dual%and%mul)ple%frame%survey%% - european commission frame survey.pdf · combining landline and...

Dual and Mul)ple Frame Survey

Fulvia Meca4 University of Milano-‐Bicocca, Italy

Fulvia Meca4

•  Professor of Sta)s)cs Dpt Sociology & Social Research, UniMiB

•  Director of the PhD program in Sta)s)cs, UniMiB (since 2010)

•  Past-‐President of S2G-‐ Survey Sampling Group of the Italian Sta)s)cal Society (2009-‐2011)

Fulvia Meca4

•  Research on DF & MF survey since 2003

•  Joint with AC Singh since 2007 Sta)s)cs Research & Innova)on Division since 2010 Center of Excellence in Survey Research •  Joint with C. Ferraz since 2014 Federal University of Pernambuco – Brazil FAO – Rome, Italy

Wednesday, 16 April 2008

Fulvia Mecatti, PhD Department of Statistics University of Milan-Bicocca Via Bicocca degli Arcimboldi, 8 20126 Milano, Italy Dear Professor Mecatti: I would like to invite you to visit Statistics Canada from July 7 to August 1, 2008 to continue your joint research with Avi Singh. I understand that your expenses will be totally covered by your Department at the University of Milan-Bicocca. Statistics Canada would provide you office space during your stay. Sincerely yours Mike Hidiroglou Director, Statistics Research and Innovation Division Statistics Canada 150 Tunney's Pasture Driveway Ottawa, Ontario K1A 0T6

MF vs Conven)onal Survey

•  Unique List

•  Complete coverage

•  Up-‐to-‐date

Perfect frame

UA B C Q > 2 U1 · · ·UQ

1


•  Under-‐coverage (Ex. Par)al list)

•  Over-‐coverage (Ex Rare Pop’s)

•  Unavailable direct frame (Ex. Difficult-‐to-‐Sample-‐Pop’s, Hidden, Elusive, Mobile …)

Perfect frame Imperfect frame

UA B C Q > 2 U1 · · ·UQ

1


Conven)onal unique-‐frame survey

DF & MF survey

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

•  •  Par)al

•  Overlapping (unknown)

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

MF Advantages & Applica)ons

•  @Selec)on stage of the survey

•  Improving Coverage & Response rate

•  Dealing with Under-‐Coverage Bias in telephone surveys

•  Flexibility & Blended survey modes

•  Cost effec)ve

Surveying difficult-‐to-‐sample pop’s (Hidden, Elusive, Mobile…)

•  Iacan & Dennis (1993): Homeless Survey, USA, Sampling on loca)on, 3-‐frames setup 1)Homeless shelter; 2) Soup kitchen; 3) Street loca)ons

•  Eurostat (2000) Italian Country Report: Interna)onal Migra)on Survey (legal & illegal), “Center sampling”

Etnic shops & resturants; unofficial medical facili)es; specific city areas (Chinatown, Central Rail Sta)on…)

Surveying difficult-‐to-‐sample pop’s (Hidden, Elusive, Mobile…)

•  ISTAT (2011-‐12): 1st Italian Survey on homeless people (research project 2009), Indirect Sampling (related to MF setup)

•  (2012) DF survey to assess )me– and space–related changes of the colonizing wolf popula)on in France (Office na/onal de la chasse et de la faune sauvage, Applied Research Unit on Predator and Depredator species, Parc Micropolis, F-‐05000 Gap, France)

Poten)al & Improving Precision

•  <<As the U.S., Canada, and other na)ons grow in diversity, different sampling frames may beier capture subgroups of the popula)on. […] We an)cipate that modular sampling designs using mul)ple frames will be widely used in the future.>> (Lohr & Rao, JASA 2006)

Dealing with Under-‐coverage Bias in Telephone Surveys

•  Households with cell phone only not included in the tradi)onal frame for landline telephone survey

•  Emerging evidence of significant differences between household with or/and without landline

Dealing with Under-‐coverage Bias in Telephone Surveys Combining landline and mobile phone samples: A dual frame approach 9

Table 1: Percent of households with no landlines and individuals with mobile phones, results from

the European Social Survey Round 4, weighted data, countries in alphabetical order

Country No landline phone in household (in %)

Personally have mobile phone (in %)

Belgium 23.1 89.4

Bulgaria 43.5 72.2

Croatia 12.5 85.2

Cyprus 10.5 85.2

Czech Republic 73.3 92.0

Denmark 20.1 92.5

Estonia 42.6 90.2

Finland 65.6 95.2

France 7.6 82.6

Germany 6.1 84.9

Greece 15.8 90.8

Hungary 46.2 79.5

Israel 12.3 89.9

Latvia 56.5 82.0

Netherlands 12.3 91.3

Norway 28.1 96.6

Poland 34.1 80.2

Portugal 35.0 81.7

Romania 59.4 77.4

Russian Federation 48.3 71.7

Slovakia 45.2 85.6

Slovenia 10.2 87.9

Spain 24.7 82.0

Sweden 8.9 92.0

Switzerland 10.2 88.6

Turkey 38.3 66.7

Ukraine 33.2 77.1

United Kingdom 11.7 85.3

In North America, Canada shows a slower trend with only 8% of CPO-HH as of December 2008 (Statis-

tics Canada, 2009). On the other hand, the United States is following the European trend (Blumberg &

Luke, 2009). In the first half of 2009, 22.7 percent of all households were cell phone-only households.

More importantly, differences between cell phone-only households and non cell phone-only households

Combining landline and mobile phone samples: A dual frame approach 9

Table 1: Percent of households with no landlines and individuals with mobile phones, results from

the European Social Survey Round 4, weighted data, countries in alphabetical order

Country No landline phone in household (in %)

Personally have mobile phone (in %)

Belgium 23.1 89.4

Bulgaria 43.5 72.2

Croatia 12.5 85.2

Cyprus 10.5 85.2

Czech Republic 73.3 92.0

Denmark 20.1 92.5

Estonia 42.6 90.2

Finland 65.6 95.2

France 7.6 82.6

Germany 6.1 84.9

Greece 15.8 90.8

Hungary 46.2 79.5

Israel 12.3 89.9

Latvia 56.5 82.0

Netherlands 12.3 91.3

Norway 28.1 96.6

Poland 34.1 80.2

Portugal 35.0 81.7

Romania 59.4 77.4

Russian Federation 48.3 71.7

Slovakia 45.2 85.6

Slovenia 10.2 87.9

Spain 24.7 82.0

Sweden 8.9 92.0

Switzerland 10.2 88.6

Turkey 38.3 66.7

Ukraine 33.2 77.1

United Kingdom 11.7 85.3

In North America, Canada shows a slower trend with only 8% of CPO-HH as of December 2008 (Statis-

tics Canada, 2009). On the other hand, the United States is following the European trend (Blumberg &

Luke, 2009). In the first half of 2009, 22.7 percent of all households were cell phone-only households.

More importantly, differences between cell phone-only households and non cell phone-only households


•  GESIS-‐Leibniz Ins)tut für Sozialwissenschaqen (2011): CELLA1, German Research Funda)on, DF telephone survey (Landline&Cell)

•  (2014) Australian Gambling DF telephone survey: <<[…] reliance on a tradi)onal landline telephone sampling approach effec)vely excludes dis)nct subgroups of the popula)on […]>>


•  (2002, Brick et al) Survey of America’s Facili)es, DF setup RDD households with landline telephone + area frame of households without landline telephone

•  (2011) USA Behavioral Risk Factor Surveillance System <<[…] documented that samples from both frames are needed to avoid bias in key es)mates[…]>>

•  Avoiding Screenings, before (MF è unique-‐frame) and aqer sampling (de-‐duplica)on)

•  Rare Popula)ons: surveying rare traits with eligible units confused in a larger frame (Ex: WHO TB Prevalence Survey, posi)vity to TB @na)onal level)

Cost Effec)ve

•  Complete Area Frame + supplementary List Frame (Ex: 2014 FAO guidelines for Agricultural Sta)s)cs)

Cost Effec)ve

UA B C L Q > 2 U1 · · ·UQ

1

UA B C L Q > 2 U1 · · ·UQ

1

•  Complete & naturally updated

•  Expensive to sample (local visit)

•  Par)al, overlapping & possibly quickly out-‐of-‐date

•  cheaper & easier to sample

UA B C L Q > 2 U1 · · ·UQ

1

UA B C L Q > 2 U1 · · ·UQ

1

•  ≠ Sampling Design in ≠ Frames •  ≠ Collec)on Mode in ≠ Frames Ex: Web survey with ≠ different webpages; survey combining telephone, email and face-‐to-‐face interviews

Flexibility & Blended Survey Mode

MF Challenges & Issues

UA B C L Q > 2 U1 · · ·UQ

1

•  @es)ma)on stage of the survey

•  Final sample = independent frame-‐samples

•  Overlap è mul)ple selec)on opportuni)es of the same units into the final sample

1s3s

2s

è  combining data from the independent random samples UA B C L Q > 2 U1 · · ·UQ

2

1

MF Challenges & Issues

1.  Increased inclusion probabili)es for unit overlap whether or not the unit is in fact selected in the final sample

2.  Possibly overlapping frame-‐samples (duplicated unit) whether or not duplica)ons actually occur in the final sample

èEfficient MF es)ma)on requires addi)onal frame-‐level info to be collected beside the study variable in order to address such issues

UA B C L Q > 2 U1 · · ·UQ

2

1

First proposal: (1962) Hartley

•  DF •  Op)mal es)mator

•  Frame-‐level info: Domain membership: è domain classifica)on (post-‐stra)fica)on)

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C L Q > 2 U1 · · ·UQ

2a b ab sA sB sa(A) sab(A) sb(B) sab(B)

1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

“To which frames”

Op)mal Hartley es)mator

UA B C L Q > 2 U1 · · ·UQ


YH = Ya(A) + ↵H Yab(A) + (1� ↵H)Yab(B) + Yb(B)

Y =X

i2Ayi +

X

i2ab

yi +X

i 2 byi

↵H = min↵2(0,1)

V⇣YH

⌘

1

•  Unbiased for pop total

•  Op)mal

•  Approx unbiased & sub-‐op)mal in prac)ce

UA B C L Q > 2 U1 · · ·UQ



Y =X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H = min↵2(0,1)

V⇣YH

⌘

1

UA B C L Q > 2 U1 · · ·UQ



Y =X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H = min↵2(0,1)

V⇣YH

⌘

1

Several DF es)mators followed…

•  Under ≠ approaches •  DF •  Increasing complexity & frame-‐level info needs

Several DF es)mators followed…

•  ML & Regression for SRS (Fuller&Burmeister, 1972)

•  Pseudo ML for complex design (Skinner&Rao, 1996; Lohr&Rao, 2006)

•  Modified regession (Singh&Wu 1996)

•  Empirical ML (Rao&Wu, 2010)

(1986) Kalton-‐Anderson es)mator

•  Unbiased

•  Empirical evidence of High Efficiency

•  More frame-‐level info needed (Domain classifica)on + inclusion probability from every frame)

UA B C L Q > 2 U1 · · ·UQ



Y =X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H = min↵2(0,1)

V⇣YH

⌘

YKA = Ya(A) +⇡A

⇡A + ⇡BYab(A) +

⇡B⇡A + ⇡B

Yab(B) + Yb(B)

1

Main Issues & Challenges limi)ng applica)on

•  Frame-‐level info needs •  Exten)on DF èMF (Nota)on)

•  Anali)cal&Computa)onal complexity

•  Assessing accuracy via variance es)ma)on (& CI)

Mul)plicity approach to MF es)ma)on

(Meca4 2007; Singh&Meca4 2011; Meca4&Singh 2014) •  Simplified Nota)on

•  Simple Mul)plicity MF es)mator

•  Basic frame-‐level info needed

•  Unbiased exact variance es)ma)on

Mul)plicity approach to MF es)ma)on

•  Generalized MF Mul)plicity Es)mators (GMHT) •  Simplifying & Unifying DF/MF approaches & es)mators

From the (complex) alphabe)cal nota)on ….

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ



Y =X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H = min↵2(0,1)

V⇣YH

⌘

YKA = Ya(A) +⇡A

⇡A + ⇡BYab(A) +

⇡B⇡A + ⇡B

Yab(B) + Yb(B)

ac bc abcsa(A) sb(B) sc(C)

sab(A) sab(B) sbc(C)

sac(A) sbc(B) sac(C)

sabc(A) sabc(B) sabc(C)

1

UA B C L Q > 2 U1 · · ·UQ



Y =X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H = min↵2(0,1)

V⇣YH

⌘

YKA = Ya(A) +⇡A

⇡A + ⇡BYab(A) +

⇡B⇡A + ⇡B

Yab(B) + Yb(B)





1

UA B C L Q > 2 U1 · · ·UQ



Y =X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H = min↵2(0,1)

V⇣YH

⌘

YKA = Ya(A) +⇡A

⇡A + ⇡BYab(A) +

⇡B⇡A + ⇡B

Yab(B) + Yb(B)





1

UA B C L Q > 2 U1 · · ·UQ


ˆYH =

ˆYa(A) + ↵HˆYab(A) + (1� ↵H

)

ˆYab(B) +ˆYb(B)

Y =

X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H= min

↵2(0,1)V⇣ˆYH

⌘

ˆYKA = Ya(A) +⇡A

⇡A + ⇡BˆYab(A) +

⇡B⇡A + ⇡B

ˆYab(B) +ˆYb(B)

c ac bc abcsA sB sC

sa(A) sb(B) sc(C)




Q Frames ! U1 · · ·Uq · · ·UQ

Frame size ! N1 · · ·Nq · · ·NQ

Dq Domain each ! Ud(q), d = 1 . . . Dq

Q Frame-samples ! s1 · · · sq · · · sQFrame-sample size (fixed) ! n1 · · ·nq, · · ·nQ

Dq Domain-samples each ! sd(q), d = 1 . . . Dq

Domain-sample size (random) ! nd(q), d = 1 . . . Dq

1

From the (complex) alphabe)cal nota)on ….

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C Q > 2 U1 · · ·UQ

1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ


1

UA B C L Q > 2 U1 · · ·UQ



Y =X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H = min↵2(0,1)

V⇣YH

⌘

YKA = Ya(A) +⇡A

⇡A + ⇡BYab(A) +

⇡B⇡A + ⇡B

Yab(B) + Yb(B)





1

UA B C L Q > 2 U1 · · ·UQ



Y =X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H = min↵2(0,1)

V⇣YH

⌘

YKA = Ya(A) +⇡A

⇡A + ⇡BYab(A) +

⇡B⇡A + ⇡B

Yab(B) + Yb(B)





1

UA B C L Q > 2 U1 · · ·UQ



Y =X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H = min↵2(0,1)

V⇣YH

⌘

YKA = Ya(A) +⇡A

⇡A + ⇡BYab(A) +

⇡B⇡A + ⇡B

Yab(B) + Yb(B)





1

UA B C L Q > 2 U1 · · ·UQ



Y =X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H = min↵2(0,1)

V⇣YH

⌘

YKA = Ya(A) +⇡A

⇡A + ⇡BYab(A) +

⇡B⇡A + ⇡B

Yab(B) + Yb(B)

ac bc abcsA sB sC

sa(A) sb(B) sc(C)




1

UA B C L Q > 2 U1 · · ·UQ


ˆYH =


)

ˆYab(B) +ˆYb(B)

Y =

X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H= min

↵2(0,1)V⇣ˆYH

⌘

ˆYKA = Ya(A) +⇡A


⇡B⇡A + ⇡B

ˆYab(B) +ˆYb(B)

c ac bc abcsA sB sC

sa(A) sb(B) sc(C)






Dq Domain each ! Ud(q), d = 1 . . . Dq




1

… To a (simpler) indexed nota)on

UA B C L Q > 2 U1 · · ·UQ


ˆYH =


)

ˆYab(B) +ˆYb(B)

Y =

X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H= min

↵2(0,1)V⇣ˆYH

⌘

ˆYKA = Ya(A) +⇡A


⇡B⇡A + ⇡B

ˆYab(B) +ˆYb(B)

c ac bc abcsA sB sC

sa(A) sb(B) sc(C)






Dq Domains each ! Ud(q), d = 1 . . . Dq




1

Mul)plicity: a simple no)on for MF es)ma)on

UA B C L Q > 2 U1 · · ·UQ


ˆYH =


)

ˆYab(B) +ˆYb(B)

Y =

X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H= min

↵2(0,1)V⇣ˆYH

⌘

ˆYKA = Ya(A) +⇡A


⇡B⇡A + ⇡B

ˆYab(B) +ˆYb(B)

c ac bc abcsA sB sC

sa(A) sb(B) sc(C)










mi = # {Uq 3 i} =

QX

q=1

1i(q)

md = #

�Uq ◆ Ud(·)

i 2 Ud(q)

mi ⌘ md

1

•  Unit mul)plicity: •  Domain mul)plicity:

“How many frames”

UA B C L Q > 2 U1 · · ·UQ


ˆYH =


)

ˆYab(B) +ˆYb(B)

Y =

X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H= min

↵2(0,1)V⇣ˆYH

⌘

ˆYKA = Ya(A) +⇡A


⇡B⇡A + ⇡B

ˆYab(B) +ˆYb(B)

c ac bc abcsA sB sC

sa(A) sb(B) sc(C)










mi = # {Uq 3 i} =

QX

q=1

1i(q)

md = #

�Uq ◆ Ud(·)

i 2 Ud(q)

mi ⌘ md

1

Mul)plicity: a simple no)on for MF es)ma)on

•  Rela)on: all share the same mul)plicity è

•  Pop total:

UA B C L Q > 2 U1 · · ·UQ


ˆYH =


)

ˆYab(B) +ˆYb(B)

Y =

X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H= min

↵2(0,1)V⇣ˆYH

⌘

ˆYKA = Ya(A) +⇡A


⇡B⇡A + ⇡B

ˆYab(B) +ˆYb(B)

c ac bc abcsA sB sC

sa(A) sb(B) sc(C)










mi = # {Uq 3 i} =

QX

q=1

1i(q)

md = #

�Uq ◆ Ud(·)

i 2 Ud(q)

mi ⌘ md

1

UA B C L Q > 2 U1 · · ·UQ


ˆYH =


)

ˆYab(B) +ˆYb(B)

Y =

X

i2ayi +

X

i2ab

yi +X

i2b

yi

↵H= min

↵2(0,1)V⇣ˆYH

⌘

ˆYKA = Ya(A) +⇡A


⇡B⇡A + ⇡B

ˆYab(B) +ˆYb(B)

c ac bc abcsA sB sC

sa(A) sb(B) sc(C)










mi = # {Uq 3 i} =

QX

q=1

1i(q)

md = #

�Uq ◆ Ud(·)

i 2 Ud(q)

mi ⌘ md

1Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

2

Simple Mul)plicity MF es)mator •  Conven)onal (unbiased) unique-‐frame HT es)mator:

•  Simple Mul)plicity (unbiased) MF es)mator:

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

2

Exact Unbiased Variance es)ma)on

•  Fixed frame-‐sample sizes

•  Customary SYG form

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yj

mj ⇡j(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq

d=1 1i2Ud(q)PQq0=1 ⇡i(q0)1i2Ud(q0)

!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3 yi, i = 1 . . . 12

2

GMHT: a class of MF es)mator •  Generalized mul)plicity-‐adjustment

•  GMHT class:

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

↵i(q) 2 (0, 1),QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yialphai(q) ⇡i(q)

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

↵i(q) 2 (0, 1),QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

2

Simplified & Unified MF es)ma)on •  with

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)PDq


ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)PDq


ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

2

Numerical Illustra)on

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

1

2

34

5

67

89

10 11

12

123456789

10

11

12

units k€ monthly income

1.23.54.8

2.3

2.61.8

0.8

5.31.51.72.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

3

1.23.5

4.8

2.3

2.61.8

0.8

5.3

1.51.7

2.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

3

1.23.5

4.2

2.3

2.61.8

0.8

5.3

1.51.7

2.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

SRS n = 7 n1 = 2 n2 = 2 n3 = 3

s1 ! s2 ! s3 !i yi mi ⇡i(1)

yimi ⇡i(1)

9 1.5 1 2/6 4.5

4 2.3 3 2/6 2.3

3


1.23.54.8

2.3

2.61.8

0.8

5.3

1.51.7

2.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

SRS n = 7 n1 = 2 n2 = 2 n3 = 3

i yi mi ⇡i(1)yi

mi ⇡i(1)

9 1.5 1 2/6 4.5

4 2.3 3 2/6 2.3

3

1.23.54.8

2.32.61.80.85.31.51.72.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

SRS n = 7 n1 = 2 n2 = 2 n3 = 3

i yi mi ⇡i(1)yi

mi ⇡i(1)

9 1.5 1 2/6 4.5

4 2.3 3 2/6 2.3

3

i yi mi ⇡i(q)yi

mi ⇡i(q)

9 1.5 1 2/6 4.5

4 2.3 3 2/6 2.3

2 3.5 2 2/6 5.25

8 5.3 2 2/6 7.95

10 1.7 1 3/7 3.97

6 1.8 2 3/7 2.1

7 0.8 3 3/7 0.62

= 26.69

4

1.23.5

4.2

2.3

2.61.8

0.8

5.3

1.51.7

2.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

SRS n = 7 n1 = 2 n2 = 2 n3 = 3

s1 !

s2 !

s3 !

3

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

1

2

34

5

67

89

10 11

12

Numerical Illustra)on •  SM es)mator è basic info: “how many frames

i yi mi ⇡i(q)yi

mi ⇡i(q)

9 1.5 1 2/6 4.5

4 2.3 3 2/6 2.3

2 3.5 2 2/6 5.25

8 5.3 2 2/6 7.95

10 1.7 1 3/7 3.97

6 1.8 2 3/7 2.1

7 0.8 3 3/7 0.62

= 26.69

4

1.23.5

4.2

2.3

2.61.8

0.8

5.3

1.51.7

2.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

SRS n = 7 n1 = 2 n2 = 2 n3 = 3

s1 !

s2 !

s3 !

3

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

1

2

34

5

67

89

10 11

12


Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

2

•  SM es)mator:

i yi mi ⇡i(1)yi

mi ⇡i(1)

9 1.5 1 2/6 4.5

4 2.3 3 2/6 2.3

i yi mi ⇡i(2)yi

mi ⇡i(2)

2 3.5 2 2/6 5.25

8 5.3 2 2/6 7.95

i yi mi ⇡i(3)yi

mi ⇡i(3)

10 1.7 1 3/7 3.97

6 1.8 2 3/7 2.1

7 0.8 3 3/7 0.62

= 26.69

4

i yi mi ⇡i(q)yi

mi ⇡i(q)

9 1.5 1 2/6 4.5

4 2.3 3 2/6 2.3

2 3.5 2 2/6 5.25

8 5.3 2 2/6 7.95

10 1.7 1 3/7 3.97

6 1.8 2 3/7 2.1

7 0.8 3 3/7 0.62

= 26.69

4

1.23.5

4.2

2.3

2.61.8

0.8

5.3

1.51.7

2.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

SRS n = 7 n1 = 2 n2 = 2 n3 = 3

s1 !

s2 !

s3 !

3

Numerical Illustra)on •  Other MF es)mators è more info: “to which frames”

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

1

2

34

5

67

89

10 11

12

Numerical Illustra)on •  Other MF es)mators è more info: “to which frames

•  (sample) domain classifica)on

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

246

7

89

10

Numerical Illustra)on •  Other MF es)mators è more info: “to which frames

1.23.5

4.2

2.3

2.61.8

0.8

5.3

1.51.7

2.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

SRS n = 7 n1 = 2 n2 = 2 n3 = 3

s1 !

s2 !

s3 !

3

i yi domain ⇡i(q)

⇣↵KAi(q)

⌘�1yi

↵KAi(q) ⇡i(q)

9 1.5 U1(1) 2/6 1 4.5

4 2.3 U1T

U2T

U3 2/6

⇡i(1)

⇡i(1)+⇡i(1)+⇡i(3)= 0.30 2.1

2 3.5 U2T

U3 2/6

⇡i(2)

⇡i(2)+⇡i(3)= 0.44 7.95

8 5.3 U2T

U1 2/6

⇡i(2)

⇡i(2)+⇡i(1)= 0.50 4.59

10 1.7 U1(3) 3/7 1 3.97

6 1.8 U3T

U1 3/7

⇡i(3)

⇡i(3)+⇡i(1)= 0.56 2.36

7 0.8 U3T

U1T

U2 3/7

⇡i(3)

⇡i(3)+⇡i(1)+⇡i(2)= 0.39 0.73

= 26.20

5

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

246

7

89

10

Numerical Illustra)on •  KA es)mator è “to which frames”& all prob

1.23.5

4.2

2.3

2.61.8

0.8

5.3

1.51.7

2.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

SRS n = 7 n1 = 2 n2 = 2 n3 = 3

s1 !

s2 !

s3 !

3

i yi domain ⇡i(q)

⇣↵KAi(q)

⌘�1yi

↵KAi(q) ⇡i(q)

9 1.5 U1(1) 2/6 1 4.5

4 2.3 U1T

U2T

U3 2/6

⇡i(1)

⇡i(1)+⇡i(1)+⇡i(3)= 0.30 2.1

2 3.5 U2T

U3 2/6

⇡i(2)

⇡i(2)+⇡i(3)= 0.44 7.95

8 5.3 U2T

U1 2/6

⇡i(2)

⇡i(2)+⇡i(1)= 0.50 4.59

10 1.7 U1(3) 3/7 1 3.97

6 1.8 U3T

U1 3/7

⇡i(3)

⇡i(3)+⇡i(1)= 0.56 2.36

7 0.8 U3T

U1T

U2 3/7

⇡i(3)

⇡i(3)+⇡i(1)+⇡i(2)= 0.39 0.73

= 26.20

5

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

U1 U2 U3

2

246

7

89

10


Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)PDq


ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

2

Y =

QX

q=1

DqX

d=1

Yd(q) =

QX

q=1

NqX

i=1

yimi

ˆYHT =

X

i2s

yi⇡i

ˆYSM =

QX

q=1

X

i2sq

yimi ⇡i(q)

Q = 1 ) mi = 1 ) ˆYSM ⌘= YHT

ˆV⇣ˆYSM

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

mi ⇡i(q)� yi

mi ⇡i(q)

◆2

↵i(q) 2 (0, 1),

QX

q=1

↵i(q) = 1

ˆYGMHT =

QX

q=1

X

i2sq

yi↵i(q) ⇡i(q)

ˆYKA 2 GMHT

↵KAi(q) =

⇡i(q)

PDq


!�1

ˆYKA =

QX

q=1

X

i2sq

yi↵KAi(q) ⇡i(q)

2

i yi domain ⇡i(q)⇣↵KAi(q)

⌘�1yi

↵KAi(q) ⇡i(q)

9 1.5 U1(1) 2/6 1 1.5

4 2.3 U1TU2

TU3 2/6

⇡i(1

⇡i(1)+⇡i(1)+⇡i(3)

= 0.30 2.3

2 3.5 U2TU3 2/6

⇡i(2)

⇡i(2)+⇡i(3)= 0.44 3.5

8 5.3 U2TU1 2/6

⇡i(2)

⇡i(2)+⇡i(1)= 0.50 3.5

10 1.7 U1(3) 3/7 1 1.7

6 1.8 U3TU1 3/7

⇡i(3)

⇡i(3)+⇡i(1)= 0.56 1.8

7 0.8 U3TU1

TU2 3/7

⇡i(3)

⇡i(3)+⇡i(1)+⇡i(2)= 0.39 0.8

= 26.20

5

1.23.5

4.2

2.3

2.61.8

0.8

5.3

1.51.7

2.1

3.2

N = 12

Y = 30.2

N1 = 6

N2 = 6

N3 = 7

SRS n = 7 n1 = 2 n2 = 2 n3 = 3

s1 !

s2 !

s3 !

3

i yi domain ⇡i(q)

⇣↵KAi(q)

⌘�1yi

↵KAi(q) ⇡i(q)

9 1.5 U1(1) 2/6 1 4.5

4 2.3 U1T

U2T

U3 2/6

⇡i(1)

⇡i(1)+⇡i(1)+⇡i(3)= 0.30 2.1

2 3.5 U2T

U3 2/6

⇡i(2)

⇡i(2)+⇡i(3)= 0.44 7.95

8 5.3 U2T

U1 2/6

⇡i(2)

⇡i(2)+⇡i(1)= 0.50 4.59

10 1.7 U1(3) 3/7 1 3.97

6 1.8 U3T

U1 3/7

⇡i(3)

⇡i(3)+⇡i(1)= 0.56 2.36

7 0.8 U3T

U1T

U2 3/7

⇡i(3)

⇡i(3)+⇡i(1)+⇡i(2)= 0.39 0.73

= 26.20

5

Numerical Illustra)on •  (unbiased) Variance Es)ma)on

i yi domain ⇡i(q)

⇣↵KAi(q)

⌘�1yi

↵KAi(q) ⇡i(q)

9 1.5 U1(1) 2/6 1 4.5

4 2.3 U1T

U2T

U3 2/6

⇡i(1)

⇡i(1)+⇡i(1)+⇡i(3)= 0.30 2.1

2 3.5 U2T

U3 2/6

⇡i(2)

⇡i(2)+⇡i(3)= 0.44 7.95

8 5.3 U2T

U1 2/6

⇡i(2)

⇡i(2)+⇡i(1)= 0.50 4.59

10 1.7 U1(3) 3/7 1 3.97

6 1.8 U3T

U1 3/7

⇡i(3)

⇡i(3)+⇡i(1)= 0.56 2.36

7 0.8 U3T

U1T

U2 3/7

⇡i(3)

⇡i(3)+⇡i(1)+⇡i(2)= 0.39 0.73

= 26.20

ˆV⇣ˆYGMHT

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi

↵i(q) ⇡i(q)� yj

↵j(q) ⇡j(q)

◆2

⇡ij(q) =nq(nq � 1)

Nq(Nq � 1)

5

•  SRS joint inclusion probability :

i yi domain ⇡i(q)

⇣↵KAi(q)

⌘�1yi

↵KAi(q) ⇡i(q)

9 1.5 U1(1) 2/6 1 4.5

4 2.3 U1TU2

TU3 2/6

⇡i(1)

⇡i(1)+⇡i(1)+⇡i(3)= 0.30 2.1

2 3.5 U2TU3 2/6

⇡i(2)

⇡i(2)+⇡i(3)= 0.44 7.95

8 5.3 U2TU1 2/6

⇡i(2)

⇡i(2)+⇡i(1)= 0.50 4.59

10 1.7 U1(3) 3/7 1 3.97

6 1.8 U3TU1 3/7

⇡i(3)

⇡i(3)+⇡i(1)= 0.56 2.36

7 0.8 U3TU1

TU2 3/7

⇡i(3)

⇡i(3)+⇡i(1)+⇡i(2)= 0.39 0.73

= 26.20

ˆV⇣ˆYGMHT

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi


↵j(q) ⇡j(q)

◆2

⇡ij(q) =nq(nq � 1)

Nq(Nq � 1)

5


i yi domain ⇡i(q)

⇣↵KAi(q)

⌘�1yi

↵KAi(q) ⇡i(q)

9 1.5 U1(1) 2/6 1 4.5

4 2.3 U1TU2

TU3 2/6

⇡i(1)

⇡i(1)+⇡i(1)+⇡i(3)= 0.30 2.1

2 3.5 U2TU3 2/6

⇡i(2)

⇡i(2)+⇡i(3)= 0.44 7.95

8 5.3 U2TU1 2/6

⇡i(2)

⇡i(2)+⇡i(1)= 0.50 4.59

10 1.7 U1(3) 3/7 1 3.97

6 1.8 U3TU1 3/7

⇡i(3)

⇡i(3)+⇡i(1)= 0.56 2.36

7 0.8 U3TU1

TU2 3/7

⇡i(3)

⇡i(3)+⇡i(1)+⇡i(2)= 0.39 0.73

= 26.20

ˆV⇣ˆYGMHT

⌘=

QX

q=1

X

i<j2sq

⇡i(q) ⇡j(q) � ⇡ij(q)⇡ij(q)

✓yi


↵j(q) ⇡j(q)

◆2

⇡ij(q) =nq(nq � 1)

Nq(Nq � 1)

⇡ij(q)⇡i(q) ⇡j(q)�⇡ij(q)

⇡ij(q)

⇣yi


↵j(q) ⇡j(q)

⌘2product

SMs1 ! 2/30 0.67 4.84 3.23

s2 ! 2/30 0.67 7.29 4.86

s3 ! 6/49 0.29 3.48 1

11.19 3.2

2.18 0.62

KAs1 ! 2/30 0.67 5.76 3.84

s2 ! 2/30 0.67 11.26 7.51

s3 ! 6/49 0.29 2.57 0.74

10.47 2.99

2.66 0.76

5

ˆV⇣ˆYSM

⌘= 12.90(SE = 3.59)

ˆV⇣ˆYKA

⌘= 15.84(SE = 3.98)

6

ˆV⇣ˆYSM

⌘= 12.90(SE = 3.59)

ˆV⇣ˆYKA

⌘= 15.84(SE = 3.98)

6


ˆYSM = 26, 69 ˆV = 12.90 (SE = 3.59)

ˆYKA = 26, 20 ˆV = 15.84 (SE = 3.98)

6

Q&A

Ques)ons ?

dual%and%mul)ple%frame%survey%% - european commission frame survey.pdf · combining landline and...

Documents