﻿ 基于β似然函数的参数估计方法
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1. 北京航空航天大学可靠性与系统工程学院, 北京 100083;
2. 北京航空航天大学无人驾驶飞行器研究所, 北京 100083

Parameter estimation method based on beta-likelihood function
WANG Xiaohong1 , LI Yuxiang1, YU Chuang1, WANG Lizhi2
1. School of Reliability and Systems Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100083, China;
2. Research Institute of Unmanned Aerial Vehicle, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
Abstract: Distribution parameter estimation is a common method which is used in reliability data analysis to study the change rules of product reliability and evaluate the reliability level of the product. Learning from the beta distribution which is used to describe the product reliability in life test and evaluate the reasonable degree of estimator of reliability, we thought that reliability estimator is more reasonable if its probability density function is bigger in the beta distribution, raised a beta likelihood function to evaluate the reasonable level of the estimate in reliability analysis, discussed the distribution parameter estimation method when using this beta likelihood function, verified the method by simulation under the exponential distribution case and Weibull distribution case, and gave the corresponding application examples. The estimation method is based on abundant theoretic evidence, and is suitable for all kinds of distribution types. From the application examples, we know that estimation results are reasonable and believable. Maximum likelihood estimation method takes the samples' probability density function in the distribution to be estimated as the evaluation criteria, while, on the contrast, our method takes the cumulative incidence estimator as the evaluation criteria. So it is more applicable in the research on reliability and survival problems when concerning the cumulative occurrences.
Key words: parameter estimation     beta distribution     reliability data analysis     life distribution     beta-likelihood function

1 β似然函数的构造

2 β似然估计方法

 图 1 β分布的概率密度曲线 Fig. 1 Probability density curves of β distribution

3 仿真分析

1) 随机确定分布参数(指数分布的故障率λ、威布尔分布的形状参数β和尺度参数α)。

2) 使用仿真的方法分别获取该参数下服从指数分布和威布尔分布的失效数据。

3) 分别选择指数分布和威布尔分布作为寿命分布模型,使用极大似然估计和β似然估计对这失效数据进行参数估计,得到指数分布的故障率λ1λ2及威布尔分布的参数α1β1α2β2

4) 比较2个估计得到的故障率λ1λ2与原始故障率λ的接近程度、比较参数α1β1α2β2与原始参数αβ之差的绝对值之和的大小关系。

5) 重复以上步骤,直到记录下足够多次的比较结果。 使用计算机进行1 000次上述仿真比较后,统计的结果为使用极大似然估计得到的λ1较使用β似然估计λ2更接近原始λ的次数为568次,反之432次;使用极大似然估计得到的|α1α|+|β1β|较使用β似然估计得到的|α2α|+|β2β|更大的次数为383次,且2种方法对各组数据的估计结果相差往往都很小。图 2为指数分布和威布尔分布下的其中各一次仿真再估计得到的可靠度曲线。

 图 2 仿真结果对比 Fig. 2 Comparison of simulation results

4 应用举例

 试件序号 工作时间/h 试验终止原因 1 216 故障 2 50 故障 3 514 撤离 4 565 故障 5 298 故障 6 460 故障 7 1 000 截尾 8 183 故障 9 131 故障 10 940 故障

 共同排序号 j 故障排序号 i 工作时间 t i/h 平均秩次 A i 1 1 50 1 2 2 131 2 3 3 183 3 4 4 216 4 5 5 298 5 6 6 460 6 7 / 514 / 8 7 565 7.25 9 8 940 8.5

 图 3 估计结果对比 Fig. 3 Comparison of estimation results
5 结 论

1) 具有较强的理论支撑:β似然函数推导过程使用的“β分布法”和平均秩次的计算都非常成熟,前者可被严格证明、后者在无法得知准确故障秩次时使用秩次的期望值也是合理的。

2) 适用于各种寿命分布类型:无论分布的数学形式如何,β似然函数的取值都在一个有限的范围内,不会出现类似极大似然函数取值可能无穷大的情况,因此β似然估计都是存在的(仅要求样本量n > 1);另外随着分布参数的增加,β似然函数作为多元函数,其求极值在数值计算中容易实现,其难度低于矩估计中需要进行的多元非线性方程组求解。

3) 估计结果合理可信:β似然函数的推导就是以评价可靠度估计值的合理性为出发点,故本方法适用于可靠度、生存率等关注事件发生比例的研究领域。 基于以上特点,本方法在浴盆曲线模型等较为复杂的分布参数估计中有较为突出的优点,克服了极大似然估计结果不存在的问题。

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#### 文章信息

WANG Xiaohong, LI Yuxiang, YU Chuang, WANG Lizhi

Parameter estimation method based on beta-likelihood function

Journal of Beijing University of Aeronautics and Astronsutics, 2016, 42(1): 41-46.
http://dx.doi.org/10.13700/j.bh.1001-5965.2015.0071