Nonparametric confidence intervals for monotone functions

TL;DR

This paper develops nonparametric confidence intervals for monotone functions, extending existing methods to new models, providing theoretical proofs, and comparing with smoothed estimators using bootstrap techniques.

Contribution

It introduces a generalized approach for constructing confidence intervals for monotone functions across various models, including new proofs and computational tools.

Findings

Limit distribution of LR test matches current status model.

Confidence intervals outperform smoothed MLE in simulations.

Lagrange-modified cusum diagrams facilitate computation and theory development.

Abstract

We study nonparametric isotonic confidence intervals for monotone functions. In Banerjee and Wellner (2001) pointwise confidence intervals, based on likelihood ratio tests for the restricted and unrestricted MLE in the current status model, are introduced. We extend the method to the treatment of other models with monotone functions, and demonstrate our method by a new proof of the results in Banerjee and Wellner (2001) and also by constructing confidence intervals for monotone densities, for which still theory had to be developed. For the latter model we prove that the limit distribution of the LR test under the null hypothesis is the same as in the current status model. We compare the confidence intervals, so obtained, with confidence intervals using the smoothed maximum likelihood estimator (SMLE), using bootstrap methods. The `Lagrange-modified' cusum diagrams, developed here, are an essential tool both for the computation of the restricted MLEs and for the development of the theory for the confidence intervals, based on the LR tests.