STAT 210B HWK #5 SOLUTIONS
GARVESH RASKUTTI
(1) Let X1 , ., Xn be i.i.d according to the Pareto distribution with density
f (x) = c x(+1) ,
where 0 < and 0 < c < x. Determine the Wald, Rao and likelihood ratio
tests of H0 : = 0 against = 0 .
The log-likel
Stat 210B Homework Assignment 2 (due February 15)
n
1. Consider the V-statistic Vn = 1/n2 n
i=1
j =1 h(xi , xj ) for a symmetric kernel h such that
2 < . Show that V is asymptotically normal.
Eh
n
2. Consider the kernel h(x1 , x2 ) = Ix1 +x2 >0 (where I d
STAT 210B HWK #1 SOLUTIONS
GARVESH RASKUTTI
(1) Given densities pn and qn with respect to some measure , dene the
likelihood ratio Ln (x) as Ln (x) = qn (x)/pn (x) for pn (x) > 0, Ln (x) = 1 if pn (x) =
qn (x) = 0 and Ln (x) = otherwise. Show that the lik
Stat 210B Homework Assignment 3 (due March 3)
1. Show that for all 1 p < , we have:
Hp (, Q, F ) Hp,B (, Q, F )
for all . Show that if Q is a probability measure, we have:
Hp,B (, Q, F ) H (/2, F ).
2. Show that the Cramr-von Mises statistic, n (Fn F )2 d
STAT 210B HWK #2 SOLUTIONS (DUE FEBRUARY 17)
GARVESH RASKUTTI
1
(1) Consider the V-statistic Vn = n2 n=1 j =1 h(xi , xj ) for a symmetric kernel
i
h such that1 Eh2 < . Show that Vn is asymptotically normal.
By van der Vaart Thm 12.3 (p. 162), if a U-stati
Stat 210B Homework Assignment 4 (due March 29)
1. Given densities p1 and p2 with respect to some -nite measure , dene that the Hellinger
distance as follows:
1/2
1
2
h(p1 , p2 ) =
( p1 p2 ) d
2
and dene the variation distance as follows:
p1 p2
1
=
|p1 p2
STAT 210B HWK #3 SOLUTIONS (DUE MARCH 13)
GARVESH RASKUTTI
(1) Show that for all 1 p < , we have
Hp (, Q, F ) Hp,B (, Q, F )
for all . Show that if Q is a probability measure, we have
Hp,B (, Q, F ) H (/2, F )
More generally, in the rst claim we can repla
Stat 210B Homework Assignment 5 (due April 19)
1. Let X1 , . . . , Xn be i.i.d. according to the Pareto distribution with density
f (x) = c x+1 ,
where 0 < and 0 < c < x. Determine the Wald, Rao and likelihood ratio tests of H0 : = 0
against = 0 .
2. Calc
STAT 210B HWK #4 SOLUTIONS (DUE MARCH 29)
GARVESH RASKUTTI
(1) Given densities p1 and p2 with respect to some nite measure , dene
that the Hellinger distance as follows:
h(p1 , p2 ) =
( p1 p2 )2 d
1
2
1/2
,
and dene the variation distance as follows:
|p1
Stat 210B Homework Assignment 1 (due February 1)
1. Given densities pn and qn with respect to some measure , let X be distributed according to
the distribution with density pn . Dene the likelihood ratio Ln (X ) as Ln (X ) = qn (X )/pn (X )
for pn (X ) >