There are often situations where two or more regression functions are ordered over a range of covariate values. In this paper, we develop efficient constrained estimation and testing procedures for such models. Specifically, necessary and sufficient conditions for ordering generalized linear regressions are given and shown to unify previous results obtained for simple linear regression, for polynomial regression and in the analysis of covariance models. We show that estimating the parameters of ordered linear regressions requires either quadratic programming or semi-infinite programming, depending on the shape of the covariate space. A distance-type test for order is proposed. Simulations demonstrate that the proposed methodology improves the mean square error and power compared with the usual, unconstrained, estimation and testing procedures. Improvements are often substantial. The methodology is extended to order generalized linear models where convex semi-infinite programming plays a role. The methodology is motivated by, and applied to, a hearing loss study.
Bibliographical noteFunding Information:
The research of Ori Davidov was partially supported by the Israeli Science Foundation Grant 1256/13. We thank Professor Shiri Artstein of Tel Aviv University for introducing us to the literature on convex bodies.
© 2017 Board of the Foundation of the Scandinavian Journal of Statistics
- constrained inference
- quadratic programming
- semi-infinite programming
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty