Stat 700 HW6 Solutions,
11/30/09
Additional Problem (a). Here we have method-of-moments estimation
with g(Xi ) = I[Xi >1] for iid Expon() data, leading to E (g(X1 ) = e
and = ln(n1 n I[Xi >1] ).
i=1
(b). Similarly, with N (, 2 data and g(Xi ) = (I[Xi >1]
Stat 700 HW1 Solutions,
9/14/0909
Bickel-Doksum, #B.1.8. In all three problem parts (i, iv, v), the random
variable Y and the conditional distribution of Y given Z are symmetric (for
any value Z), so that E(Y ) = E(Y |Z) = 0 = E(E(Y |Z).
(i). Using the gi
Eric Slud
9/15/2014
Partial Solutions to Stat 700 HW1, Fall 2014
(1). It is easy to check that the density function is even, i.e., that it is exactly the
same when its real argument x is replaced by x. This is the same (for a random
variable with density)
Topics & Sample Problems for Stat 700 In-Class Test, Fall 2008
(I) [Multivariate normal and transformations]
In the theory of simple linear regression, data Y = (Y1 , Y2 , . . . , Yn ) satisfy
Yi = a + bXi +
,
i
iid
i
N (0, 2 )
where Xi are treated as kn
Stat 700 HW2 Solutions,
9/25/09
k
(1). By the spectral theorem, B =
are an
j=1 j vj vj , where vj
orthonormal basis of eigenvectors of B with corresponding eigenvalues j .
Now, since
j vj = Bvj = B 2 vj = B(j vj ) = 2 vj
j
and j 0, it follows that j = 0 o
Eric Slud
October 22, 2013
Partial Solutions of HW4 Problems in Stat 700
(1). Algorithm #1 : Generate successive iid pairs (Ui1 , Ui2 ) of Uniform[0, 1]
random variables and dene V = 2 Ui1 , W = Ui2 for the rst such pair to
satisfy Ui2 < 2 min(Ui1 , 1 Ui1
Eric Slud
9/29/2014
Partial Solutions to Stat 700 HW2, Fall 2014
(1). The easiest solution is to take F 1 (U ) by the probability integral theorem,
where U Uniform[0, 1] and for 0 r 1 the right-continuous inverse F 1 (r)
of F is see from the denition to b
Eric Slud
October 6, 2013
Partial Solutions of HW3 Problems in Stat 700
(1). One way to do this is to write (for t = 0)
P (Z2 max(t, z1 s), t Z3 sz1 | Z1 = z)(z1 ) dz1
P (Y1 s, Y2 t) =
=
(s z) (t) I[zst] (1 (max(t, z s) (z) dz
= I[t>0] (1 (t)
st
(s z) (
Wednesday, 10/28/09
56:20 p.m.
Stat 700 In-Class Test, Fall 2009
Instructions. This test is closed-book, but you may bring up to two
written or typed 8.5 11 notebook sides of formulas, denitions, etc. as
memory aids. You may use a calculator, although you
Solutions, Fall 08 Stat 700 Take-Home
(I) (a) The log of the joint density of cfw_(Xi , Yi )n (most conveniently
i=1
written as the product of the marginal density fX by fYX ) is
n log(
n
1
2 2
2) n log( 2 )
(Xi )2 + 2(Yi Xi )2
iu=1
This log density has
Stat 700, Fall 01
10/12/01
Sample of Problems for Stat 700 In-Class Test
The test will be closed-book, but you are permitted to bring 1 or 2 notebook sheets of formulas and notes for references if you want. You are allowed
to use calculators if you want.
Eric Slud
November 7, 2013
Partial Solutions of HW5 Problems in Stat 700
(1). The log-likelihood for the n = 100 pairs (ei , Ti ) can be written
log L(A , B ; e, T) = nA log(3/5) + (n nA ) log(2/5) + nA log(A )
+ (n nA ) log(B ) A SA B SB
from which sucie
Stat 700 HW 7 Solutions, F09
Bickel-Doksum, #2.2.35. Dierntiate the log-likelihood with respect to
and multiply through by (1 + (x1 )2 )(1 + (x2 )2) to get the likelihood
equation
0 = 2(x1) (1+(x2)2 ) + 2(x2) (1+(x1 )2 ) = 2() (1 + (x1)(x2 )
x
= 2( ) (1
Stat 700 HW4 Solutions,
10/22/09
(1). Extra Problem In this problem, the actions are pairs of points interpreted as endpoints of an interval estimate (a, b], and the statistical
procedures are pairs of functions, (X) = (a(X), b(X).
Lemma (i). P (|Z k| > )
Stat 700 HW5 Solutions,
11/15/09
#1.6.3. (a). By inspection, the probability mass functions p(x, ) =
exp(x ln(1 ) + ln ), x 0, form an exponential family.
(b). Now p(x, ) = p(x1 , . . . , xn , ) = exp( n Xi ln(1)+n ln().
i=1
As in Theorem 1.6.1 with sucie
Stat 700, Fall 01
12/12/01
Sample Problems for Stat 700 Final
The test will be closed-book, but you are permitted to bring 1 or 2 notebook sheets of formulas and notes for references if you want. You are allowed
to use calculators if you want.
The followi