Stat 210B Homework Assignment 2
(due February 13)
1. Let
ˆ
U
n
denote the H´
ajek projection of the Ustatistic
U
n
:
ˆ
U
n
=
n
X
i
=1
E
[
U
n

X
i
]

(
n

1)
θ.
Show that
ˆ
U
n

U
n
can itself be written as a Ustatistic.
2. Consider the Vstatistic
V
n
= 1
/n
2
∑
n
i
=1
∑
n
j
=1
h
(
x
i
, x
j
) for a symmetric kernel
h
such that
Eh
2
<
∞
. Show that
V
n
is asymptotically normal.
3. Consider the kernel
h
(
x
1
, x
2
) =
I
x
1
+
x
2
>
0
(where
I
denotes an indicator function). Evaluate
θ
=
E
[
h
(
X
1
, X
2
)] for:
(a) the mixture
F
= (1

)
N
(0
,
1) +
N
(
α, β
)
(b) the uniform distribution
F
= Un(
θ

1
2
, θ
+
1
2
).
4. Given a probability space, consider a sequence of sigmafields
{F
n
}
on that space such that
F
1
⊃ F
2
⊃ F
3
⊃ · · ·
.
Consider a sequence of random variables
{
X
n
}
such that
X
n
is
F
n
measurable and
E

X
n

<
∞
. The sequence
{
X
n
}
is called a
reverse martingale
if
E
[
X
n
 F
n
+1
] =
X
n
+1
,
(wp1, for all
n
)
.
Let the symbol
X
(
n
)
denote the order statistics of the sequence (
X
1
, X
2
, . . . , X
n
).
Define
F
n
=
σ
(
X
(
n
)
, X
n
+1
, X
n
+2
, . . .
). (Intuitively, this sigmafield is the information contained in
the infinite sequence (
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 Fall '05
 MICHAELJORDAN
 Probability, Probability theory

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