若干广义信息与散度生成函数：性质、估计、验证及应用
Some Generalized Information and Divergence Generating Functions: Properties, Estimation, Validation and Applications
AI总结：
该研究提出Rényi信息生成函数与Rényi散度信息生成函数，推导其性质与关联，给出两类估计量并通过仿真和真实数据验证其性能更优，还探索了其在系统不确定性度量、模型选择、混沌映射验证等场景的应用。
摘要：
我们提出了Rényi信息生成函数并讨论了其性质。针对离散型随机变量，我们建立了Rényi信息生成函数与多样性指数之间的关联。本文推导了Rényi信息生成函数与阶数$q>0$的Shannon熵之间的关系，并得到了若干界值。我们还导出了伴随分布（escort distribution）的Rényi信息生成函数。此外，我们引入了Rényi散度信息生成函数，并讨论了其在单调变换下的效应。\n本文提出了Rényi信息生成函数的非参数估计量与参数估计量。我们开展了仿真研究，并分析了一组与电子元件失效时间相关的真实数据。我们从标准差、绝对偏差和均方误差三个维度，对非参数估计量与参数估计量进行了对比研究，结果显示新提出的估计量性能更优。\n本文还给出了所提出的Rényi信息生成函数与Rényi散度信息生成函数的若干应用。针对三个相干系统，我们计算了Rényi信息生成函数的取值以及其他成熟的不确定性度量值，观察到Rényi信息生成函数表现出相似的行为。此外，我们还研究了Rényi散度信息生成函数与Rényi信息生成函数作为模型选择准则的实用性。最后，我们考虑了三种混沌映射，并利用它们对所提出的信息生成函数进行了验证。
英文摘要：
We propose Rényi information generating function and discuss its properties. A connection between the Rényi information generating function and the diversity index is proposed for discrete type random variables. The relation between the Rényi information generating function and Shannon entropy of order $q>0$ is established and several bounds are obtained. The Rényi information generating function of escort distribution is derived. Furthermore, we introduce Rényi divergence information generating function and discuss its effect under monotone transformations. We present non-parametric and parametric estimators of the Rényi information generating function. A simulation study is carried out and a real data relating to the failure times of electronic components is analyzed. A comparison study between the non-parametric and parametric estimators is made in terms of the standard deviation, absolute bias, and mean square error. We have observed superior performance for the newly proposed estimators. Some applications of the proposed Rényi information generating function and Rényi divergence information generating function are provided. For three coherent systems, we calculate the values of the Rényi information generating function and other well-established uncertainty measures and similar behaviour of the Rényi information generating function is observed. Further, a study regarding the usefulness of the Rényi divergence information generating function and Rényi information generating function as model selection criteria is conducted. Finally, three chaotic maps are considered and then used to establish a validation of the proposed information generating function.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
