unstructuredio/SCORE-Bench · Datasets at Hugging Face

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(2.11)
for some kernel function k(.), leading to the usual kernel LRV estimator. Thus, by substituting the smoothing estimator of the particular kernel function, we obtain the following test statistic
--------------------------------------------------- Unstructured Formula Begin
Q T (θ)= 1/2 [1/√T ∑ T s=1 f(Y t ,θ)]′ Vˆ -1 ff [1/√T ∑ T s=1 f(Y t ,θ)] .
--------------------------------------------------- Unstructured Formula End
(2.12)
Then, the K statistic is based on the first-order derivative of Q T (θ). Define as below the gradients
--------------------------------------------------- Unstructured Formula Begin
g j (Y t ,θ)=∂f(Y t ,θ)/∂θ j ∈R m×1, j∈{1,....,d},
--------------------------------------------------- Unstructured Formula End
(2.13)
--------------------------------------------------- Unstructured Formula Begin
g(Y t ,θ) = ∂f(Y t ,θ)/∂θ′ = (g 1 (Y t ,θ),....,g d (Y t ,θ)) ∈ R m×d ,
--------------------------------------------------- Unstructured Formula End
(2.14)
--------------------------------------------------- Unstructured Formula Begin
ḡ(Y t ,θ)=1/T ∑ T t=1 ∑ T t=1 ∂f(Y t ,θ)/∂θ′ ∈ R m×d .
--------------------------------------------------- Unstructured Formula End
(2.15)
Taking the first-order and second-order derivatives of


--------------------------------------------------- Unstructured Formula Begin
Vˆ f f (θ)
--------------------------------------------------- Unstructured Formula End


with respect to θ j , we obtain
--------------------------------------------------- Unstructured Formula Begin
V^ gjf (θ)= 1/T ∑ T t=1 ∑ T s=1 ω h (t/T,t/T) [g j (Y t ,θ)−ḡ j (Y t ,θ)] [f(Y t ,θ)− fˉ(Y t ,θ)]′
--------------------------------------------------- Unstructured Formula End
(2.16)
--------------------------------------------------- Unstructured Formula Begin
Vˆ g j g j (θ) = 1/T ∑ T t=1 ∑ T s=1 ω h (t/T,t/T) [g j (Y t ,θ)−ḡ j (Y t ,θ)] [g j (Y t ,θ)-ḡ j (Y t ,θ)]'
--------------------------------------------------- Unstructured Formula End


(2.17)


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Then, it follows that
--------------------------------------------------- Unstructured Formula Begin
∂Q T (θ)/∂θ = D T (θ) V -1 ff (θ) [1/√T T ∑ t=1 f(Y t ,θ)]
--------------------------------------------------- Unstructured Formula End
, (2.18)
Denote with D T (θ) = [D T,1 (θ),....,D T,d (θ)] ∈ Rᵐ ˣ ᵈ, such that
--------------------------------------------------- Unstructured Formula Begin
D T,j (θ) = [1/√T T ∑ t=1 g j (Y t ,θ)] −V^ g,f (θ)V^ -1 ff (θ)[1/√T T ∑ t=1 f j (Y t ,θ)] ∈ R mx1
--------------------------------------------------- Unstructured Formula End
. (2.19)
Then, the K statistic for testing the null hypothesis H₀ : θ = θ₀ against the alternative hypothesis given
by H₁:θ ≠ θ₀ is given by
--------------------------------------------------- Unstructured Formula Begin
K T (θ 0 )=(∂Q T (θ 0 )/∂θ)′ [D T (θ 0 )′V- -1 ff (θ 0 )D T (θ 0)]−1(∂Q T (θ 0 )/∂θ)
--------------------------------------------------- Unstructured Formula End
. (2.20)
where for any concave function φ(θ), ∂φ(θ₀)/∂θ is defined to be
--------------------------------------------------- Unstructured Formula Begin
∂φ(θ 0 )/∂θ = ∂φ(θ)/∂θ \| θ=θ 0 ∈ R d×1
--------------------------------------------------- Unstructured Formula End
. (2.21)
Thus, to consider fixed-smoothing asymptotics, we employ the orthonormal series LRV estimator
--------------------------------------------------- Unstructured Formula Begin
ω h (t/T, s/T) = 1/G G ∑ l=1 Φ l (t/T) Φ l (s/T)
--------------------------------------------------- Unstructured Formula End
, (2.22)
where G is a smoothing parameter for this estimator and Φ l (.) is a set of a basis functions on L²[0,1].
The weighting function is expressed with respect to a set of basis functions on the space of L²[0,1].
Therefore, the LRV estimator takes the following form
--------------------------------------------------- Unstructured Formula Begin
V- ff (θ)=1/G G ∑ l=1 { 1/√T T ∑ t=1 Φ l (t/T) [f(Y t ,θ)-f-(Y t ,θ)]} {1/√T T ∑ t=1 Φ l (t/T)[f(Y t ,θ)-f-(Y t ,θ)]}'
--------------------------------------------------- Unstructured Formula End
.
Then, an updated estimator for J T (θ₀) needs to be obtained from the sample such that
--------------------------------------------------- Unstructured Formula Begin
J T (θ 0 )= [1/√T T ∑ t=1 f~(Y t ,θ 0 )]′V^ -1 ff (θ 0 )[1/√T T ∑ t=1 f~(Y t ,θ 0 )]
--------------------------------------------------- Unstructured Formula End
(2.23)
Remark 1. The modified statistic is not the same as the original statistic due to the projection of the function into the space which is induced by the transformation of the column vector space. This property allows us to obtain a consistent estimator θˆ of θ₀. However, to obtain an unbiased estimator for the variance of the estimator, we also need to obtain unbiased estimators for each partial derivative that the covariance matrix is composed to.
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--------------------------------------------------- Unstructured Title Begin
Key Issues for Importing Countries
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Reduction in crop and livestock yields as well as fish catch
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(2.11)

for some kernel function k(.), leading to the usual kernel LRV estimator. Thus, by substituting the smoothing estimator of the particular kernel function, we obtain the following test statistic

--------------------------------------------------- Unstructured Formula Begin

Q T (θ)= 1/2 [1/√T ∑ T s=1 f(Y t ,θ)]′ Vˆ -1 ff [1/√T ∑ T s=1 f(Y t ,θ)] .

--------------------------------------------------- Unstructured Formula End

(2.12)

Then, the K statistic is based on the first-order derivative of Q T (θ). Define as below the gradients

--------------------------------------------------- Unstructured Formula Begin

g j (Y t ,θ)=∂f(Y t ,θ)/∂θ j ∈R m×1, j∈{1,....,d},

--------------------------------------------------- Unstructured Formula End

(2.13)

--------------------------------------------------- Unstructured Formula Begin

g(Y t ,θ) = ∂f(Y t ,θ)/∂θ′ = (g 1 (Y t ,θ),....,g d (Y t ,θ)) ∈ R m×d ,

--------------------------------------------------- Unstructured Formula End

(2.14)

--------------------------------------------------- Unstructured Formula Begin

ḡ(Y t ,θ)=1/T ∑ T t=1 ∑ T t=1 ∂f(Y t ,θ)/∂θ′ ∈ R m×d .

--------------------------------------------------- Unstructured Formula End

(2.15)

Taking the first-order and second-order derivatives of

--------------------------------------------------- Unstructured Formula Begin

Vˆ f f (θ)

--------------------------------------------------- Unstructured Formula End

with respect to θ j , we obtain

--------------------------------------------------- Unstructured Formula Begin

V^ gjf (θ)= 1/T ∑ T t=1 ∑ T s=1 ω h (t/T,t/T) [g j (Y t ,θ)−ḡ j (Y t ,θ)] [f(Y t ,θ)− fˉ(Y t ,θ)]′

--------------------------------------------------- Unstructured Formula End

(2.16)

--------------------------------------------------- Unstructured Formula Begin

Vˆ g j g j (θ) = 1/T ∑ T t=1 ∑ T s=1 ω h (t/T,t/T) [g j (Y t ,θ)−ḡ j (Y t ,θ)] [g j (Y t ,θ)-ḡ j (Y t ,θ)]'

--------------------------------------------------- Unstructured Formula End

(2.17)

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10

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Then, it follows that

--------------------------------------------------- Unstructured Formula Begin

∂Q T (θ)/∂θ = D T (θ) V -1 ff (θ) [1/√T T ∑ t=1 f(Y t ,θ)]

--------------------------------------------------- Unstructured Formula End

, (2.18)

Denote with D T (θ) = [D T,1 (θ),....,D T,d (θ)] ∈ Rᵐ ˣ ᵈ, such that

--------------------------------------------------- Unstructured Formula Begin

D T,j (θ) = [1/√T T ∑ t=1 g j (Y t ,θ)] −V^ g,f (θ)V^ -1 ff (θ)[1/√T T ∑ t=1 f j (Y t ,θ)] ∈ R mx1

--------------------------------------------------- Unstructured Formula End

. (2.19)

Then, the K statistic for testing the null hypothesis H₀ : θ = θ₀ against the alternative hypothesis given

by H₁:θ ≠ θ₀ is given by

--------------------------------------------------- Unstructured Formula Begin

K T (θ 0 )=(∂Q T (θ 0 )/∂θ)′ [D T (θ 0 )′V- -1 ff (θ 0 )D T (θ 0)]−1(∂Q T (θ 0 )/∂θ)

--------------------------------------------------- Unstructured Formula End

. (2.20)

where for any concave function φ(θ), ∂φ(θ₀)/∂θ is defined to be

--------------------------------------------------- Unstructured Formula Begin

∂φ(θ 0 )/∂θ = ∂φ(θ)/∂θ | θ=θ 0 ∈ R d×1

--------------------------------------------------- Unstructured Formula End

. (2.21)

Thus, to consider fixed-smoothing asymptotics, we employ the orthonormal series LRV estimator

--------------------------------------------------- Unstructured Formula Begin

ω h (t/T, s/T) = 1/G G ∑ l=1 Φ l (t/T) Φ l (s/T)

--------------------------------------------------- Unstructured Formula End

, (2.22)

where G is a smoothing parameter for this estimator and Φ l (.) is a set of a basis functions on L²[0,1].

The weighting function is expressed with respect to a set of basis functions on the space of L²[0,1].

Therefore, the LRV estimator takes the following form

--------------------------------------------------- Unstructured Formula Begin

V- ff (θ)=1/G G ∑ l=1 { 1/√T T ∑ t=1 Φ l (t/T) [f(Y t ,θ)-f-(Y t ,θ)]} {1/√T T ∑ t=1 Φ l (t/T)[f(Y t ,θ)-f-(Y t ,θ)]}'

--------------------------------------------------- Unstructured Formula End

.

Then, an updated estimator for J T (θ₀) needs to be obtained from the sample such that

--------------------------------------------------- Unstructured Formula Begin

J T (θ 0 )= [1/√T T ∑ t=1 f~(Y t ,θ 0 )]′V^ -1 ff (θ 0 )[1/√T T ∑ t=1 f~(Y t ,θ 0 )]

--------------------------------------------------- Unstructured Formula End

(2.23)

Remark 1. The modified statistic is not the same as the original statistic due to the projection of the function into the space which is induced by the transformation of the column vector space. This property allows us to obtain a consistent estimator θˆ of θ₀. However, to obtain an unbiased estimator for the variance of the estimator, we also need to obtain unbiased estimators for each partial derivative that the covariance matrix is composed to.

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11

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--------------------------------------------------- Unstructured Plain Text Format 1.0.4

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7

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--------------------------------------------------- Unstructured Title Begin

Key Issues for Importing Countries

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Reduction in crop and livestock yields as well as fish catch

--------------------------------------------------- Unstructured Sub-Title End