代写STAT5610、代写Python/Java程序

日期：2022-11-04 08:47

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|) θ?n satisfies

√

n

(

θ?n ? 1

)

= ?a?1

{

n?1/2

n∑

i=1

m(Xi)

}

+ op(1) .

Use this to derive the limiting distribution of

√

n

(

θ?n ? 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 ??(·) of any asymptotically linear estimator θ?n which is regular at θ = 0

must satisfy two conditions:

1.

∫∞

?∞ ψ0(x)??(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 ??(·) and describe this

influence function. Note: by “essentially” we mean that for any two such influence functions ??1(·)

and ??2(·) we have

∫∞

?∞

{

??1(x)? ??2(x)

}2

f0(x) dx = 0.

2