Robust Parameter Estimation for the Lee-Carter Family: A Probabilistic Principal Component Approach

Abstract

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.