
Something is wrong somewhere!
Exercise #4 is harder and if you have already done it and want to hand it in then that is fine and I will treat it as a bonus question.
- For the take-home exam I won't expect a deep understanding
of this material but hopefully at least some understanding
of what it means for a group to leave a decision problem
invariant. A hint here is to read the notes while imagining that
everything is finite, the sample space, the parameter space
and the group. Then all the computations (integrals) are just sums
so the main ideas are simpler to see, I understand this material
is hard to grasp for first timers and I won't be too ambitious with respect to
what I ask.
- The comprehensive exam is, I believe, delayed until August.
If so, that presents the possibility that I could lecture on this
materlal and, provided there is at least one student in attendance,
I will do this as soon as we are allowed to meet. I think a class
where questions can be asked is important for understanding this stuff.
Let me know if you will attend such a class.
- If you have any comments or suggestions let me know. Also if you
would like a Zoom meeting where I could answer questions I'm willing to
do that. The easiest solution for all would be to just forget this material
but I would rather try and do the best we can on it. But I certainly
don't want to add tons of stress either.
March 27 - Point of confusion caused by my notation
In the notes on p. 103 I defined a 1-parameter exponential family as taking the form
(1) f_theta (x) = exp{theta*T(x) -A(theta)}
but this family is also commonly written as
(2) f_theta (x) = gamma(theta)exp{theta*T(x)}
so gamma(theta)=exp{-A(theta)}.
Unfortunately in the proof of Theorem 5, I used the representation (2)
and didn't catch that I had used (1) in the definition. To add to the confusion
I also used gamma_1 and gamma_2 when determining the size of the test to
specify the randomization probabilities and these have no direct relationship
with gamma(theta).
March 25 - Final Exam Update
The instructors for STA2201, STA2211 and STA3000 have discussed the timing
of their final exams. These will be held in the first, second and third weeks
after the end of classes with some days in between two exams. Consult with the
other instructors concerning their specific dates. The STA3000
exam will begin on Monday, April 20 and is to be handed in (scanned and
emailed to me) on Friday, April 24. Let me know of any concerns.
March 23 - Final Exam
There has been a request that this be held off until at least April 17.
I will try to accommodate this but in part this depends on the School of
Graduate Studies. So for now consider the date to be TBA.
If this causes anyone problems please contact me. Also it would be good to know
when the exams are for the other courses being taken by the Ph.D. students
so we can schedule the STA3000 exam appropriately.
March 22
- I added question 6 to Exercises 4 and put up the solution to question 5.
So Assignment 4 is questions 4 and 6 and my suggestion is that you do #6
first as this is an easy extension of an example in the notes.
- I won't hold office hours tomorrow in person due to the current problems
but I will be available via a Zoom meeting if you wish. Just email me
and I will set it up.
March 20
For #2 in Exercises 4 you don't need closed form expressions. Rather, indcate
how you would compute the test and confidence region using R. You don't need
to write a program just indicate how you would go about designing such a program.
For the confidence region indicate how you would form a region based on a
grid of theta values equispaced in [0,1].
March 15
I have now put up all the notes on Invariance (earlier notes modified).
So the notes up encompass all the material that will be covered in the course.
I planned to do some more on inference but, given current circumstances,
it seems best to end on this topic.
The material on invariance is perhaps the most complicated topic
coverecd in the course, I recognize that this may make for some difficulties
in trying to learn this by yourself but it is an important topic and it
is necessary to cover the topic for a credible course on decision theory.
Given the difficulty of the material I will modify Assignment 4.
Exercise #4 will still be on but not #5 which will be replaced
by another question. For both questions there are examples in the notes
that can serve as templates for doing the questions. I think this will
make the Assignment easier,
March 15
- Given the disruption caused by the pandemic we have to change
how we conduct the course as there are no more classes.
There are two assignments up and you have the
notes for Assignment 3. I will put up the notes that allow you to complete
Assignment 4 in the next few days. The assignments can be handed in
by scanning and sending to me. The due dates are as posted (next point and note changes).
I will put up a take-home exam on April 6 which will be due a week later.
My office hours will be Mondays 12-3. I thnk it will be okay if
I handle at most two students at a time so if you want to meet please
email me and I will schedule some times. It is okay to just show up but you
may have to wait.
Let me know if you have any thoughts on this. Check here periodically
for any changes to this.
- Last week I got part way through the proof of Theorem 5 in the notes
labelled Unbiasedness II. Please read over the rest of the proof and
the example that follows. This will give you enough to complete Assignment 3.
As far as the rest of this section, read up to the statement of Theorem 8
(don't worry about the proof unless you want to!). This will give a pretty
good coverage of unbiased testing.
- Assignment 3 will now be just #1 and #2 from Exercises 4 and due March 30.
Assignment 4 is still #4 and #5 but due April 6.
- I would have lectured on Invariance I on March 16 so please read these
notes over. I will shortly add Invariance II and that will complete the course.
I would have also lectured on some topics
on inference but I've chosen not to, given the circumstances.
Office Hours
Immediately after class on Mondays 1-3 in SS6026D.
Syllabus
Roughly speaking I plan to cover the following topics with some things covered in greater depth than others.
- Decision theory - basic formulation of a decision problem, reduction
principles, minimax, Bayes, admissibility.
- Asymptotics
- Bayesian inference
Evaluation
There will be 3-4 assignments for 60% of the mark and a final worth 40%.
This will comprise 50% of the mark in the course.
References
I won't use a specific text but students may find the following references useful:
- J. Berger - Statistical Decision Theory - Springer
- M. Schervish - Theory of Statistics - Springer
- G. Casella and R. Berger - Statistical Inference - Duxbury
- C. Robert - The Bayesian Choice, Second Edition - Springer
- E. Lehmann - Theory of Point Estimation - Wiley
- E. Lehmann - Testing Statistical Hypotheses - Wiley
- J. Kadane - Principles of Uncertainty - CRC
Website
The course website is
http://www.utstat.utoronto.ca/mikevans/sta3000/sta3000.html
Class Notes
Please note that these are very rough notes.
Exercises