小样本下Weibull分布参数的贝叶斯估计研究

Research on Bayesian Estimation of Weibull Distribution Parameters with Small Sample Sizes

摘要: Weibull分布是可靠性工程中刻画产品寿命特征的重要模型，其参数的精确估计直接关系到寿命预测与维修决策的可靠性。本文在形状参数已知的假设下，系统研究Weibull分布尺度参数的贝叶斯估计方法。首先给出尺度参数的极大似然估计；其次，分别在无信息先验和共轭先验下，结合平方损失函数推导贝叶斯估计的解析表达式；最后通过蒙特卡洛模拟，在不同样本量和参数真值下比较三种估计方法的均方误差与偏差。结果表明，无信息先验下的贝叶斯估计与极大似然估计在点估计上完全等价；共轭先验通过引入合理的先验信息，在小样本下能够显著降低估计的均方误差与偏差，且样本量越小提升幅度越明显。

Abstract: The Weibull distribution is a crucial model in reliability engineering for characterizing product lifespan features, where accurate parameter estimation directly impacts the reliability of lifespan prediction and maintenance decisions. Under the assumption of a known shape parameter, this study systematically investigates Bayesian estimation methods for the scale parameter of the Weibull distribution. First, the maximum likelihood estimation of the scale parameter is derived. Second, analytical expressions for Bayesian estimation are obtained under both non-informative and conjugate priors, combined with the squared error loss function. Finally, Monte Carlo simulations are employed to compare the mean squared error and bias of three estimation methods under different sample sizes and true parameter values. The results indicate that Bayesian estimation under non-informative priors is fully equivalent to maximum likelihood estimation in point estimation; conjugate priors, by incorporating reasonable prior information, can significantly reduce the mean squared error and bias of estimation with small samples, and the improvement becomes more pronounced as the sample size decreases.