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

TL;DR

This paper develops a Bayesian method for estimating parameters of the left-truncated log-logistic distribution, improving stability and uncertainty quantification in time-to-event data analysis with truncated samples.

Contribution

It introduces a Bayesian approach using MCMC for LTLL parameter estimation, addressing challenges of irregular likelihood surfaces due to truncation.

Findings

Bayesian estimates are more stable than classical methods.

The approach effectively quantifies uncertainty in parameter estimates.

Simulation and real data show improved reliability in truncated data contexts.

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 \( X \sim LL(\alpha, \beta; x_L) \), where \( \alpha > 0 \) is the scale parameter and \( \beta > 0 \) 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 \( \hat{\alpha} \) and \( \hat{\beta} \). 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.