The well-known Lee-Carter model uses a bilinear form $\log(m_{x,t})=a_x+b_xk_t$ to represent the log mortality rate and has been widely researched and developed over the past thirty years. However, there has been little attention being paid to the robustness of the parameters against outliers, especially when estimating $b_x$. In response, we propose a robust estimation method for a wide family of Lee-Carter-type models, treating the problem as a Probabilistic Principal Component Analysis (PPCA) with multivariate $t$-distributions. An efficient Expectation-Maximization (EM) algorithm is also derived for implementation. The benefits of the method are threefold: 1) it produces more robust estimates of both $b_x$ and $k_t$, 2) it can be naturally extended to a large family of Lee-Carter type models, including those for modelling multiple populations, and 3) it can be integrated with other existing time series models for $k_t$. Using numerical studies based on United States mortality data from the Human Mortality Database, we show the proposed model performs more robust compared to conventional methods in the presence of outliers.

翻译：著名的Lee-Carter模型采用双线性形式$\log(m_{x,t})=a_x+b_xk_t$表示对数死亡率，并在过去三十年间得到广泛研究和拓展。然而，参数对异常值的稳健性问题——特别是$b_x$的估计——鲜少受到关注。为此，我们提出一种适用于Lee-Carter型模型的稳健估计方法，将问题转化为基于多元t分布的概率主成分分析（PPCA）。同时推导了高效的期望最大化（EM）算法实现。该方法具有三重优势：1）能同时更稳健地估计$b_x$和$k_t$；2）可自然扩展至包括多群体建模在内的大规模Lee-Carter型模型族；3）能与$k_t$的现有时间序列模型进行集成。基于人类死亡率数据库中美国死亡率数据的数值研究表明，所提方法在异常值存在时比传统方法具有更强的稳健性。