Nonparametric maximum likelihood estimation mle for bivariate censored data
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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:

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Nonparametric maximum likelihood estimation mle for bivariate censored data

Nonparametric maximum likelihood estimation (MLE) for bivariate censored data

Marloes H. Maathuis

advisors:

Piet Groeneboom and Jon A. Wellner


Motivation
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


Approach
Approach

  • Use MLE to estimate the bivariate distribution

  • Integrate over diagonal strips: P(Y-X ≤ z)

Y (AIDS)

z

X (HIV)


Main focus of the project
Main focus of the project

  • MLE for bivariate censored data:

    • Computational aspects

    • (In)consistency and methods to repair the inconsistency


Main focus of the project1
Main focus of the project

  • MLE for bivariate censored data:

    • Computational aspects

    • (In)consistency and methods to repair the inconsistency


Y (AIDS)

1996

Interval of

onset of AIDS

1992

1980

1980

1983

1986

X (HIV)

Interval of

HIV infection


Observation rectangle Ri

Y (AIDS)

1996

Interval of

onset of AIDS

1992

1980

1980

1983

1986

X (HIV)

Interval of

HIV infection


Observation rectangle Ri

Y (AIDS)

X (HIV)


Observation rectangle Ri

Maximal intersections

Y (AIDS)

X (HIV)


Observation rectangle Ri

Maximal intersections

Y (AIDS)

X (HIV)


Observation rectangle Ri

Maximal intersections

Y (AIDS)

X (HIV)


Observation rectangle Ri

Maximal intersections

Y (AIDS)

X (HIV)


Observation rectangle Ri

Maximal intersections

Y (AIDS)

X (HIV)


Observation rectangle Ri

Maximal intersections

Y (AIDS)

α1

α2

α3

α4

s.t.

and

X (HIV)


Observation rectangle Ri

Maximal intersections

Y (AIDS)

2

5

s.t.

and

X (HIV)

3/5

0

0

The αi’s are not always uniquely determined: mixture non uniqueness


Computation of the mle
Computation of the MLE

  • Reduction step:

    determine the maximal intersections

  • Optimization step:

    determine the amounts of mass assigned to the maximal intersections


Computation of the mle1
Computation of the MLE

  • Reduction step:

    determine the maximal intersections

  • Optimization step:

    determine the amounts of mass assigned to the maximal intersections


Existing reduction algorithms
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).


New algorithms
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)


Height map algorithm o n 2
Height map algorithm: O(n2)

1

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0

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1

1

1

0

1

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1

2

1

0

1

2

2

2

1

0

1

0

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0

0


Main focus of the project2
Main focus of the project

  • MLE of bivariate censored data:

    • Computational aspects

    • (In)consistency and methods to repair the inconsistency


u1

u2

Time of HIV infection is interval censored case 2

AIDS

HIV


u1

u2

Time of HIV infection is interval censored case 2

AIDS

HIV


u1

u2

Time of HIV infection is interval censored case 2

AIDS

HIV


t = min(c,y)

u1

u2

Time of onset of AIDS is right censored

AIDS

HIV


u1

u2

Time of onset of AIDS is right censored

AIDS

t = min(c,y)

HIV


u1

u2

Time of onset of AIDS is right censored

AIDS

t = min(c,y)

HIV


t = min(c,y)

AIDS

u1

u2

HIV


t = min(c,y)

AIDS

u1

u2

HIV


t = min(c,y)

AIDS

u1

u2

HIV


t = min(c,y)

AIDS

u1

u2

HIV






Methods to repair inconsistency
Methods to repair inconsistency

  • Transform the lines into strips

  • MLE on a sieve of piecewise constant densities

  • Kullback-Leibler approach


X = time of HIV infection

Y = time of onset of AIDS

Z = Y-X = incubation period

  • cannot be estimated consistently


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:


Conclusions 1
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.


Conclusions 2
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: .


Acknowledgements
Acknowledgements

  • Piet Groeneboom

  • Jon Wellner


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