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Nonparametric maximum likelihood estimation (MLE) for bivariate censored data

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Title: Nonparametric maximum likelihood estimation (MLE) for bivariate censored data


1
Nonparametric maximum likelihood estimation (MLE)
for bivariate censored data
  • Marloes H. Maathuis
  • advisors
  • Piet Groeneboom and Jon A. Wellner

2
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

3
Approach
  • Use MLE to estimate the bivariate distribution
  • Integrate over diagonal strips P(Y-X z)

Y (AIDS)
z
X (HIV)
4
Main focus of the project
  • MLE for bivariate censored data
  • Computational aspects
  • (In)consistency and methods to repair the
    inconsistency

5
Main focus of the project
  • MLE for bivariate censored data
  • Computational aspects
  • (In)consistency and methods to repair the
    inconsistency

6
Y (AIDS)
1996
Interval of onset of AIDS
1992
1980
1980
1983
1986
X (HIV)
Interval of HIV infection
7
Y (AIDS)
1996
Interval of onset of AIDS
1992
1980
1980
1983
1986
X (HIV)
Interval of HIV infection
8
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15
3/5
0
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The ais are not always uniquely determined
mixture non uniqueness
16
Computation of the MLE
  • Reduction step
  • determine the maximal intersections
  • Optimization step
  • determine the amounts of mass assigned to the
    maximal intersections

17
Computation of the MLE
  • Reduction step
  • determine the maximal intersections
  • Optimization step
  • determine the amounts of mass assigned to the
    maximal intersections

18
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).

19
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)

20
Height map algorithm O(n2)
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Main focus of the project
  • MLE of bivariate censored data
  • Computational aspects
  • (In)consistency and methods to repair the
    inconsistency

23
Time of HIV infection is interval censored case 2
AIDS
HIV
24
Time of HIV infection is interval censored case 2
AIDS
HIV
25
Time of HIV infection is interval censored case 2
AIDS
HIV
26
Time of onset of AIDS is right censored
AIDS
HIV
27
Time of onset of AIDS is right censored
AIDS
t min(c,y)
HIV
28
Time of onset of AIDS is right censored
AIDS
t min(c,y)
HIV
29
AIDS
u1
u2
HIV
30
AIDS
u1
u2
HIV
31
AIDS
u1
u2
HIV
32
AIDS
u1
u2
HIV
33
Inconsistency of the naive MLE
34
Inconsistency of the naive MLE
35
Inconsistency of the naive MLE
36
Inconsistency of the naive MLE
37
Methods to repair inconsistency
  • Transform the lines into strips
  • MLE on a sieve of piecewise constant densities
  • Kullback-Leibler approach

38
X time of HIV infection Y time of onset of
AIDS Z Y-X incubation period
  • cannot be estimated consistently

39
X time of HIV infection Y time of onset of
AIDS Z Y-X incubation period
  • An example of a parameter we can estimate
    consis- tently is

40
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.

41
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 .

42
Acknowledgements
  • Piet Groeneboom
  • Jon Wellner
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