The University of Sydney 
School of Mathematics and Statistics 
Practice Quiz 2: Introduction to semiparametric methods 
STAT5610: Advanced Inference Semester 1, 2022 
Lecturers: Rachel Wang and Michael Stewart 
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|) θˆn satisfies 
√ 
n 
( 
θˆn − 1 
) 
= −a−1 
{ 
n−1/2 
n∑ 
i=1 
m(Xi) 
} 
+ op(1) . 
Use this to derive the limiting distribution of 
√ 
n 
( 
θˆn − 1 
) 
. 
3. Suppose f(·) is the Cauchy density (see the previous question) and 
b(x) = 2 
[ 
1 {x ≤ 0} − 1 
2 
] 
. 
Define the parametric family of densities {q(·; η) : |η| ≤ 1} according to 
q(x; η) = f(x) [1 + ηb(x)] . 
Copyright© 2022 The University of Sydney 1 
(a) Show that if Qnη is the joint distribution of Y1, . . . , Yn with common density q(x; η) then 
the LAN condition holds for the family {Qnη} at η = 0. You may make use of the fact that 
for |x| ≤ ε ≤ 12 , ∣∣∣∣log(1 + x)− [x− x22 
]∣∣∣∣ ≤ 8ε33 . 
(b) Define p(x; θ, η) = q(x−θ; η) and let Pnθη denote the joint distribution ofX1, . . . , Xn iid with 
common density p(x; θ, η). Show that the LAN condition holds at θ = 0, η = 0. State clearly 
the score functions and information matrix. Note: it is known that 
∫∞ 
−∞ 
f ′(x)2 
f(x) dx = 
1 
2 . 
4. Consider the semiparametric, integral-constrained location model where n iid observations have 
common density given by 
p(x; θ) = f0(x− θ) 
where the “centred” density f0(·) satisfies∫ ∞ 
−∞ 
w(x)f0(x) dx = 0 
for a constraint function w(·) satisfying ∫∞−∞ w2(x)f0(x) dx = 1. 
Assume that 
• f0(·) is differentiable and write ψ0(x) = −f ′0(x)/f0(x) for the location score function; 
• there exists a complete orthonormal basis for L2(f0) = {g : 
∫∞ 
−∞ g 
2(x)f0(x) dx < ∞} of the 
form 
{1} ∪ {w} ∪ {bj : j = 1, 2, . . .} 
and that for each k = 1, 2, . . . it is possible to construct a parametric family of densities 
Fk = {f(·; η1, . . . , ηk) : |ηj | ≤ εk} (for some εk > 0) satisfying∫ ∞ 
−∞ 
f(x)w(x) dx = 0 
for all f ∈ Fk, f(·; 0, 0, ..., 0) = f0(·) and that the larger parametric model with common 
density 
q(x; θ, η1, . . . , ηk) = f(x− θ; η1, . . . , ηk) 
satisfies the LAN condition at θ = η1 = · · · = ηk = 0 with score function vector (ψ, b1, . . . , bk)T . 
The influence function ℓ˜(·) of any asymptotically linear estimator θ˜n which is regular at θ = 0 
must satisfy two conditions: 
1. 
∫∞ 
−∞ ψ0(x)ℓ˜(x)f0(x) dx = 1; 
2. it must be orthogonal to the scores for any nuisance parameters for any (regular, so LAN 
holds) parametric submodel that includes the true density. 
Explain why there is (essentially) only one possible such influence function ℓ˜(·) and describe this 
influence function. Note: by “essentially” we mean that for any two such influence functions ℓ˜1(·) 
and ℓ˜2(·) we have 
∫∞ 
−∞ 
{ 
ℓ˜1(x)− ℓ˜2(x) 
}2 
f0(x) dx = 0. 
2