Texts: required Peter Bickel and Kjell Doksum, Mathematical
Statistics, vol.I, 2nd ed., Pearson Prentice Hall, 2007.
(recommended)
V. Rohatgi and A.K. Saleh, An Introduction to Probability and Statistics,
2nd ed., Wiley.
For all homework assignments listed as being due on a day when
we do not have class: you may hand the
hard-copy HW in to me
personally, or under my door or in my Math Department mailbox by 5pm
of the indicated
day; or if you will not be on campus, you may
email or fax the HW. If you send via email, to document timely
completion, please bring the hard-copy to hand in at the next class
session.
Problem Set 1. The first assignment
consists of 7 problems, due Friday, September 11.
In Bickel and Doksum
text, do: B.1.8 [(a)-(c) for (i),(iv), (v) only], pp.524-5;
and B.3.2, B.3.6,
B.3.12, B.3.15, pp. 529-531;
Also, in Rohatgi and
Saleh text, do: 7.4.16, pp. 333-334 and
7.7.1, pp. 351-352.
Note that in problem 7.7.1, the statistic R
is not the usual correlation coefficient.
Problem Set 2. The second assignment consists
of 7 problems, due either in class
Wednesday, September 23 or by 4:30pm Friday, 9/25.
(1). Show that if W is a MVN(0,B) random
k-vector, where B is a kxk nonnegative
definite
symmetric matrix such that B2 = B ,
then the scalar random variable W'W
is χr2 distributed,
where
r = rank(B).
(2). Show that if Y is a MVN(0,A)
-distributed random n-vector with A nonsingular,
and
R = (Y' A-1 Y)1/2, then
V = Y/R is independent of R, and find the
distributions of the
n-vector V and the random
variable R.
(3)-(7): Bickel and Doksum, #1.1.3, p.67; #1.1.6,
p.68; #1.1.8(a)-(b), pp. 68-69;
and #1.1.10, p.69; #1.2.3, p. 72.
Optional Problem. This problem is suggested by the
topic of singular-covariance multivariate normal
distributions
that we discussed in our last class (9/14/09). It is optional in the
sense that it counts
10 points extra but does not contribute
to the denominator of your HW score: so 70 points
from
the previous 7 problems count as a perfect score on this HW,
but 80 points are possible.
Suppose that a random 3-vector Y is
multivariate normal with mean vector μ = (1,2,3)
and
covariance matrix Σ
defined by symmetry and
&Sigma1,1 = &Sigma2,2 = 5,
&Sigma1,2 = -3, &Sigmaj,3 = 1
for j=1,2,3.
(a) Verify that the covariance matrix Σ is
singular, and find its rank and the hyperplane where all its
values lie.
(b) Find the probability that Y1 >0
and Y2 >0 and Y3 >0.
(The event of interest is the 3-way intersection.)
You should find the answer as a bivariate probability
integral, but you may check your answer by simulation.
Problem Set 3. The third assignment consists of 7 problems, due in class Wednesday, October 7.
(1) Suppose that X1, ..., X10
are random variables which are conditionally iid
Exponential(θ)
(i.e. with density θ
e-x θ for x>0) given the
hazard-parameter θ , where θ
has prior probability
mass function π(θ)
with π(1) = π(2) = 0.5.
(a). Find the (unconditional) correlation of X1 and X2.
(b). Find the posterior probability mass function of θ if X1+...+X10 = 13.
(c). Find the posterior predictive density of X11
given X1,...,X10 if
X1+...+X10 = 13.
(2)-(7): Bickel and Doksum, #1.2.12, 1.2.14, p.74; 1.3.3, 1.3.11, 1.3.19, pp. 75-80; 1.4.18, pp.82-83.
Problem Set 4. The fourth assignment consists of 7 problems, due in class Wednesday, October 21.
(1) Suppose that you observe data X1,
X2,..., Xn known to be
iid ~ N(θ, 1), with
parameter space Θ (b). Find the Bayes-optimal decision rule in the decision
problem stated above, with prior density Hint. The proof in (b) involves first obtaining the
posterior as in #1.2.14 (from HW3). Then the rπ (2) Do the following problems from Bickel and Doksum:
Sec.4 (pp. 82-84): #14(a),(c),(e),(f), #20.
Problem Set 5. The fifth assignment
consists of 8 problems, due Friday, November 13, by 5pm. Problem Set 6. The sixth assignment
consists of 6 problems, due Wednesday, November 25 IN CLASS.
equal to the real line, action
space A equal to the set of intervals (a,b]
with a ≤ b , and loss function
L(&theta, (a,b]) =
(b-a)/10 + I( θ ≤ a or θ > b).
The objective of this problem is to show that the usual two-sided
confidence interval based on the average of
the X
as a Bayesian
decision problem.
(a). Establish the following two Lemmas:
(i)
For a standard normal random variable Z, and any
constant ε >0, the probability P(|Z-k| > ε)
is minimized over k when k=0.
(ii)
For a standard normal random variable Z , the
probability P(Z < -W) is minimized over
all nonnegative
random variables W which are independent of Z and which have a fixed
finite expectation c > 0,
by the degenerate random variable W=c.
θ ~
N(τ, ν) , as a function of τ
and ν. Explain what your procedure becomes in the
limit as ν goes
to infinity while τ is fixed. (Here
n is arbitrary and fixed.)
risk
function is exhibited as an integral over fX of an
inner integral which can be interpreted as the
conditioal risk given
the data X. This integral is minimized first over
(a(X)+b(X))/2 for fixed
ε =
ε(X) = (b(X)-a(X))/2 using Lemma (i).
Then you show using Lemma (ii) that the optimal choice
for
ε(X) is a constant. Finally, you evaluate the
constant by solving a calculus minimum problem.
Sec.5 (pp. 84-87): #3, , #4, #9,
#14.
Do problems #1.6.3, 1.6.5, 1.6.11(b),(e), 1.6.12,
1.6.18, 1.6.22, 1.6.28, and
1.6.32,
from Bickel and Doksum pages 87-93.
(1). Do the following problems from Bickel and Doksum:
#1.6.36, p. 94;
and also #2.1.5, 2.1.9, 2.2.12, 2.2.17, pp. 142-145.
(2). Additional Problem. (required, not optional). (a).
If X1, X2, ... ,
Xn are i.i.d. Expon(λ) random variables, then
find the generalized method of moments estimator for λ
based on these observations using g(Xi) =
I[Xi > 1].
(b). If X1,
X2, ... , Xn are a sample from N(μ,
σ2), then find the generalized method of moments
estimator
of θ = (μ, σ2) based on the sample using
g(Xi) = (I[Xi > 1],
I[Xi > 2]).
(c). Use the
Delta Method to find the (approximate large-sample) variances of your
estimators found
in (a) and (b), as a function of n and the true parameters.