Research Projects

The likelihood function plays an important role in statistical decision theory. The key concepts (namely, sufficiency, completeness, UMP test, etc.) used in statistical inference are fundamentally based on the usual likelihood function. One of the inherent limitations of the usual likelihood-based approaches is their high sensitivity to model misspecification and the presence of outliers. Thus, it is necessary to consider some generalized likelihood-based methods that do not suffer from this drawback. 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 concept of usual sufficiency is defined based on the usual likelihood function, and the underlying estimation is the maximum likelihood estimation. In many robust estimation methods based on different divergences, the classical notion of sufficiency is not adequate. Recently, the notion of generalized sufficiency based on a generalized likelihood function was introduced in the lierature. It is important to note that the concept of sufficiency alone does not necessarily produce optimal statistics (estimators). This emphasizes the necessity of a generalized definition of completeness that is compatible with the notion of generalized sufficiency. Thus, we focus to study a generalized notion of completeness with respect to generalized likelihood functions. We then characterize families of probability distributions that possess completeness in the context of the Basu et al. and Jones et al. likelihood functions. Further, we extend the Lehmann-Scheffe theorem for both Basu et al. and Jones et al. estimations and find UMVUE for the power-law families. Next, we aim to study a robust estimation method based on the norm relative entropy (a reformulation of S-divergence) and study its associated inference problems. In hypothesis testing, one of the primary goals is to find the most powerful test to detect deviations from the null hypothesis. The Neyman-Pearson lemma provides a key theoretical foundation for developing the most powerful tests. Note that UMP, UMPU, and UMPB tests for the exponential family of distributions are well established. However, when the null and alternative hypotheses are formed for some generalized power-law families such as B(α) and M(α)-families, current approaches frequently fail to produce the most powerful tests. Thus, we focus to introduce the notions of UMP, UMPU and UMPB tests based on a generalizedlikelihood function, and study the existence of these tests for power-law families. Robustness is also an important aspect in classical Bayesian estimation because the posterior distribution is constructed using the usual likelihood function. Divergence-based methods have been used as an alternative for obtaining robust Bayesian estimators. A crucial feature of such approaches is that the posterior distribution is often generated on the basis of a likelihood function corresponding to a specified divergence. This understanding motivates the development of a unified Bayesian method based on generalized likelihood functions, offering enhanced flexibility in model specification. In particular, we aim to study Bayesian inference based on norm relative entropy (NRE) and logarithmic norm relative entropy (LNRE).