基于最小二乘迭代威布尔分布的三参数估计

Three-Parameter Estimation of the Weibull Distribution Based on Least Squares Iteration

威布尔分布因其数学处理的便利性和适应性而被广泛用于描述产品寿命,引入了位置参数的三参数威布尔分布是研究机械零部件可靠性最适合的模型之一,尤其适用于长寿命、高可靠性的产品。三参数威布尔分布的参数估计一直是关注的焦点,本文中提出了一种基于最小二乘的迭代方法对其进行参数估计,将初始位置参数设置为0,使用最小二乘法得到初始形状参数和尺度参数,将其代入位置参数的无偏估计得到新的位置参数,进行多次迭代,在此过程中形状参数和尺度参数逐渐变小,位置参数逐渐变大,最终获得稳定的形状参数、尺度参数和位置参数,即为最终的参数估计值,并计算可靠度为99%的寿命。通过蒙特卡洛(Monte Carlo)仿真证明此方法是收敛的,并在不同的威布尔模型、不同中小样本量(样本量为10、15、20、25和30)下,使用偏差(Bias)和均方差根(RMSE)两个指标与相关系数法进行对比,此方法估计的3个参数及可靠度为99%的寿命更加准确。通过两个实例分析,表明该方法具有可行性和有效性,估计结果与相关系数法相比更加保守,更适于工程应用。

The three-parameter Weibull distribution is widely used to describe product longevity because of the convenience and adaptability of its mathematical processing.The three-parameter Weibull distribution with location parameter is one of the most suitable models for studying the reliability of mechanical components,especially for longlife and high-reliability products.Parameter estimation of three-parameter Weibull distribution has always been the focus of attention.This paper proposed an iterative method based on least squares to estimate the parameters of the three-parameter Weibull distribution.The initial location parameter was set to 0,the initial shape parameter and scale parameter were obtained by using least squares,and the new location parameter was obtained by substituting them into the unbiased estimation of the location parameter,and multiple iterations were performed.In this process,the shape parameters and scale parameters gradually become smaller and the location parameters gradually become larger,and finally the stable shape parameters,scale parameters and location parameters were obtained,which are the final parameter estimates,and the lifetime of 99%reliability was calculated.The method was proved to be convergent by Monte Carlo simulation.Compared with the correlation coefficient method by two metrics including Bias and Root Mean Square Error(RMSE)for different Weibull models with different small and medium sample sizes(10,15,20,25 and 30),the three estimated parameters and the 99%reliability of the lifetime of the proposed method are more accurate.The analysis of two examples shows that the method is feasible and valid.Compared with the correlation coefficient method,the estimation results are more conservative and more suitable for engineering application.