Nonparametric Nelson-Aalen reliability evaluation model based on random censored data
-
摘要:
针对可靠性工程试验中的随机截尾数据，从累积失效率函数的分析角度出发，基于Nelson-Aalen（NA）估计理论，实现了对产品的非参数化可靠性评估。基于所获离散样本，给出累积失效率在连续和离散形式下的非参数极大似然估计，并推导出随机截尾样本下累积失效率函数的NA估计形式；由NA估计所得的可靠度衍生完全非参数化置信评估模型；构建广义加权滑动平均模型，实现了对样本最大观测时间之后的可靠度估计。算例分析表明：在对寿命分布信息完全未知时，NA模型实现了基于随机截尾受测型寿命数据对产品可靠性的有效置信评估，估计相对偏差率控制在
0.9787 %以下，且估计精度随着样本量的增加和截尾比例的减小而显著提高。结果验证了NA可靠性计算的有效性和评估精准性。-
关键词:
- Nelson-Aalen估计 /
- 随机截尾数据 /
- 非参数极大似然估计 /
- 置信评估 /
- 可靠性分析
Abstract:For the random censored data in the reliability engineering test, a nonparametric reliability evaluation of the product was realized based on the Nelson-Aalen (NA) estimation theory from the perspective of the analysis of cumulative failure rate function. The nonparametric maximum likelihood estimation of the cumulative failure rate in continuous and discrete forms was given by using the obtained discrete samples, and the NA estimation form of the cumulative failure rate function under the random censored samples was derived. A completely nonparametric confidence evaluation model was derived from the reliability by NA estimation. A generalized weighted moving average model was constructed to estimate the reliability after the maximum observation time of the sample. Finally, example analysis showed that when the life distribution information was completely unknown, the NA model realized the effective confidence evaluation of product reliability based on random censored life data, and the estimated relative bias rate was controlled below
0.9787 %, and the estimation accuracy was improved significantly with the increase of sample size and the decrease of censoring ratio. The results verified the validity and evaluation accuracy of NA reliability calculation. -
图 1 不同样本量和截尾比下NAE可靠度与真实可靠度曲线
Figure 1. Curves of NAE reliability and true reliability under different sample sizes and censoring ratios
表 1 常用可靠性指标
Table 1. Common reliability indicators
指标 定义 与$F （ t ） $的关系 $F （ t ） $ $P （ {T \leqslant t} ） $ $F （ t ） $ $R （ t ） $ $P （ {T > t} ） $ $ 1 - F （ t ） $ $ \lambda （ t ） $ $ \mathop {\lim }\limits_{\Delta t \to 0} \dfrac{{P （ {t < T \leqslant t + \Delta t|T > t} ） }}{{\Delta t}} $ $ \dfrac{{F' （ t ） }}{{1 - F （ t ） }} $ $E （ T ） $ $ \displaystyle\int_0^{ + \infty } {tf （ t ） {\mathrm{d}}t} $ $ \displaystyle\int_0^{ + \infty } {\left[ {1 - F （ t ） } \right]{\mathrm{d}}t} $ 
下载: 导出CSV
表 2 模型参数选取
Table 2. Selection of model parameters
参数 数值 截尾比$S$/% 5，10，20 样本量$n$ 50，100，200 寿命分布参数 $m = 1.5$，$\eta = 2\;000$
下载: 导出CSV
表 3 随机截尾数据试样（部分）
Table 3. Random censored data sample （part）
序号 失效时间$t$/h（“+”表示随机截尾） n=50 n=100 n=200 S=5% S=10% S=20% S=5% S=10% S=20% S=5% S=10% S=20% 1 105.4 152.7 105.4 54.7 36.3 43.0 56.9 85.5 7.0+ 2 154.9 182.9 154.9 64.4 49.1 111.2+ 77.8 112.9 76.2+ 3 403.8 260.1 269.9+ 199.8 152.7 176.1 102.7 145.5 85.5 4 431.3 357.0 380.6+ 221.3 181.8+ 212.1 149.5 168.5 107.9+ 5 434.4 390.0 403.8 222.3 182.9 272.6 173.2 182.3 112.9 6 474.3 463.5 431.3 227.2 189.5 293.9+ 183.2 185.8 120.9 7 487.8 577.8 434.4 227.3 327.0 344.1 188.4 200.4 138.2+ 8 491.9 630.4 474.3 260.4 332.4 392.2+ 190.3 238.7 139.6+ 9 568.3 637.7 487.8 330.7 357.0 420.6+ 228.3 239.6 145.5 10 720.7 643.6 491.9 390.5 374.0 447.0 247.7 245.2 168.4+ 11 734.9+ 872.1 568.3 398.1 390.0 447.2 251.4 248.6 168.5 12 746.3 901.0 648.6+ 439.9 426.0 452.8 283.2 277.2 182.3 13 793.0 922.5 687.3+ 442.4 459.0 458.0 289.8 283.6 185.8 14 887.5 923.8 720.7 451.0 463.5 484.1 290.2 295.5 200.4 15 948.9 965.5 734.9+ 495.2+ 487.7 488.7 328.3 301.4 238.7 16 965.5 969.4 746.3 506.5 615.0 494.2 337.9 316.4 239.6 17 971.8 976.0 793.0 516.0 630.4 500.0 409.1 320.0 245.2 18 1009.2 1020.0 887.5 574.8 637.7 568.7 410.9 331.6+ 248.6 19 1060.4 1106.5 945.6+ 581.6 643.6 599.3+ 413.8 361.0 283.6 20 1062.0 1112.5 948.9 611.4 653.5 625.0+ 420.5 363.5 295.5
下载: 导出CSV
表 4 不同样本量和截尾比下估计量的MSE
Table 4. MSE of estimators under different sample sizes and censoring ratios
S/% MSE n=50 n=100 n=200 5 0.002775 0.000547 0.000177 10 0.003632 0.000788 0.000656 20 0.003688 0.001546 0.000877
下载: 导出CSV
表 5 不同样本量和截尾比下估计的相对偏差率
Table 5. Relative bias rates of estimators under different sample sizes and censoring ratios
S/% 相对偏差率/% n=50 n=100 n=200 5 0.7450 0.1528 0.0537 10 0.9787 0.2078 0.1693 20 0.8447 0.3893 0.2053
下载: 导出CSV
表 6 某型机载设备随机截尾数据（“+”表示随机截尾）
Table 6. Random censored data of a certain type of airborne equipment （“+” indicates random censoring）
序号 失效时间t/h 序号 失效时间t/h 序号 失效时间t/h 1 21 9 270+ 17 346 2 35 10 288 18 350 3 100 11 290 19 411 4 128 12 311 20 476 5 150 13 321 21 497 6 152 14 330 22 499 7 205 15 334 8 264 16 343+
下载: 导出CSV
表 7 基于NAE的可靠度估计值及置信水平为95%的置信限
Table 7. Reliability estimated value and confidence level are 95% confidence limits based on NAE
时间$t$/h $ {\hat R_{\rm{NA}}} （ t ） $ $ {R_{\rm{l}}} （ t ） $ $ {R_{\rm{u}}} （ t ） $ 时间$t$/h $ {\hat R_{\rm{NA}}} （ t ） $ $ {R_{\rm{l}}} （ t ） $ $ {R_{\rm{u}}} （ t ） $ 21 0.955563 0.783446 0.992237 311 0.501367 0.308386 0.693942 35 0.911126 0.724422 0.975599 321 0.453656 0.268472 0.652618 100 0.866690 0.670241 0.954119 330 0.405949 0.230077 0.609783 128 0.822255 0.619363 0.929337 334 0.358249 0.193254 0.565387 150 0.777820 0.571004 0.902037 343 0.358249 0.193254 0.565387 152 0.733385 0.524708 0.872675 346 0.303251 0.152875 0.512123 205 0.688952 0.480183 0.841543 350 0.248281 0.114982 0.456419 264 0.644519 0.437239 0.808832 411 0.193361 0.079998 0.397891 270 0.644519 0.437239 0.808832 476 0.138550 0.048651 0.335911 288 0.596800 0.392731 0.772088 497 0.084035 0.022314 0.269425 290 0.549082 0.349799 0.733772 499 0.030915 0.004127 0.197166
下载: 导出CSV
表 8 基于GWMA-NAE模型的可靠度估计值
Table 8. Reliability estimated value based on GWMA-NAE model
${t_{ （ i ） }}$ ${t_{ （ {i - 1} ） }}$动态权重 ${t_{ （ {i - 2} ） }}$动态权重 ${t_{ （ {i - 3} ） }}$动态权重 ${t_{ （ {i - 4} ） }}$动态权重 修正的可靠度值 150 0.408228 0.363924 0.158228 0.069620 0.861768791 152 0.600000 0.266667 0.123077 0.010256 0.801974087 205 0.362069 0.265517 0.189655 0.182759 0.786400470 264 0.323040 0.270784 0.266033 0.140143 0.743306741 270 0.388350 0.381877 0.210356 0.019417 0.682768691 288 0.521073 0.318008 0.091954 0.068966 0.654733486 290 0.639098 0.195489 0.150376 0.015038 0.614689772 311 0.356061 0.310606 0.174242 0.159091 0.595715732 321 0.408000 0.264000 0.248000 0.080000 0.549083314 330 0.381818 0.363636 0.172727 0.081818 0.499199914 334 0.523810 0.273810 0.154762 0.047619 0.440594602 343 0.421053 0.289474 0.171053 0.118421 0.405324643 346 0.446429 0.285714 0.214286 0.053571 0.373581388 350 0.425532 0.340426 0.148936 0.085106 0.338905114 411 0.284133 0.250923 0.239852 0.225092 0.313203166 476 0.292952 0.286344 0.277533 0.143172 0.263192518 497 0.372840 0.362963 0.212346 0.051852 0.190285287 499 0.568702 0.335878 0.087786 0.007634 0.113196092
下载: 导出CSV
-
[1] 陈家鼎,李东风. 随机截尾情形下正态分布参数的最大似然估计[J]. 应用数学学报,2011,34(6): 961-975. CHEN Jiading,LI Dongfeng. Maximum likelihood estmators for the parameters of normal population in randomly censored data[J]. Acta Mathematicae Applicatae Sinica,2011,34(6): 961-975. (in ChineseCHEN Jiading, LI Dongfeng. Maximum likelihood estmators for the parameters of normal population in randomly censored data[J]. Acta Mathematicae Applicatae Sinica, 2011, 34(6): 961-975. (in Chinese) [2] KUMAR K,KUMAR I. Estimation in inverse weibull distribution based on randomly censored data[J]. Statistica,2019,79(1): 47-74. [3] GARG R,DUBE M,KRISHNA H. Estimation of parameters and reliability characteristics in Lindley distribution using randomly censored data[J]. Statistics,Optimization & Information Computing,2020,8(1): 80-97. [4] AJMAL M,DANISH M Y,AHMAD ARSHAD I. Objective Bayesian analysis for Weibull distribution with application to random censorship model[J]. Journal of Statistical Computation and Simulation,2022,92(1): 43-59. doi: 10.1080/00949655.2021.1931210 [5] RANJAN R,SEN R,UPADHYAY S K. Bayes analysis of some important lifetime models using MCMC based approaches when the observations are left truncated and right censored[J]. Reliability Engineering & System Safety,2021,214: 107747. [6] PAKYARI R,BAKLIZI A. On goodness-of-fit testing for Burr type X distribution under progressively type-Ⅱ censoring[J]. Computational Statistics,2022,37(5): 2249-2265. doi: 10.1007/s00180-022-01197-5 [7] GOEL N,KRISHNA H. Different methods of estimation in two parameter Geometric distribution with randomly censored data[J]. International Journal of System Assurance Engineering and Management,2022,13(4): 1652-1665. [8] ROSSA A. On the estimation of survival function under random censorship[J]. Communications in Statistics-Theory and Methods,2002,31(6): 961-975. doi: 10.1081/STA-120004192 [9] PARAST L,TIAN Lu,CAI Tianxi. Assessing the value of a censored surrogate outcome[J]. Lifetime Data Analysis,2020,26(2): 245-265. doi: 10.1007/s10985-019-09473-1 [10] ABDUSHUKUROV A A,BOZOROV S B,MANSUROV D R. Estimation of distribution function based on presmoothed relative-risk function[J]. Applied Mathematics,2022,13(2): 191-204. doi: 10.4236/am.2022.132015 [11] 沈安慰,郭基联,王卓健. 竞争性故障模型可靠性评估的非参数估计方法[J]. 航空动力学报,2016,31(1): 49-56. SHEN Anwei,GUO Jilian,WANG Zhuojian. Nonparametric estimation method of reliability evaluation in competitive fault model[J]. Journal of Aerospace Power,2016,31(1): 49-56. (in ChineseSHEN Anwei, GUO Jilian, WANG Zhuojian. Nonparametric estimation method of reliability evaluation in competitive fault model[J]. Journal of Aerospace Power, 2016, 31(1): 49-56. (in Chinese) [12] KAPLAN E L,MEIER P. Nonparametric estimation from incomplete observations[J]. Journal of the American Statistical Association,1958,53(282): 457-481. doi: 10.1080/01621459.1958.10501452 [13] NELSON W. Hazard plotting for incomplete failure data[J]. Journal of Quality Technology,1969,1(1): 27-52. doi: 10.1080/00224065.1969.11980344 [14] AALEN O. Nonparametric inference for a family of counting processes[J]. The Annals of Statistics,1978,6(4): 701-726. [15] HU Guanyu,HUFFER F. Modified kaplan-meier estimator and nelson-aalen estimator with geographical weighting for survival data[J]. Geographical Analysis,2020,52(1): 28-48. doi: 10.1111/gean.12185 [16] COLOSIMO E,FERREIRA F,OLIVEIRA M,et al. Empirical comparisons between Kaplan-Meier and Nelson-Aalen survival function estimators[J]. Journal of Statistical Computation and Simulation,2002,72(4): 299-308. doi: 10.1080/00949650212847 [17] JIANG R. A bias-corrected Nelson-Aalen estimator[J]. IOP Conference Series: Materials Science and Engineering,2021,1043(2): 022013. doi: 10.1088/1757-899X/1043/2/022013 [18] MAI Z. Empirical likelihood method in survival analysis[M]. Boca Raton,US: CRC Press,2015: 1-23. [19] 茆诗松,王静龙,濮晓龙. 高等数理统计[M]. 3版. 北京: 高等教育出版社,2022. MAO Shisong,WANG Jinglong,PU Xiaolong. Advanced mathematical statistics[M]. 3rd ed. Beijing: Higher Education Press,2022. (in ChineseMAO Shisong, WANG Jinglong, PU Xiaolong. Advanced mathematical statistics[M]. 3rd ed. Beijing: Higher Education Press, 2022. (in Chinese) [20] 周永道,王会琦,吕王勇. 时间序列分析及应用[M]. 北京: 高等教育出版社,2015: 46-63. ZHOU Yongdao,WANG Huiqi,LV Wangyong. Time series analysis and its application[M]. Beijing: Press of Advanced Education,2015: 46-63. (in ChineseZHOU Yongdao, WANG Huiqi, LV Wangyong. Time series analysis and its application[M]. Beijing: Press of Advanced Education, 2015: 46-63. (in Chinese) [21] 蔡忠义,张强,陈云翔,等. 航空产品外场使用可靠性评估方法[J]. 火力与指挥控制,2018,43(5): 44-48,53. CAI Zhongyi,ZHANG Qiang,CHEN Yunxiang,et al. Research on assessment method of field usage reliability for aviation product[J]. Fire Control & Command Control,2018,43(5): 44-48,53. (in Chinese doi: 10.3969/j.issn.1002-0640.2018.05.009CAI Zhongyi, ZHANG Qiang, CHEN Yunxiang, et al. Research on assessment method of field usage reliability for aviation product[J]. Fire Control & Command Control, 2018, 43(5): 44-48, 53. (in Chinese) doi: 10.3969/j.issn.1002-0640.2018.05.009 -
点击查看大图
计量
- 文章访问数: 1042
- HTML浏览量: 227
- PDF量: 35
- 被引次数: 0