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.

翻译：从数据约简的角度来看，最小充分统计量与完备统计量的概念共同在确定最优统计量（估计量）方面起着重要作用。经典的充分性与完备性概念在许多基于不同散度的稳健估计中并不适用。最近，文献中引入了基于广义似然函数的广义充分性概念。需要特别指出的是，仅充分性概念本身未必能产生最优统计量（估计量）。因此，与广义充分性相对应，我们引入了关于广义似然函数的广义完备性概念。随后，我们刻画了关于密度幂散度（DPD）相关联的广义似然函数具有完备性的概率分布族。此外，我们证明了与对数密度幂散度（LDPD）相关联的分布族不具备完备性。进一步地，我们将Lehmann-Scheffé定理和Basu定理推广至广义似然估计。基于此，我们得到了$\mathcal{B^{(\alpha)}}$族广义一致最小方差无偏估计量（UMVUE）。此外，我们推导了渐近期望亏损（AED）的公式，用于比较$\mathcal{B^{(\alpha)}}$族的最小密度幂散度估计量（MDPDE）与广义UMVUE之间的性能。最后，我们将所发展的结果应用于应力-强度可靠性模型。