Stat 5102 (Geyer) Spring 2012
Homework Assignment 2
Due Wednesday, February 1, 2012
Solve each problem. Explain your reasoning. No credit for answers with
no explanation. If the problem is a proof, then you need words as well as
formulas. Explain why your formulas follow one from another.
21.
Suppose
X
1
,
X
2
,
. . .
are IID Unif(0
, θ
).
As usual
X
(
n
)
denotes the
n
th order statistic, which is the maximum of the
X
i
.
(a) Show that
X
(
n
)
P
→
θ,
as
n
→ ∞
.
(b) Show that
n
(
θ

X
(
n
)
)
D
→
Exp(1
/θ
)
,
as
n
→ ∞
.
Hints
This is a rare problem (the only one of its kind we will meet in
this course) when we can’t use the LLN or the CLT to get convergence
in probability and convergence in distribution results (obvious because the
problem is not about
X
n
and the asymptotic distribution we seek isn’t
normal). Thus we need to derive convergence in distribution directly from
the characterization as convergence of distribution functions (5101 deck 6,
slide 4), that is,
X
n
D
→
X
if and only if
F
n
is the DF of
X
n
and
F
is the DF of
X
and
F
n
(
x
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 Spring '03
 Staff
 Laplace, Probability theory, lim, Cauchy, asymptotic distribution, Slutsky

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