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.

翻译：针对连续结果的回归模型通常需要对结果变量进行变换，这类变换由使用者预先指定或从参数族中估计得到。累积概率模型（CPM）能够非参数地估计变换函数，因此成为分析连续结果的一种灵活方法。然而，由于变换函数的可能无界性，建立CPM的渐近性质存在困难。本文证明了将CPM应用于经过轻微修正的数据（即对结果两端进行删失处理）时的渐近性质。我们证明了非删失区域内回归系数估计量和变换函数估计量的相合性，并描述了它们的联合渐近分布。通过模拟研究显示，当删失数据占比很小时，这种删失方法得到的结果与原始数据上CPM的结果相近。我们利用CPM重新分析了HIV阳性患者数据集，以说明和比较不同方法。