Studying the estimates of gamma distribution parameters

S. M. Shebanov
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Abstract

The main goal of the work is to obtain additional information about the experimentally obtained sample with a previously known theoretical distribution, the point estimates of the parameters of which are considered known. At the same time, the laws of distribution of these parameters remain unknown, whereas they could provide a researcher with additional information about both the material and technological processes. Hence, it is necessary to obtain an additional number of samples, which is not always possible experimentally. Here we used data on the service life of cutters (GOST 11.011–83 «Rules for determining estimates and confidence limits for gamma distribution parameters») as an experimental sample. The experimental sample contains the results of 50 measurements. The mean was 57.88 hours CI [50.74:65.01]. The confidence probability is taken to be 0.95. Bootstrap was used as a way to obtain additional samples. The universal mathematical package MATLAB is used in the study. Bootstrap allows generation of a large number of samples that require certain selection rules to be applied to them. The first obvious requirement is the significance of the correlation coefficient of the generated sample with the original one. Even at this stage, the bootstrap showed certain limitations in performing the task set in the study. For 1000 samples generated by the standard bootstrap routine, the mean for the population of all mean bootstrap samples was 57.80 hours, and the confidence interval was [50.59:58.08]. The result is good. Though the nonparametric hypothesis regarding an agreement between the bootstrap samples for the gamma distribution and the parameters characteristic of the original experimentally obtained sample was not rejected, the statistically significant correlation coefficient was observed only for 29 bootstrap samples. As a result of meeting these obvious requirements, less than 3% of the generated bootstrap samples remained for further consideration. This fact requires the introduction of additional conditions when using the bootstrap to obtain samples that are close to the original experimental sample, which can be rather specific. To determine the parameters of the gamma distribution for bootstrap samples, the method of moments and the one-step method were used.
研究伽马分布参数的估计值
这项工作的主要目标是获取实验所得样本的更多信息,这些样本具有先前已知的理论分 布,其参数的点估计被认为是已知的。同时,这些参数的分布规律仍是未知的,而它们可以为研究人员提供有关材料和技术过程的更多信息。因此,有必要获取更多的样本,而这在实验中并不总是可行的。在这里,我们使用有关刀具使用寿命的数据(GOST 11.011-83 "确定伽马分布参数的估计值和置信区间的规则")作为实验样本。实验样本包含 50 次测量结果。平均值为 57.88 小时 CI [50.74:65.01]。置信概率取 0.95。使用 Bootstrap 方法获得更多样本。研究中使用了通用数学软件包 MATLAB。Bootstrap 可以生成大量样本,这些样本需要应用某些选择规则。第一个显而易见的要求是生成的样本与原始样本的相关系数的显著性。即使在这一阶段,自举法在完成研究任务时也显示出一定的局限性。对于标准引导程序生成的 1000 个样本,所有平均引导样本的总体平均值为 57.80 小时,置信区间为[50.59:58.08]。结果不错。虽然没有拒绝关于伽马分布的 bootstrap 样本与原始实验样本特征参数一致的非参数假设,但只在 29 个 bootstrap 样本中观察到了具有统计意义的相关系数。由于满足了这些明显的要求,生成的自举样本中只有不到 3%的样本可以进一步考虑。这就要求在使用 bootstrap 时引入额外的条件,以获得与原始实验样本接近的样本。为了确定自举样本的伽马分布参数,使用了矩量法和一步法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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