nonparametric maximum likelihood estimation (mle) for bivariate censored data
DESCRIPTION
Nonparametric maximum likelihood estimation (MLE) for bivariate censored data. Marloes H. Maathuis advisors: Piet Groeneboom and Jon A. Wellner. Motivation. Estimate the distribution function of the incubation period of HIV/AIDS: Nonparametrically Based on censored data: - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/1.jpg)
Nonparametric maximum likelihood estimation (MLE)
for bivariate censored data
Marloes H. Maathuis
advisors:
Piet Groeneboom and Jon A. Wellner
![Page 2: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/2.jpg)
Motivation
Estimate the distribution function of the
incubation period of HIV/AIDS:– Nonparametrically– Based on censored data:
• Time of HIV infection is interval censored
• Time of onset of AIDS is interval censored
or right censored
![Page 3: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/3.jpg)
Approach
• Use MLE to estimate the bivariate distribution
• Integrate over diagonal strips: P(Y-X ≤ z) X (HIV)
Y (AIDS)
z
![Page 4: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/4.jpg)
Main focus of the project
• MLE for bivariate censored data:– Computational aspects– (In)consistency and methods to repair the
inconsistency
![Page 5: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/5.jpg)
Main focus of the project
• MLE for bivariate censored data:– Computational aspects– (In)consistency and methods to repair the
inconsistency
![Page 6: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/6.jpg)
1980
1992
1996
1980 1983 1986 X (HIV)
Y (AIDS)In
terv
al o
f on
set
of A
IDS
Interval ofHIV infection
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1980
1992
1996
1980 1983 1986 X (HIV)
Y (AIDS)In
terv
al o
f on
set
of A
IDS
Interval ofHIV infection
Observation rectangle Ri
![Page 8: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/8.jpg)
X (HIV)
Y (AIDS)
Observation rectangle Ri
1
max log ,n
F F i i ii
P X Y R
F
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X (HIV)
Y (AIDS)Maximal intersections
Observation rectangle Ri
1
max log ,n
F F i i ii
P X Y R
F
![Page 10: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/10.jpg)
X (HIV)
Y (AIDS)Maximal intersections
Observation rectangle Ri
1
max log ,n
F F i i ii
P X Y R
F
![Page 11: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/11.jpg)
X (HIV)
Y (AIDS)Maximal intersections
Observation rectangle Ri
1
max log ,n
F F i i ii
P X Y R
F
![Page 12: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/12.jpg)
X (HIV)
Y (AIDS)Maximal intersections
Observation rectangle Ri
1
max log ,n
F F i i ii
P X Y R
F
![Page 13: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/13.jpg)
X (HIV)
Y (AIDS)Maximal intersections
Observation rectangle Ri
1
max log ,n
F F i i ii
P X Y R
F
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α1 α2
α3 α4
X (HIV)
Y (AIDS)Maximal intersections
Observation rectangle Ri
1
max log ,n
F F i i ii
P X Y R
F
s.t. and
4 1 1 3max log( ) log( )
1 2 2 4log( ) log( )
3 4log( )
0, 1, , 4,i i 4
1
1ii
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3/5 0
0 25
X (HIV)
Y (AIDS)Maximal intersections
Observation rectangle Ri
The αi’s are not always uniquely determined: mixture non uniqueness
1
max log ,n
F F i i ii
P X Y R
F
s.t. and
4 1 1 3max log( ) log( )
1 2 2 4log( ) log( )
3 4log( )
0, 1, , 4,i i 4
1
1ii
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Computation of the MLE
• Reduction step:
determine the maximal intersections
• Optimization step:
determine the amounts of mass assigned to the maximal intersections
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Computation of the MLE
• Reduction step:
determine the maximal intersections
• Optimization step:
determine the amounts of mass assigned to the maximal intersections
![Page 18: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/18.jpg)
Existing reduction algorithms
• Betensky and Finkelstein (1999, Stat. in Medicine) • Gentleman and Vandal (2001, JCGS) • Song (2001, Ph.D. thesis) • Bogaerts and Lesaffre (2003, Tech. report)
The first three algorithms are very slow,
the last algorithm is of complexity O(n3).
![Page 19: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/19.jpg)
New algorithms
• Tree algorithm
• Height map algorithm: – based on the idea of a height map of the
observation rectangles– very simple– very fast: O(n2)
![Page 20: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/20.jpg)
1
11
1000
0
11
33
2110
0
21
33
2121
0
21
22
1011
0
10
11
0011
0
00
21
1122
1
00
11
0011
0
00
00
0011
0
0
Height map algorithm: O(n2)
1
22
2110
0
2
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Main focus of the project
• MLE of bivariate censored data:– Computational aspects – (In)consistency and methods to repair the
inconsistency
![Page 23: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/23.jpg)
HIV
AIDS
u1 u2
Time of HIV infection is interval censored case 2
![Page 24: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/24.jpg)
HIV
AIDS
u1 u2
Time of HIV infection is interval censored case 2
![Page 25: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/25.jpg)
HIV
AIDS
u1 u2
Time of HIV infection is interval censored case 2
![Page 26: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/26.jpg)
HIV
AIDS
t = min(c,y)
u1 u2
Time of onset of AIDS is right censored
![Page 27: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/27.jpg)
HIV
AIDS
t = min(c,y)
u1 u2
Time of onset of AIDS is right censored
![Page 28: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/28.jpg)
HIV
AIDS
t = min(c,y)
u1 u2
Time of onset of AIDS is right censored
![Page 29: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/29.jpg)
t = min(c,y)
HIV
AIDS
u1 u2
![Page 30: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/30.jpg)
HIV
AIDS
u1 u2
t = min(c,y)
![Page 31: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/31.jpg)
HIV
AIDS
u1 u2
t = min(c,y)
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HIV
AIDS
u1 u2
t = min(c,y)
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Inconsistency of the naive MLE
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Inconsistency of the naive MLE
![Page 35: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/35.jpg)
Inconsistency of the naive MLE
![Page 36: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/36.jpg)
Inconsistency of the naive MLE
![Page 37: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/37.jpg)
Methods to repair inconsistency
• Transform the lines into strips
• MLE on a sieve of piecewise constant densities
• Kullback-Leibler approach
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• cannot be estimated
consistently
X = time of HIV infectionY = time of onset of AIDSZ = Y-X = incubation period
( )P Z z
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X = time of HIV infectionY = time of onset of AIDSZ = Y-X = incubation period
1 2( )P Z z x X x
• An example of a parameter we can estimate consis-tently is:
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Conclusions (1)
• Our algorithms for the parameter reduction step are significantly faster than other existing algorithms.
• We proved that in general the naive MLE is an inconsistent estimator for our AIDS model.
![Page 41: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/41.jpg)
Conclusions (2)
• We explored several methods to repair the inconsistency of the naive MLE.
• cannot be estimated consistently without additional assumptions. An alternative parameter that we can estimate consistently is:
. 1 2( )P Z z x X x
( )P Z z
![Page 42: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data](https://reader033.vdocuments.mx/reader033/viewer/2022061608/56815904550346895dc638a2/html5/thumbnails/42.jpg)
Acknowledgements
• Piet Groeneboom
• Jon Wellner