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Robust LH stratified sampling strategy Maria Caterina Bramati Sapienza University of Rome Southampton Research Seminar - July 15th 2014 - ( Southampton Research Seminar ) Robust LH stratified sampling strategy 1 / 37 Introduction Outline

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SLIDE 1

Robust LH stratified sampling strategy

Maria Caterina Bramati

Sapienza University of Rome

Southampton Research Seminar

  • July 15th 2014 -

( Southampton Research Seminar ) Robust LH stratified sampling strategy 1 / 37

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SLIDE 2

Introduction

Outline

Motivation Robustness issues in Stratified design Some proposals Simulation Study Further issues: Time-dependent survey variables Agenda

( Southampton Research Seminar ) Robust LH stratified sampling strategy 2 / 37

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SLIDE 3

Introduction

Key features on surveying firms

1 Population has a skewed distribution (small number of units accounts

for a large share of the study variables)

2 Availability of administrative information, providing a list of the

statistical units of the target population (i.e. tax declaration, social security registers)

3 Survey burdens for firms and costs for NSIs 4 Data quality (administrative sources and survey collection) 5 @EU: Compliance requirements established by EUROSTAT ( Southampton Research Seminar ) Robust LH stratified sampling strategy 3 / 37

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SLIDE 4

Introduction

Key features on surveying firms

1 Population has a skewed distribution (small number of units accounts

for a large share of the study variables)

2 Availability of administrative information, providing a list of the

statistical units of the target population (i.e. tax declaration, social security registers)

3 Survey burdens for firms and costs for NSIs 4 Data quality (administrative sources and survey collection) 5 @EU: Compliance requirements established by EUROSTAT ( Southampton Research Seminar ) Robust LH stratified sampling strategy 3 / 37

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SLIDE 5

Introduction

Key features on surveying firms

1 Population has a skewed distribution (small number of units accounts

for a large share of the study variables)

2 Availability of administrative information, providing a list of the

statistical units of the target population (i.e. tax declaration, social security registers)

3 Survey burdens for firms and costs for NSIs 4 Data quality (administrative sources and survey collection) 5 @EU: Compliance requirements established by EUROSTAT ( Southampton Research Seminar ) Robust LH stratified sampling strategy 3 / 37

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SLIDE 6

Introduction

Key features on surveying firms

1 Population has a skewed distribution (small number of units accounts

for a large share of the study variables)

2 Availability of administrative information, providing a list of the

statistical units of the target population (i.e. tax declaration, social security registers)

3 Survey burdens for firms and costs for NSIs 4 Data quality (administrative sources and survey collection) 5 @EU: Compliance requirements established by EUROSTAT ( Southampton Research Seminar ) Robust LH stratified sampling strategy 3 / 37

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SLIDE 7

Introduction

Key features on surveying firms

1 Population has a skewed distribution (small number of units accounts

for a large share of the study variables)

2 Availability of administrative information, providing a list of the

statistical units of the target population (i.e. tax declaration, social security registers)

3 Survey burdens for firms and costs for NSIs 4 Data quality (administrative sources and survey collection) 5 @EU: Compliance requirements established by EUROSTAT ( Southampton Research Seminar ) Robust LH stratified sampling strategy 3 / 37

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SLIDE 8

Sampling Strategy

Sampling design: 3 main problems

1 choice of the sampling design 2 sample size determination 3 sample allocation

under some constraints

  • costs related to the surveying process and statistical burdens
  • statistical precision
  • legal obligations and requirements (EUROSTAT, NBB, . . .)
  • availability of auxiliary information

( Southampton Research Seminar ) Robust LH stratified sampling strategy 4 / 37

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SLIDE 9

Sampling Strategy

Sampling design: 3 main problems

1 choice of the sampling design 2 sample size determination 3 sample allocation

under some constraints

  • costs related to the surveying process and statistical burdens
  • statistical precision
  • legal obligations and requirements (EUROSTAT, NBB, . . .)
  • availability of auxiliary information

( Southampton Research Seminar ) Robust LH stratified sampling strategy 4 / 37

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SLIDE 10

Sampling Strategy

Sampling design: stratified sample

  • population is divided into subgroups (or strata) in order to maximize the

intra-group ‘homogeneity’ (according to a chosen target variable) and to minimize the inter-group ‘homogeneity’.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 5 / 37

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SLIDE 11

Sampling Strategy

Sampling design: stratified sample

  • population is divided into subgroups (or strata) in order to maximize the

intra-group ‘homogeneity’ (according to a chosen target variable) and to minimize the inter-group ‘homogeneity’. It requires mutually exclusive strata: 1 unit can belong to 1 stratum only collectively exhaustive strata: no population unit excluded

( Southampton Research Seminar ) Robust LH stratified sampling strategy 5 / 37

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SLIDE 12

Sampling Strategy

Sampling design: stratified sample

  • population is divided into subgroups (or strata) in order to maximize the

intra-group ‘homogeneity’ (according to a chosen target variable) and to minimize the inter-group ‘homogeneity’. It requires mutually exclusive strata: 1 unit can belong to 1 stratum only collectively exhaustive strata: no population unit excluded The choice of 1) − 3) should be linked to quality issues of the final statistical product, balancing costs and benefits. = ⇒ Target statistical precision is the constraint under which choices are made

( Southampton Research Seminar ) Robust LH stratified sampling strategy 5 / 37

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Sampling Strategy

HL sampling algorithm

The HT estimator for the total ^ tystrat = L

h=1 Nh nh

  • k∈Sh yk

has variance estimated by ^ Var(^ tystrat) =

L

  • h=1

Nh (1 − ah) ah s2

yh

(1) where s2

yh =

1 nh − 1

  • k∈Sh

(yk − ^ yh)2, and ^ yh is the sample mean of Y within stratum h.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 6 / 37

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SLIDE 14

Sampling Strategy

HL sampling algorithm

The HL algorithm with Neyman allocation represents an optimal solution for the three problems. n^

tystrat

= NL + L−1

h=1 W 2

h s2 yh

ah

(cY /N)2 + L−1

h=1 Wh N s2 yh

(2) ah = nh Nh = Whsyh L−1

k=1 Wksyk

(3)

( Southampton Research Seminar ) Robust LH stratified sampling strategy 7 / 37

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SLIDE 15

Sampling Strategy

HL sampling algorithm

The idea of HL algorithm is to find the optimal strata boundaries b1, . . . , bL−1 which minimize the size n^

tystrat subject to a required precision

c, with some appropriate sampling allocation (Neyman, proportional...).

( Southampton Research Seminar ) Robust LH stratified sampling strategy 8 / 37

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SLIDE 16

Sampling Strategy

HL sampling algorithm

The idea of HL algorithm is to find the optimal strata boundaries b1, . . . , bL−1 which minimize the size n^

tystrat subject to a required precision

c, with some appropriate sampling allocation (Neyman, proportional...). However

( Southampton Research Seminar ) Robust LH stratified sampling strategy 8 / 37

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SLIDE 17

Sampling Strategy

HL sampling algorithm

The idea of HL algorithm is to find the optimal strata boundaries b1, . . . , bL−1 which minimize the size n^

tystrat subject to a required precision

c, with some appropriate sampling allocation (Neyman, proportional...). However

1 s2

yh is unknown=

⇒ use of auxiliary information X for Y

2 number L of strata is selected by the user 3 low quality of the administrative records: outliers? ( Southampton Research Seminar ) Robust LH stratified sampling strategy 8 / 37

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SLIDE 18

Sampling Strategy

HL sampling algorithm

The idea of HL algorithm is to find the optimal strata boundaries b1, . . . , bL−1 which minimize the size n^

tystrat subject to a required precision

c, with some appropriate sampling allocation (Neyman, proportional...). However

1 s2

yh is unknown=

⇒ use of auxiliary information X for Y

2 number L of strata is selected by the user 3 low quality of the administrative records: outliers?

BUT auxiliary information X = Y target variable. = ⇒ modified HL algorithm (Rivest, 2002): the discrepancy between Y and X is estimated

( Southampton Research Seminar ) Robust LH stratified sampling strategy 8 / 37

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SLIDE 19

Sampling Strategy

The effects of outliers in the HL sampling algorithm

Type of anomalies erroneous records in the surveyed data (Y ) (vertical outliers) quality issues in the administrative registers (X) (leverage)

  • utliers in both variables (X, Y ) (good/bad leverages)

= ⇒ Unreliable conditional mean and variance of Y |X, affecting sample size strata bounds sample allocation

( Southampton Research Seminar ) Robust LH stratified sampling strategy 9 / 37

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SLIDE 20

Sampling Strategy

The effects of outliers in the HL sampling algorithm

Type of anomalies erroneous records in the surveyed data (Y ) (vertical outliers) quality issues in the administrative registers (X) (leverage)

  • utliers in both variables (X, Y ) (good/bad leverages)

= ⇒ Unreliable conditional mean and variance of Y |X, affecting sample size strata bounds sample allocation

( Southampton Research Seminar ) Robust LH stratified sampling strategy 9 / 37

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SLIDE 21

Sampling Strategy

The effects of outliers in the HL sampling algorithm

Type of anomalies erroneous records in the surveyed data (Y ) (vertical outliers) quality issues in the administrative registers (X) (leverage)

  • utliers in both variables (X, Y ) (good/bad leverages)

= ⇒ Unreliable conditional mean and variance of Y |X, affecting sample size strata bounds sample allocation

( Southampton Research Seminar ) Robust LH stratified sampling strategy 9 / 37

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SLIDE 22

Sampling Strategy

The effects of outliers in the HL sampling algorithm

Type of anomalies erroneous records in the surveyed data (Y ) (vertical outliers) quality issues in the administrative registers (X) (leverage)

  • utliers in both variables (X, Y ) (good/bad leverages)

= ⇒ Unreliable conditional mean and variance of Y |X, affecting sample size strata bounds sample allocation

( Southampton Research Seminar ) Robust LH stratified sampling strategy 9 / 37

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SLIDE 23

Sampling Strategy

The effects of outliers in the HL sampling algorithm

Type of anomalies erroneous records in the surveyed data (Y ) (vertical outliers) quality issues in the administrative registers (X) (leverage)

  • utliers in both variables (X, Y ) (good/bad leverages)

= ⇒ Unreliable conditional mean and variance of Y |X, affecting sample size strata bounds sample allocation

( Southampton Research Seminar ) Robust LH stratified sampling strategy 9 / 37

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SLIDE 24

Sampling Strategy

The effects of outliers in the HL sampling algorithm

5.5 6 6.5 7 15 20 25 30 35 40 45 50 55 60 65 5.5 6 6.5 7 15 20 25 30 35 40 45 5 6 7 8 9 10 11 12 13 14 15 20 25 30 35 40 45 5 6 7 8 9 10 11 12 13 14 15 15 20 25 30 35 40 45

( Southampton Research Seminar ) Robust LH stratified sampling strategy 10 / 37

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SLIDE 25

Sampling Strategy

Robust HL sampling strategy

One might clean X from outliers (use some robust measures to identify them) Main concern: how to disentangle extreme values from outliers? = ⇒ robust modified HL algorithm is based on robust regression estimates!

( Southampton Research Seminar ) Robust LH stratified sampling strategy 11 / 37

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SLIDE 26

Sampling Strategy

Generalization of HL sampling strategy

Consider the loglinear model log yi = α + β log xi + εi, i = 1, . . . , N εi ∼ N(0, σ2); X, Y continuous random variables; data {xi, i = 1, . . . , N} are N independent realizations of X, with density f (x), x ∈ R.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 12 / 37

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SLIDE 27

Sampling Strategy

Generalization of HL sampling strategy

Solve: find the optimal size n and strata bounds bh, h = 1, . . . , L compatible with the fixed precision c. The HL stratification process uses Var[Y |bh ≥ X > bh−1] and E[Y |bh ≥ X > bh−1].

( Southampton Research Seminar ) Robust LH stratified sampling strategy 13 / 37

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SLIDE 28

Sampling Strategy

Robust HL sampling strategy

Idea: write moments weighting for the presence of contaminated data using a S-estimator for regression and scale. S-estimators of regression S(x, y) = arg min

β s(r1(β), ..., rN(β))

with ri(β) regressions residuals and s scale measure which solves 1 N

N

  • i=1

ρ(ri(β) s ) = b for a conveniently chosen ρ function and a constant b. Robust w.r.t. both vertical outliers and leverage points.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 14 / 37

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SLIDE 29

Sampling Strategy

Robust HL sampling strategy

Then use the approximation (expanding ρ(·)) Var[Y |bh ≥ X > bh−1] ≈ eσ2ψh/Wh − (φh/Wh)2 where Wh = bh

bh−1

ω(xβ)f (x)dx φh = bh

bh−1

xβω(xβ)f (x)dx ψh = bh

bh−1

x2βω(xβ)f (x)dx ω(x) = ρ′(x)/x weighting function.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 15 / 37

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Sampling Strategy

Robust HL sampling strategy

Some choices of ρ(x) and ω(x) = ρ′(x)/x

( Southampton Research Seminar ) Robust LH stratified sampling strategy 16 / 37

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SLIDE 31

Sampling Strategy

Robust HL sampling strategy

Solve for bounds b1, . . . , bh, . . . , bL minimizing n in the Neyman allocation scheme n^

tystrat

= NL + (L−1

h=1(eσ2ψhWh − φ2 h)1/2)2

(c xβ

i /N)2 + L−1 h=1 (eσ2ψh−φ2

h/Wh)

N

where β and σ are the parameters of the log-linear model estimated robustly. Use Sethi’s algorithm, for given L and precision c.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 17 / 37

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SLIDE 32

Simulation study

Simulation study

( Southampton Research Seminar ) Robust LH stratified sampling strategy 18 / 37

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SLIDE 33

Simulation study

Simulation study

Description of the experiment: Build a stratified sample for surveying value added (=y) in the industry of Constructions, by

( Southampton Research Seminar ) Robust LH stratified sampling strategy 19 / 37

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SLIDE 34

Simulation study

Simulation study

Description of the experiment: Build a stratified sample for surveying value added (=y) in the industry of Constructions, by h = economic-size class

( Southampton Research Seminar ) Robust LH stratified sampling strategy 19 / 37

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SLIDE 35

Simulation study

Simulation study

Description of the experiment: Build a stratified sample for surveying value added (=y) in the industry of Constructions, by h = economic-size class Population is generated from log yi = β log xi + εi

( Southampton Research Seminar ) Robust LH stratified sampling strategy 19 / 37

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SLIDE 36

Simulation study

Simulation study

Description of the experiment: Build a stratified sample for surveying value added (=y) in the industry of Constructions, by h = economic-size class Population is generated from log yi = β log xi + εi X = turnover (VAT register), β = .75

( Southampton Research Seminar ) Robust LH stratified sampling strategy 19 / 37

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SLIDE 37

Simulation study

Simulation study

Design of the experiment

1 design 1: εi ∼ N(0, 1) no outliers 2 design 2: εi ∼ Cauchy1 long-tailed errors 3 design 3: εi ∼ t3 long-tailed errors 4 design 4: δ% of εi ∼ N(5

  • χ2

1;0.99, 1.5) vertical outliers

5 design 5: δ% of εi ∼ N(10, 10) and corresponding X ∼ N(−10, 10)

bad leverage points Contamination δ = 15% and 30%.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 20 / 37

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SLIDE 38

Simulation study

Simulation study

Design of the experiment

1 design 1: εi ∼ N(0, 1) no outliers 2 design 2: εi ∼ Cauchy1 long-tailed errors 3 design 3: εi ∼ t3 long-tailed errors 4 design 4: δ% of εi ∼ N(5

  • χ2

1;0.99, 1.5) vertical outliers

5 design 5: δ% of εi ∼ N(10, 10) and corresponding X ∼ N(−10, 10)

bad leverage points Contamination δ = 15% and 30%.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 20 / 37

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SLIDE 39

Simulation study

Simulation study

Design of the experiment

1 design 1: εi ∼ N(0, 1) no outliers 2 design 2: εi ∼ Cauchy1 long-tailed errors 3 design 3: εi ∼ t3 long-tailed errors 4 design 4: δ% of εi ∼ N(5

  • χ2

1;0.99, 1.5) vertical outliers

5 design 5: δ% of εi ∼ N(10, 10) and corresponding X ∼ N(−10, 10)

bad leverage points Contamination δ = 15% and 30%.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 20 / 37

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SLIDE 40

Simulation study

Simulation study

6 strata bounds on variable X, following the SBS practice (= 1 take-all + 5 take-some)

( Southampton Research Seminar ) Robust LH stratified sampling strategy 21 / 37

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SLIDE 41

Simulation study

Simulation study

6 strata bounds on variable X, following the SBS practice (= 1 take-all + 5 take-some)

1 generalized HL method 2 naive robust generalized HL method 3 robust generalized HL method

at 1% precision using X as auxiliary information

( Southampton Research Seminar ) Robust LH stratified sampling strategy 21 / 37

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SLIDE 42

Simulation study

Simulation study

6 strata bounds on variable X, following the SBS practice (= 1 take-all + 5 take-some)

1 generalized HL method 2 naive robust generalized HL method 3 robust generalized HL method

at 1% precision using X as auxiliary information Stratification performance is evaluated by MSE comparisons.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 21 / 37

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SLIDE 43

Simulation study

Robust generalization of HL design vs LS-based stratification

HT Estimator for the mean

Design MSE(y NR−GLH)/MSE(y GLH) MSE(y R−GLH)/MSE(y GLH) No outliers 4.52 0.10 Long-tailed Cauchy 0.07 0.00 Long-tailed t 6.79 0.08 Vertical outliers 15% 1.01 0.99 Leverage points 15% 4.52 0.00 Vertical outliers 30% 0.99 0.99 Leverage points 30% 0.11 0.00 Table: MSE comparisons of NR-GLH and R-GLH methods versus GLH (Rivest, 2002), target precision: 1%

( Southampton Research Seminar ) Robust LH stratified sampling strategy 22 / 37

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SLIDE 44

Simulation study

Time dependent survey variables

( Southampton Research Seminar ) Robust LH stratified sampling strategy 23 / 37

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SLIDE 45

Stratified Sampling for time dependent survey variables

Stratified Sampling for time dependent survey variables

Surveys where target variables are characterized by a strong time dependence → not only the dispersion across population units is taken into account, but also the variability through time and the ability of forecasting future values of the target variable conditional to the values observed in the past.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 24 / 37

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SLIDE 46

Stratified Sampling for time dependent survey variables

Short-term statistics on industry

Example Monthly survey on producer prices for industrial products sold on the domestic market Type of mean Model from 31-01-2003 to 31-12-2008 Paper ISTAT W/142BIS since 01-01-2009 Paper/electronic P311 Observed units: Industrial products sold on the domestic market (producer prices)

( Southampton Research Seminar ) Robust LH stratified sampling strategy 25 / 37

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SLIDE 47

Stratified Sampling for time dependent survey variables

The survey design

( Southampton Research Seminar ) Robust LH stratified sampling strategy 26 / 37

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SLIDE 48

Stratified Sampling for time dependent survey variables

The survey design

( Southampton Research Seminar ) Robust LH stratified sampling strategy 27 / 37

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SLIDE 49

Stratified Sampling for time dependent survey variables

The questionnaire

( Southampton Research Seminar ) Robust LH stratified sampling strategy 28 / 37

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SLIDE 50

Stratified Sampling for time dependent survey variables

Objective

Include the temporal pattern of the target variable in order to improve the sampling design.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 29 / 37

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SLIDE 51

Stratified Sampling for time dependent survey variables

2 main settings

1 Strata bounds (b(t)

1 , . . . , b(t) L ) at time t are fixed

and equal to (b(t−1)

1

, . . . , b(t−1)

L

). Then, only points 2 and 3 are considered.

2 Design is reviewed: strata bounds at time t are obtained according to

an optimization algorithm (HL with some modification for time dependency structure)

( Southampton Research Seminar ) Robust LH stratified sampling strategy 30 / 37

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SLIDE 52

Stratified Sampling for time dependent survey variables

2 main settings

1 Strata bounds (b(t)

1 , . . . , b(t) L ) at time t are fixed

and equal to (b(t−1)

1

, . . . , b(t−1)

L

). Then, only points 2 and 3 are considered.

2 Design is reviewed: strata bounds at time t are obtained according to

an optimization algorithm (HL with some modification for time dependency structure)

( Southampton Research Seminar ) Robust LH stratified sampling strategy 30 / 37

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SLIDE 53
  • 1. Fixed strata bounds

Time dependent target variable

Let yt = βxt + ut where ut = ρut−1 + εt is an AR(1), εt ∼ WN(0, σ2). Set the transformations y∗

t = yt − ρyt−1

x∗

t = xt − ρxt−1

Then s2

y∗

h,t = s2

yh,t −

nh nh − 1ρ2 s2

yh,t−1 + (yt − yt−1)2

, s2

yh,t =

1 nh − 1

  • k∈Sh

(yk − ^ yh)2

( Southampton Research Seminar ) Robust LH stratified sampling strategy 31 / 37

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SLIDE 54
  • 1. Fixed strata bounds

Sample size determination

From the usual result (variance minimization of the HT estimator of total in stratified design) n^

tystrat

= NL + L−1

h=1 W 2

h s2 yh

ah

(cY /N)2 + L−1

h=1 Wh N s2 yh

with Neyman allocation rule, then n∗

^ tystrat

= NL + (L−1

h=1 Nhsy∗

h,t)2

(cY

∗)2 + L−1 h=1 Nhs2 y∗

h,t

with s2

y∗

h,t obtained from the previous formula. ( Southampton Research Seminar ) Robust LH stratified sampling strategy 32 / 37

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SLIDE 55
  • 1. Fixed strata bounds

Caveat

When autocorrelation is not taken into account, sample sizes are unduly inflated. → Need adjustment for AR errors to reduce sample sizes.

n∗ = N∗

L +

L−1

h=1 N2 hs2 yht /ah − ρ2

L−1

h=1 N2 h^

s2

yh,t−1|t + L−1 h=1 Nh ahNh−1 (yTOT t

− yTOT

t−1 )2

  • (cy)2 + L−1

h=1 Nhs2 yht − ρ2

L−1

h=1 Nh^

s2

yh,t−1|t + L−1 h=1 ah ahNh−1 (yTOT t

− yTOT

t−1 )2

  • ( Southampton Research Seminar )

Robust LH stratified sampling strategy 33 / 37

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SLIDE 56
  • 2. Time-dependent Stratified design

Main idea

Stratification at time t − 1: (b(t−1)

1

, . . . , b(t−1)

L

) Let log yit = β log xit + uit with uit = ρui,t−1 + εit, εit ∼ WN(0, σ2) Define y∗

it =

yit yρ

i,t−1

x∗

it =

xit xρ

i,t−1

( Southampton Research Seminar ) Robust LH stratified sampling strategy 34 / 37

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SLIDE 57
  • 2. Time-dependent Stratified design

Strata bounds determination

Then use Var[Y ∗|b(t−1)

h

≥ X ∗ > b(t−1)

h−1 ] ≈ eσ2

∗ψ∗

h/W ∗ h − (φ∗ h/W ∗ h )2

where W ∗

h

= b(t−1)

h

b(t−1)

h−1

b(t)

h

b(t)

h−1

f (u, v)dudv φ∗

h

= b(t−1)

h

b(t−1)

h−1

b(t)

h

b(t)

h−1

(u, v−ρ)βf (u, v)dudv ψ∗

h

= b(t−1)

h

b(t−1)

h−1

b(t)

h

b(t)

h−1

(u, v−ρ)2βf (u, v)dudv

( Southampton Research Seminar ) Robust LH stratified sampling strategy 35 / 37

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SLIDE 58
  • 2. Time-dependent Stratified design

Solution

Solve for bounds b(t)

1 , . . . , b(t) h , . . . , b(t) L

minimizing n∗ in the Neyman allocation scheme n∗

^ tystrat

= NL + (L−1

h=1(Var[Y ∗|b(t−1) h

≥ X ∗ > b(t−1)

h−1 ])1/2)2

(c xβ

i /N)2 + L−1 h=1 Var[Y ∗|b(t−1)

h

≥X ∗>b(t−1)

h−1 ]

N

where β and σ are the parameters of the log-linear model. Use Sethi’s algorithm, for given L and precision c.

( Southampton Research Seminar ) Robust LH stratified sampling strategy 36 / 37

slide-59
SLIDE 59
  • 2. Time-dependent Stratified design

Agenda

1 Simulation study 2 Example in Short Term Statistics 3 Further issues on estimation and robustness, AR order choice...

Other issues: stratification based on more than 1 target variable: multiple survey

  • ptimal stratification

. . .

( Southampton Research Seminar ) Robust LH stratified sampling strategy 37 / 37

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SLIDE 60
  • 2. Time-dependent Stratified design

Agenda

1 Simulation study 2 Example in Short Term Statistics 3 Further issues on estimation and robustness, AR order choice...

Other issues: stratification based on more than 1 target variable: multiple survey

  • ptimal stratification

. . .

( Southampton Research Seminar ) Robust LH stratified sampling strategy 37 / 37