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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:
- Time of HIV infection is interval censored
- Time of onset of AIDS is interval censored

or right censored

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

- MLE for bivariate censored data:
- Computational aspects
- (In)consistency and methods to repair the inconsistency

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

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

- Reduction step:

determine the maximal intersections

- Optimization step:

determine the amounts of mass assigned to the maximal intersections

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

- 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

- 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(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

Methods to repair inconsistency

- Transform the lines into strips
- MLE on a sieve of piecewise constant densities
- Kullback-Leibler approach

Y = time of onset of AIDS

Z = Y-X = incubation period

- cannot be estimated consistently

Y = time of onset of AIDS

Z = Y-X = incubation period

- An example of a parameter we can estimate consis- tently is:

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)

- 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

- Piet Groeneboom
- Jon Wellner

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