Abstract The expectation–maximization (EM) algorithm is a seminal method to calculate the maximum likelihood estimators (MLEs) for incomplete data. However, one drawback of this algorithm is that the asymptotic variance–covariance matrix of the MLE is not automatically produced. Although there are several methods proposed to resolve this drawback, limitations exist for these methods. In this paper, we propose an innovative interpolation procedure to directly estimate the asymptotic variance–covariance matrix of the MLE obtained by the EM algorithm. Specifically we make use of the cubic spline interpolation to approximate the first-order and the second-order derivative functions in the Jacobian and Hessian matrices from the EM algorithm. It does not require iterative procedures as in other previously proposed numerical methods, so it is computationally efficient and direct. We derive the truncation error bounds of the functions theoretically and show that the truncation error diminishes to zero as the mesh size approaches zero. The optimal mesh size is derived as well by minimizing the global error. The accuracy and the complexity of the novel method is compared with those of the well-known SEM method. Two numerical examples and a real data are used to illustrate the accuracy and stability of this novel method.

中文翻译：

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EM算法中方差-协方差矩阵的有效和直接估计与插值方法

摘要 期望最大化 (EM) 算法是计算不完整数据的最大似然估计量 (MLE) 的开创性方法。然而，该算法的一个缺点是 MLE 的渐近方差-协方差矩阵不是自动生成的。尽管提出了几种方法来解决这个缺点，但这些方法存在局限性。在本文中，我们提出了一种创新的插值程序来直接估计由 EM 算法获得的 MLE 的渐近方差-协方差矩阵。具体来说，我们利用三次样条插值来逼近来自 EM 算法的 Jacobian 和 Hessian 矩阵中的一阶和二阶导数函数。它不需要像以前提出的其他数值方法那样的迭代程序，所以它在计算上是高效和直接的。我们从理论上推导出函数的截断误差界限，并表明随着网格尺寸接近零，截断误差减小到零。最佳网格尺寸也是通过最小化全局误差得出的。将新方法的准确性和复杂性与众所周知的 SEM 方法的准确性和复杂性进行了比较。两个数值例子和一个真实的数据被用来说明这种新方法的准确性和稳定性。