Smooth Shape-restricted Regression

Estimation of a function under shape restriction is of considerable interest in many practical applications. It is not uncommon that in many elds, researchers are in the position of having strong presumptions about certain relationships satisfying qualitative restrictions, such as monotonicity and convexity (concavity). Typical examples include the study of utility functions, cost functions, and pro t functions in economics (Gallant, 1984; Terrell, 1996), the study of dose response curve in medicine, growth curves of animals and plants in ecology, and the estimation of the hazard rate in survival analysis (Chang et al., 2007). Imposing shape restrictions can improve the predictive performance and reduce overfitting if the underlying regression function takes the specific form. The classic least squares solutions for shape-restricted estimation are typically neither smooth nor parsimonious. There are many researches investigating the smooth shape-restricted regressors in recent years (Meyer, 2008; Wang and Ghosh, 2012). In this thesis, we propose new nonparametric estimators for univariate regression subject to monotonicity and convexity constraints with simple structures, which are acquired by replacing the discrete measures in the non-smooth least squares solutions with continuous ones. Our estimators are composed as the linear combinations of several well-constructed component functions which satisfy corresponding shape constraints. The smoothness of our models is controlled by one tuning parameter. A fast gradient-based iterative algorithm is used to find the least squares estimates with efficiency (Wang, 2007). Finite sample properties and asymptotic behaviours including the existence, the uniqueness, the equivalence, and the consistency have been investigated. Numerical studies with simulated and real-world data show that our estimators have better prediction performance comparing to other shape-restricted estimators in most scenarios.

Name thesis:degree_name

PhD

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Statistics

Grantor dc:publisher

ResearchSpace@Auckland

Year dc:date.issued

2019

Author dc:creator

• Guo, Hongbin

Advisors dc:contributor.advisor

• Wang, Yong
• Yee, Thomas

Statement dc:rights

• Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher.

Licence dc:rights.uri

• https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm

Handle dc:identifier.uri

https://hdl.handle.net/2292/52497

OAI identifier oai:identifier

oai:researchspace.auckland.ac.nz:2292/52497