Correcting an estimator of a multivariate monotone function with isotonic regression

Area of Science:

• Statistics
• Nonparametric Statistics
• Statistical Inference

Background:

• Monotone function estimation is crucial in various statistical problems.
• Existing estimators can lose monotonicity, impacting reliability.
• Correcting non-monotone estimators is essential for accurate inference.

Purpose of the Study:

• To develop and analyze a projection-based method for correcting non-monotone estimators of monotone functions.
• To evaluate the impact of this correction on estimation error and confidence bands.
• To explore the asymptotic properties and practical applicability of the corrected estimator.

Main Methods:

• Projection onto the space of monotone functions over a finite grid.
• Supremal estimation error analysis.
• Asymptotic equivalence under stochastic equicontinuity and Lipschitz conditions.
• Application to G-computed distribution function and local linear estimators.

Main Results:

• The corrected estimator has no worse supremal estimation error than the initial estimator.
• Corrected confidence bands maintain coverage and bandwidth.
• Asymptotic equivalence is shown under weaker stochastic equicontinuity conditions.
• Projection can yield significant practical improvements, especially in bivariate cases.

Conclusions:

• The projection method effectively corrects non-monotone estimators without compromising statistical properties.
• The approach offers a more flexible alternative to existing correction procedures.
• This method enhances the reliability and practical utility of monotone function estimation.