A Generalized Notion of Completeness and Its Application
Himanshi Singh, Tanmay Sahoo, Nil Kamal Hazra
TL;DR
This paper develops a generalized notion of completeness tied to generalized likelihoods induced by divergence measures ($DPD$ and $LDPD$), enabling robust estimation beyond the classical ML framework. It extends fundamental results—Lehmann–Scheffé and Basu's theorems—to this generalized setting and derives a generalized UMVUE for the $\mathcal{B}^{(\alpha)}$-family, including the identification of a generalized complete sufficient statistic. The authors show that the analogous $\mathcal{M}^{(\alpha)}$-family may lack completeness, underscoring limits of generalized sufficiency alone, and provide explicit AED-based comparisons between the MDPDE and generalized UMVUE, with an application to stress–strength reliability. Collectively, the work links divergence-based estimation, generalized sufficiency/completeness, and practical reliability modeling to offer robust estimators and a framework for estimator comparison using AED in generalized settings.
Abstract
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-Scheffé theorem and the Basu's theorem for the generalized likelihood estimation. Subsequently, we obtain the generalized uniformly minimum variance unbiased estimator (UMVUE) for the $\mathcal{B^{(α)}}$-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 $\mathcal{B^{(α)}}$-family. Finally, we provide an application of the developed results in stress-strength reliability model.