STAT5610: Advanced Inference Semester 1, 2022
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Practice Quiz 2: Introduction to semiparametric methods
STAT5610: Advanced Inference
Semester 1, 2022
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 − 
(1 + o(1)).
2. The Cauchy density given by
f(x) = 
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} − 
]− 2 [1{x ≤ 1} − 
] .
Show that for some constant a,
n −1/2 
[m ( 
)− m(Xi )] 
ah
uniformly in bounded h and determine the constant a. You may use the result that for all 0 < C < ∞ , ω(Cn−1/2) 
0, where ω(δ) is the modulus of continuity of the uniform empirical process:
ω(δ) = sup |Hn (u) − Hn (v)| ,
and Hn (u) = n−1/2 z
[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 
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} − 
] .
Define the parametric family of densities {q(· ;η): |η| ≤ 1} according to
q(x;η) = f(x)[1 + ηb(x)] .
(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 ≤ 
,
图log(1 + x) − [x − 
] 图 ≤ 
.
(b) Define p(x;θ,η) = q(x− θ;η) and let Pnθη denote the joint distribution of X1 , . . . ,Xn iid with
the score functions and information matrix. Note: it is known that l
dx = 
.
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(·) satisfyingl
w2 (x)f0 (x) dx = 1.
Assume that
• f0 ( ·) is differentiable and write ψ0 (x) = −f
(x)/f0 (x) for the location score function;
• there exists a complete orthonormal basis for L2 (f0 ) = {g : l
g2 (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 = η 1 = ··· = 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. l
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
{ℓ˜1 (x) − ℓ˜2 (x)}2 f0 (x) dx = 0.
2022-11-03