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Please write out answers to the questions below and submit to the appropriate Canvas
Assignment portal. If you believe the question is a special case of a general problem
that has already been solved in lectures, tutorials or homeworks, you may refer to that
result to obtain your answer rather than deriving from first principles, if you prefer.
In that case make sure you verify any conditions required for the general result to
hold.
1. Show that if Qnθ denotes the joint distribution of X1, . . . , Xn iid Poisson with rate θ that the LAN
condition holds at θ = 1. Identify the score function and information. You may use the fact that
as z → 0, log(1 + z) = z ? z22 (1 + o(1)).
2. The Cauchy density given by
f(x) =
1
π(1 + x2)
is known to have median zero and quartiles equal to ±1. Suppose X1, . . . , Xn are iid Cauchy.
(a) A version of the function sign(|x| ? 1) is given by
m(x) = 2
[
1 {x ≤ ?1} ? 1
4
]
? 2
[
1 {x ≤ 1} ? 3
4
]
.
Show that for some constant a,
n?1/2
n∑
i=1
[
m
(
Xi
1 + n?1/2h
)
?m(Xi)
]
P→ ah
uniformly in bounded h and determine the constant a. You may use the result that for
all 0 < C < ∞, ω(Cn?1/2) P→ 0, where ω(δ) is the modulus of continuity of the uniform
empirical process:
ω(δ) = sup
|u?v|≤δ
|Hn(u)?Hn(v)| ,
andHn(u) = n
?1/2∑n
i=1 [1 {Ui ≤ u} ? u] for independent U(0, 1) random variables U1, . . . , Un.
(b) The previous part implies that the sample median absolute value (the sample median of
|X1|, . . . , |Xn|) θ?n satisfies