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

doi:10.12141/j.issn.1000-565X.220231

Abstract

Abstract:

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 long-life 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.

Key words: three-parameter Weibull distribution, least square iterative method, parameter estimation, correlation coefficient method

CLC Number:

TB114.3

YANG Xiaoyu, SONG Jiaxin, XIE Liyang, et al. Three-Parameter Estimation of the Weibull Distribution Based on Least Squares Iteration[J]. Journal of South China University of Technology(Natural Science Edition), 2023, 51(2): 20-26.

Figures/Tables 14

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序号	抽取的样本W（2，1 000，1 000）					估计的β	估计的η	估计的γ	可靠度为99%的寿命
1	1 586.98	1 742.30	2 342.56	2 434.73	2 462.58	1.84	1 105.98	1 177.39	1 268.06
2	1 477.99	1 493.02	1 579.96	2 526.79	2 751.75	0.66	558.71	1 412.47	1 413.00
3	1 428.92	1 529.55	1 784.55	1 957.81	2 214.92	1.80	714.52	1 168.76	1 224.42
4	1 456.74	1 474.92	2 037.50	2 124.71	2 245.08	0.95	647.80	1 335.22	1 340.28
5	1 768.98	1 801.86	2 089.93	2 176.56	2 242.71	1.27	432.39	1 655.77	1 667.39
6	1 324.59	1 365.93	1 426.28	1 956.05	2 590.91	0.80	505.13	1 247.88	1 249.50

方法	估计的β	估计的η	估计的γ	99%可靠度的寿命	K-S检验p值
迭代方法	1.000 1	440.810 3	64.955 5	69.385 2	0.996 7
相关系数法	0.935 9	430.550 9	71.590 2	74.754 6	0.985 0

方法	估计的β	估计的η	估计的γ	99%可靠度的寿命	K-S检验p值
迭代方法	2.382 9	86.387 2	30.005 7	42.538 8	0.945 2
相关系数法	2.081 4	78.779 8	37.172 2	45.813 5	0.916 5

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