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.
Problem Set 1. The first assignment consists of 7 problems, due
in class Monday, Feb. 9.
Do the following problems in Rohatgi & Saleh:
#2 in Sec 9.5, p.486; #2 in Sec. 9.6, p.489; and #4 in Sec. 10.2, p.499.
Do problems #4.9.1, 4.9.2, 4.9.4, and 4.9.11, pp. 290-293 in Bickel and Doksum.
Problem Set 2. The second assignment consists of 7 problems, due
in class Wednesday, Feb. 18.
First do the
worksheet related to best similar tests
(counts as 3 problems).
Then do the following
problems in Bickel & Doksum: #4.4.6, 4.4.7, 4.4.9, 4.6.1
Problem Set 3. The third assignment consists of 7 problems, due
in class Wednesday, Mar. 4.
NOTE that due to the campus closure
on March 2, the due-date is extended to
Friday March 6 at 5pm. (You could
email or fax or drop off the HW.)
Also note the hint for #5.2.4 below, and the
modification of problem (IV) [Part (ii) is now dropped,
and for part (iii)
you should try to use the Theorem in Bickel-Doksum on global consistency of
minimum contrast estimators to prove the assertion.]
(I) (Counts as 2)
(i) Solve Bickel-Doksum #5.2.4 parts (a)-(b). Then (ii) compare the
large-sample
(asymptotic) variance of the estimator of part (b)
with that of the maximum-likelihood estimator of ρ .
Hint:
It is not easy to integrate directly to calculate the quadrant probability
P(U>0, V>0)
to be equal to an expression involving the arcsin. But
you will find that if you write the quadrant
probability as a univariate integral
with integrand involving the standard-normal cdf, then its
derivative with respect
to ρ is easily integrated analytically and can be shown equal
to the
derivative of the epxression you want to equate it to. In
part (b), there is a missing term -1/4
following the average of
products of indicators, and to do part (b) you should first estimate
that a
negligible-in-probability error is made by replacing each of
Xbar and Ybar by their respective
means in the
indicator-expressions. Also, in the last part of the problem, you may
get the
asymptotic variance either by the delta-method applied
to a formula for the MLE or, more
conveniently, by using the
Fisher Information.
(II) Bickel and Doksum
#5.2.5.
(III) Bickel and
Doksum #5.3.25, 5.3.27, 5.3.33.
NOTE: the last symbol in the first
line of 5.3.33(b) should be sigma-hat squared, not mu-hat squared.
(IV) Suppose that
X1,...,Xn are i.i.d. with Xi =
(Yi,Zi) where Y's and Z's are scalar and
Yi-a - b Zi = εi,
where the variable εi is
independent of Zi, and the variable
εi/σ has a known
symmetric (i.e. even)
density with mean 0 and variance 1. Here θ = (a, b, &sigma)
is the unknown parameter vector.
(i) Show that
ρ(X,θ) =
&Sigmai=1n {
(Yi-a-bZi)2 + (
(Yi-a-bZi)2 -
σ2)2 } is a
contrast function.
(ii) Find the minimum
contrast estimators in this problem. NOTE: these estimators cannot be
found
in closed form , so you should simply omit this part.
(iii) Show that
the minimum contrast estimators in (ii) are globally (weakly) consistent.
Problem Set 4. The fourth assignment consists of 5 problems, due
in class Monday, Mar. 23.
(A) (Counts as 3) Do the
Problems (I)-(III) on the 2x2 Table Handout/Worksheet.
(B) (Counts as 2) Suppose that U and
W are respectively mean-0 random-vectors of dimensions p and q, with
E(UU') = A ;
(pxp matrix) , E(UW') = B
(pxq matrix) ,
E(WW') = C (qxq matrix)
and assume that the p+q dimensional random vector concatenating U and W has
nonsingular variance-covariance matrix M .
(i) Prove that the upper-left pxp
block of M-1 is K =
(A - B C-1 B')-1 .
(ii) Prove that for
all p-dimensional non-zero vectors c ,
c'K-1c = min b in
Rq
E(c'U-b'W)2 .
(iii) Use (ii) to conclude that
K-1 is the minimum over pxq matrices H of
variance-covariance matrices
(in the sense of positive-definite
matrix ordering, under which A < B iff B-A is positive-definite) of
random p-vectors of
the form U - HW .
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Problem Set 5. The fifth assignment consists of 4 problems, due
Monday, April 20 in class. Problem Set 6. The sixth assignment consists of 5 problems, due
Wednesday, May 6 in class:
(A) Do # 5.4.14 p. 360. Recall that
the inverse Gaussian density has exponential-family properties
which were
established in Stat 700 in Problem 1.6.36 (with solution given in Stat 700 -- see
Stat 700 web-page.)
(B) Suppose that iid
data pairs (Xi,Zi) for i=1,...,n
are observed, where Zi ~ Binom(1,1/2) and given
Zi ,
the distribution of Xi is
Xi ~ Weibull(γ, &lambda eθZi) .
Suppose that the data are analyzed by estimating θ using the
MLE θ* for the incorrect model which postulates
given Zi that the distribution of Xi is
Xi ~ Expon(&lambda eθZi)
.
Note that the parameters θ and
γ and λ are all unknown in this
problem. The definitions of the Weibull and
Exponential distributions in this problem
are as follows: if T ~ Weibull(a,b), then for t>0,
P(T >t) = exp(-bta)
and if V ~
Expon(b), then P(V>t) = exp(-bt).
(a) Show that this misspecified model estimator θ*
has large-sample limit which is 0 when the true
value of
&theta = 0 and which otherwise (for θ non-zero) exists and has the same
sign as θ .
(b) Find the relative efficiency at θ=0 of the
misspecified model estimator
θ* of θ .
(C) Do # 6.2.4 and 6.2.5, p. 396.
#6.1.4, p.423; #6.3.1,
6.3.5, pp. 428-430; #6.4.6, p.432; #6.5.1,
p. 4334.
NOTE added 4/30/09: In Problem #6.1.4, there is a typo in the
definition of the weights aj :
the numerator of the ratio defining aj
should be the sum from i=0 to
i=n-j of (-c)i (1/(1+c)) (1 -
(-c)i+j).
The denominator of the
aj expression is exactly as given on p.423 of the
Bickel and Doksum book.
To do this problem #6.1.4, I suggest that you do part (b) first: the
hint is first to find linear combinations
b1i
Y1 + b2i
Y2 +... + bii Yi of the
Yk observations, k=1,...,i, which
are strictly independent for
different i ,
that is, which involve only ei.
NOTE added 5/4/09: In Problem #6.5.1, you should use a general
form of the GLM specificatiion, in which:
g(μ) = Zβ ,
with g 1-to-1 and differentiably invertible, with the link
possibly noncanonical, and with the
gradient of A(η)
one-to-one and differrentiably invertible. This is actually not a difficult
problem, if you
write the gradient of η with respect to
β as a nxp matrix M ,
and if you try to write the Information
and gradient of likelihood as far
as possible in terms of M.