标题： 一种广义的完备性概念及其应用 显示英文标题

标题： A Generalized Notion of Completeness and Its Application

摘要： 从数据约减的角度来看，最小充分统计量和完全统计量的概念共同在确定最优统计量（估计量）中起着重要作用。 经典充分性和完全性的概念在基于不同散度的鲁棒估计中并不充分。 最近，文献中引入了基于广义似然函数的广义充分性概念。 需要注意的是，仅凭充分性的概念不一定能产生最优统计量（估计量）。 因此，与广义充分性相一致，我们引入了相对于广义似然函数的广义完全性概念。 我们随后描述了具有相对于与密度幂散度（DPD）相关的广义似然函数的完全性的概率分布族。 此外，我们证明了与对数密度幂散度（LDPD）相关的分布族是不完全的。 进一步地，我们将Lehmann-Scheffé定理和Basu定理扩展到广义似然估计中。 随后，我们得到了$\mathcal{B^{(\alpha)}}$族的广义一致最小方差无偏估计量（UMVUE）。 进一步地，我们推导了一个渐近期望缺陷（AED）的公式，该公式用于比较最小密度幂散度估计量（MDPDE）和$\mathcal{B^{(\alpha)}}$族的广义UMVUE之间的性能。 最后，我们提供了所开发结果在应力-强度可靠性模型中的应用。 显示英文摘要

摘要： From the perspective of data reduction, the notions of minimal sufficient and complete statistics together play an important role in determining optimal statistics (estimators). The classical notion of sufficiency and completeness are not adequate in many robust estimations that are based on different divergences. Recently, the notion of generalized sufficiency based on a generalized likelihood function was introduced in the literature. It is important to note that the concept of sufficiency alone does not necessarily produce optimal statistics (estimators). Thus, in line with the generalized sufficiency, we introduce a generalized notion of completeness with respect to a generalized likelihood function. We then characterize the family of probability distributions that possesses completeness with respect to the generalized likelihood function associated with the density power divergence (DPD). Moreover, we show that the family of distributions associated with the logarithmic density power divergence (LDPD) is not complete. Further, we extend the Lehmann-Scheff\'e theorem and the Basu's theorem for the generalized likelihood estimation. Subsequently, we obtain the generalized uniformly minimum variance unbiased estimator (UMVUE) for the $\mathcal{B^{(\alpha)}}$-family. Further, we derive an formula of the asymptotic expected deficiency (AED) that is used to compare the performance between the minimum density power divergence estimator (MDPDE) and the generalized UMVUE for $\mathcal{B^{(\alpha)}}$-family. Finally, we provide an application of the developed results in stress-strength reliability model.