Bayesian Inference for Left-Truncated Log-Logistic Distributions for Time-to-event Data Analysis

2025-06-21

AI Summary

For left-truncated time-to-event data—such as precipitation extremes with known lower thresholds or censored cancer survival times—this paper proposes a Bayesian inference framework for parameter estimation of the Log-Logistic distribution (LTLL). Unlike classical maximum likelihood estimation (MLE), which suffers from instability under small sample sizes or irregular likelihood surfaces, our approach employs independent priors and conducts posterior sampling via the Metropolis–Hastings MCMC algorithm, thereby enhancing estimation robustness and uncertainty quantification. Simulation studies and empirical analyses—including extreme precipitation and clinical survival datasets—demonstrate that the proposed method consistently outperforms MLE in terms of bias, mean squared error, and nominal coverage of credible intervals. The improvement is particularly pronounced under high truncation proportions or limited sample sizes. This work provides a reliable, principled Bayesian solution for modeling left-truncated extremes and survival data.

Technology Category

Machine Learning: Bayesian LearningReasoning under Uncertainty: Probabilistic InferenceCognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsUser Modeling, Personalization and Recommendation: Large Language Models (LLM) for user modeling and recommendationSecurity and Privacy: Large-scale security measurements

Abstract

Parameter estimation is a foundational step in statistical modeling, enabling us to extract knowledge from data and apply it effectively. Bayesian estimation of parameters incorporates prior beliefs with observed data to infer distribution parameters probabilistically and robustly. Moreover, it provides full posterior distributions, allowing uncertainty quantification and regularization, especially useful in small or truncated samples. Utilizing the left-truncated log-logistic (LTLL) distribution is particularly well-suited for modeling time-to-event data where observations are subject to a known lower bound such as precipitation data and cancer survival times. In this paper, we propose a Bayesian approach for estimating the parameters of the LTLL distribution with a fixed truncation point ( x_L > 0 ). Given a random variable ( X sim LL(α, β; x_L) ), where ( α> 0 ) is the scale parameter and ( β> 0 ) is the shape parameter, the likelihood function is derived based on a truncated sample ( X_1, X_2, dots, X_N ) with ( X_i > x_L ). We assume independent prior distributions for the parameters, and the posterior inference is conducted via Markov Chain Monte Carlo sampling, specifically using the Metropolis-Hastings algorithm to obtain posterior estimates ( hatα ) and ( hatβ ). Through simulation studies and real-world applications, we demonstrate that Bayesian estimation provides more stable and reliable parameter estimates, particularly when the likelihood surface is irregular due to left truncation. The results highlight the advantages of Bayesian inference outperform the estimation of parameter uncertainty in truncated distributions for time to event data analysis.

Problem

• Bayesian parameter estimation for left-truncated log-logistic distributions
• Handling irregular likelihood surfaces due to left truncation
• Improving reliability in time-to-event data analysis

Innovation

• Bayesian estimation for left-truncated log-logistic distribution
• Markov Chain Monte Carlo sampling for posterior inference
• Metropolis-Hastings algorithm for stable parameter estimates