标题： 左截断对数逻辑分布的贝叶斯推断用于生存数据分析 显示英文标题

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

摘要： 参数估计是统计建模中的基础步骤，使我们能够从数据中提取知识并有效应用。 贝叶斯参数估计将先验信念与观测数据结合，以概率和稳健的方式推断分布参数。 此外，它提供了完整的后验分布，允许不确定性量化和正则化，特别是在小样本或截断样本中非常有用。 使用左截断对数对数逻辑（LTLL）分布特别适合建模时间至事件数据，其中观测值受已知下限的影响，例如降水数据和癌症生存时间。 在本文中，我们提出了一种贝叶斯方法来估计具有固定截断点 \( x_L > 0 \) 的 LTLL 分布的参数。 给定一个随机变量 \( X \sim LL(\alpha, \beta; x_L) \)，其中 \( \alpha > 0 \) 是尺度参数， \( \beta > 0 \) 是形状参数，似然函数基于截断样本 \( X_1, X_2, \dots, X_N \) 得出，其中 \( X_i > x_L \)。 我们假设参数的先验分布是独立的，后验推断通过马尔可夫链蒙特卡洛抽样进行，具体使用梅特罗波利斯-哈金斯算法获得后验估计\( \hat{\alpha} \)和\( \hat{\beta} \)。 通过模拟研究和实际应用，我们证明贝叶斯估计提供了更稳定和可靠的参数估计，特别是在似然面由于左截断而不规则时。 结果突显了贝叶斯推断在时间至事件数据分析中优于参数不确定性的估计。 显示英文摘要

摘要： 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.