This paper presents some results on the maximum likelihood (ML) estimation from incomplete data. Finite sample properties of conditional observed information matrices are established. They possess positive definiteness and the same Loewner partial ordering as the expected information matrices do. An explicit form of the observed Fisher information (OFI) is derived for the calculation of standard errors of the ML estimates. It simplifies Louis (1982) general formula for the OFI matrix. To prevent from getting an incorrect inverse of the OFI matrix, which may be attributed by the lack of sparsity and large size of the matrix, a monotone convergent recursive equation for the inverse matrix is developed which in turn generalizes the algorithm of Hero and Fessler (1994) for the Cram\'er-Rao lower bound. To improve the estimation, in particular when applying repeated sampling to incomplete data, a robust M-estimator is introduced. A closed form sandwich estimator of covariance matrix is proposed to provide the standard errors of the M-estimator. By the resulting loss of information presented in finite-sample incomplete data, the sandwich estimator produces smaller standard errors for the M-estimator than the ML estimates. In the case of complete information or absence of re-sampling, the M-estimator coincides with the ML estimates. Application to parameter estimation of a regime switching conditional Markov jump process is discussed to verify the results. The simulation study confirms the accuracy and asymptotic properties of the M-estimator.
翻译:本文从不完整的数据中介绍了最大可能性(ML)估算的一些结果; 确定了有条件观察到的信息矩阵的精度样本属性; 具有肯定性, 与预期的信息矩阵一样, Loewner 部分排序也与预期的信息矩阵相同。 为计算ML估计数的标准误差,将观察到的Fisher信息的明显形式(OFI)用于计算标准误差; 简化了OFI矩阵的Louis(1982年)通用公式; 为防止获得不正确的OFI矩阵的反差, 这可能是由于缺乏松散和矩阵大尺寸造成的, 开发了一个单调的反向矩阵复现公式, 反过来又将Hero和Fessler(1994年)的算法普遍化为Cram\'er-Rao较低约束的偏差值。 为了改进估算, 特别是在对不完整数据进行反复抽样时, 引入了坚固的 M 估测数据。 提议对调控矩阵的封闭表格三明治估测算仪提供了M- 测算仪的标准误差。 由此导致在缩缩缩不全数据中提供的信息丢失, 将Sandbsimestimestimestal- Lestestimestermatistration 校验算结果作为标准的标定结果。