Shape-constrained estimation for current duration data in cross-sectional studies

Area of Science:

• Statistics
• Survival Analysis
• Nonparametric Estimation

Background:

• Cross-sectional studies without follow-up present challenges for survival function estimation.
• Observed durations are length-biased and multiplicatively censored, complicating analysis.
• Existing methods like the Grenander estimator can suffer from inconsistency at time zero.

Purpose of the Study:

• To develop shape-constrained nonparametric estimators for survival functions.
• To investigate the properties of estimators under log-concavity and convexity constraints.
• To address the limitations of existing methods, particularly the Grenander estimator's inconsistency.

Main Methods:

• Nonparametric estimation of survival functions.
• Application of shape constraints, specifically log-concavity and convexity.
• Establishment of consistency and asymptotic distribution for the proposed estimators.

Main Results:

• A consistent and tuning-parameter-free estimator for the survival function under log-concavity was developed.
• The proposed log-concave estimator overcomes the inconsistency issue of the Grenander estimator at time zero.
• The versatility of the log-concavity constraint was highlighted, accommodating various density types.

Conclusions:

• Shape-constrained nonparametric estimation provides a robust approach for survival analysis in specific study designs.
• The log-concavity constraint offers significant advantages, including improved accuracy and simpler implementation.
• This method advances survival function estimation in challenging data settings.