Traditional likelihood based methods for parameter estimation get highly affected when the given data is contaminated by outliers even in a small proportion. In this paper, we consider a robust parameter estimation method, namely the minimum logarithmic norm relative entropy (LNRE) estimation procedure, and study different (generalized) sufficiency principles associated with it. We introduce a new two-parameter power-law family of distributions (namely, $\mathcal{M}^{(\alpha,\beta)}$-family), which is shown to have a fixed number of sufficient statistics, independent of the sample size, with respect to the generalized likelihood function associated with the LNRE. Then, we obtain the generalized minimal sufficient statistic for this family and derive the generalized Rao-Blackwell theorem and the generalized Cram\'{e}r-Rao lower bound for the minimum LNRE estimation. We also study the minimum LNRE estimators (MLNREEs) for the family of Student's distributions particularly in detail. Our general results reduces to the classical likelihood based results under the exponential family of distributions at specific choices of the tuning parameter $\alpha$ and $\beta$. Finally, we present simulation studies followed by a real data analysis, which highlight the practical utility of the MLNREEs for data contaminated by possible outliers. Along the way we also correct a mistake found in a recent paper on related theory of generalized likelihoods.

翻译：传统基于似然的参数估计方法在数据受到异常值污染时，即使污染比例很小，其性能也会受到显著影响。本文研究一种稳健的参数估计方法——最小对数范数相对熵（LNRE）估计程序，并探讨与之相关的多种（广义）充分性原则。我们引入一个新的双参数幂律分布族（即 $\mathcal{M}^{(\alpha,\beta)}$ 族），该分布族被证明在 LNRE 相关的广义似然函数下具有固定数量的充分统计量，且与样本量无关。随后，我们获得了该分布族的广义最小充分统计量，并推导了最小 LNRE 估计的广义 Rao-Blackwell 定理和广义 Cramér-Rao 下界。我们还特别针对 Student 分布族详细研究了最小 LNRE 估计量（MLNREE）。在特定调节参数 $\alpha$ 和 $\beta$ 的取值下，我们的广义结果可退化为指数分布族下的经典似然估计结果。最后，我们通过模拟研究和实际数据分析，展示了 MLNREE 在可能受异常值污染数据中的实际应用价值。此外，我们还修正了近期一篇关于广义似然相关理论的论文中存在的错误。