Small area quantile estimation via spline regression and empirical likelihood

Section 4. Bootstrap estimation of the mean squared errors

Table of contents

The proposed small area quantile estimators are assembled with many intermediate steps. It is difficult to analytically evaluate the variances or mean squared error (MSE) of such estimators. We follow others (Sinha and Rao (2009), Tzavidis et al. (2010) and Chen and Liu (2018)) to develop a bootstrap procedure as follows:

Step 1

Obtain estimates

\hat{β}, \hat{γ}, {\hat{σ}}_{v}^{2}

and

{\hat{m}}_{0} (x, \hat{β}, \hat{γ})

based on Model (2.1), and calculate

{\hat{G}}_{i} (u)

as in (3.7).

Step 2

Generate a bootstrap finite population

H^{*} = {y_{i j}^{*}, x_{i j}}, i =0, \dots, m, j =1, \dots, N_{i}

with

$y_{i j}^{*} = {\hat{m}}_{0} (x_{i j}, \hat{β}, \hat{γ}) + v_{i}^{*} + ε_{i j}^{*},$

where the bootstrap residuals

ε_{i j}^{*}

are sampled from CDF

{\hat{G}}_{i} (u)

, and

v_{i}^{*}

are generated from

N (0, {\hat{σ}}_{v}^{2}) .

Step 3

From the bootstrap population

H^{*},

we select

n_{i}^{*} = n_{i}

sample units from small area

i

by simple random sampling without replacement, and repeat it

L

times to get

h_{l}^{*}, l =1, \dots, L .

For each sample

h_{l}^{*},

compute the estimates

{\hat{F}}_{i}^{(a) * l} (u)

and

{\hat{F}}_{i}^{(b) * l} (u)

as in (3.8) and (3.9) respectively.

Step 4

Compute the empirical MSE estimator of

\hat{τ}

$mse (τ^{*}) = L^{- 1} \sum_{l =1}^{L} {({\hat{τ}}^{* l} - τ^{*})}^{2},$

where

{\hat{τ}}^{* l} = τ ({\hat{F}}^{* l} (u))

denotes any functional of

{\hat{F}}^{(a) * l} (u)

{\hat{F}}^{(b) * l}

and

τ^{*} = τ (F^{*} (u))

with

F^{*} (u)

being the known CDF of the bootstrap populations.

Step 5

Repeat Steps 2 to 4, B times, and define the bootstrap MSE estimate as

$B^{- 1} \sum_{b =1}^{B} mse {(τ^{*})}_{b},$

where

mse {(τ^{*})}_{b}

is the

mse (τ^{*})

calculated in the

b^{th}

repetition.

The performance of the bootstrap MSE estimator will be examined and reported in the simulation section.

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Small area quantile estimation via spline regression and empirical likelihood

Section 4. Bootstrap estimation of the mean squared errors

Small area quantile estimation via spline regression and empirical likelihood Section 4. Bootstrap estimation of the mean squared errors

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

Small area quantile estimation via spline regression and empirical likelihood

Section 4. Bootstrap estimation of the mean squared errors