标题： 基于对数范数相对熵的广义似然估计 显示英文标题

标题： On Generalized Likelihood Estimation Based on the Logarithmic Norm Relative Entropy

摘要： 基于传统似然的方法在给定数据中存在少量异常值时会受到严重影响。 在本文中，我们考虑一种稳健的参数估计方法，即最小对数范数相对熵（LNRE）估计过程，并研究与之相关的不同（广义）充分性原理。 我们引入了一个新的两参数幂律分布族（即，$\mathcal{M}^{(\alpha,\beta)}$-族），该分布族被证明相对于与LNRE相关的广义似然函数具有固定数量的充分统计量，与样本大小无关。 然后，我们得到了该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。 我们还特别详细地研究了学生氏分布族的最小LNRE估计量（MLNREEs）。 在特定选择调整参数$\alpha$和$\beta$的情况下，我们的普遍结果退化为指数分布族下的经典似然结果。 最后，我们进行了模拟研究，并随后进行了一项实际数据分析，这突显了MLNREEs在可能含有异常值的数据中的实际效用。 在此过程中，我们还纠正了一篇相关理论的近期论文中发现的一个错误。 显示英文摘要

摘要： 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.