Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes

Regression models for continuous outcomes often require a transformation of the outcome, which the user either specify {\it a priori} or estimate from a parametric family. Cumulative probability models (CPMs) nonparametrically estimate the transformation and are thus a flexible analysis approach for continuous outcomes. However, it is difficult to establish asymptotic properties for CPMs due to the potentially unbounded range of the transformation. Here we show asymptotic properties for CPMs when applied to slightly modified data where the outcomes are censored at the ends. We prove uniform consistency of the estimated regression coefficients and the estimated transformation function over the non-censored region, and describe their joint asymptotic distribution. We show with simulations that results from this censored approach and those from the CPM on the original data are similar when a small fraction of data are censored. We reanalyze a dataset of HIV-positive patients with CPMs to illustrate and compare the approaches.