Most Likely Transformations

TL;DR

This paper introduces a flexible maximum likelihood estimation framework for conditional transformation models, enabling direct distribution function estimation for various data types, including censored or truncated data, with proven asymptotic properties.

Contribution

It develops a comprehensive theoretical and computational framework for maximum likelihood estimation of transformation models, applicable to both discrete and continuous responses, with software implementation.

Findings

Establishment of asymptotic normality for estimators

Framework applicable to censored and truncated data

Software implementation demonstrating practical utility

Abstract

We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterisation of the unconditional or conditional transformation function, we establish a cascade of increasingly complex transformation models that can be estimated, compared and analysed in the maximum likelihood framework. Models for the unconditional or conditional distribution function of any univariate response variable can be set-up and estimated in the same theoretical and computational framework simply by choosing an appropriate transformation function and parameterisation thereof. The ability to evaluate the distribution function directly allows us to estimate models based on the exact likelihood, especially in the presence of random censoring or truncation. For discrete and continuous responses, we establish the asymptotic normality of the proposed estimators. A reference software implementation of maximum likelihood-based estimation for conditional transformation models allowing the same flexibility as the theory developed here was employed to illustrate the wide range of possible applications.