Stochastic orders and shape properties for a new distorted proportional odds model

TL;DR

This paper introduces two new flexible distorted proportional odds models that enhance control over shape properties and hazard functions, supported by stochastic order conditions to compare distributions.

Contribution

It develops two novel models expanding distribution classes while preserving shape characteristics, and establishes stochastic order conditions for these models.

Findings

Enhanced control over skewness and tail behaviour.

Enlarged log-logistic family capturing diverse hazard rates.

Conditions for stochastic ordering within and across models.

Abstract

Building on recent developments in models focused on the shape properties of odds ratios, this paper introduces two new models that expand the class of available distributions while preserving specific shape characteristics of an underlying baseline distribution. The first model offers enhanced control over odds and log-odds functions, facilitating adjustments to skewness, tail behaviour, and hazard rates. The second model, with even greater flexibility, describes odds ratios as quantile distortions. This approach leads to an enlarged log-logistic family capable of capturing these quantile transformations and diverse hazard behaviours, including non-monotonic and bathtub-shaped rates. Central to our study are the shape relations described through stochastic orders; we establish conditions that ensure stochastic ordering both within each family and across models under various ordering concepts, such as hazard rate, likelihood ratio, and convex transform orders