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

As a traditional and widely-adopted mortality rate projection technique, by representing the log mortality rate as a simple bilinear form $\log(m_{x,t})=a_x+b_xk_t$. The Lee-Carter model has been extensively studied throughout the past 30 years, however, the performance of the model in the presence of outliers has been paid little attention, particularly for the parameter estimation of $b_x$. In this paper, we propose a robust estimation method for Lee-Carter model by formulating it as a probabilistic principal component analysis (PPCA) with multivariate $t$-distributions, and an efficient expectation-maximization (EM) algorithm for implementation. The advantages of the method are threefold. It yields significantly more robust estimates of both $b_x$ and $k_t$, preserves the fundamental interpretation for $b_x$ as the first principal component as in the traditional approach and is flexible to be integrated into other existing time series models for $k_t$. The parameter uncertainties are examined by adopting a standard residual bootstrap. A simulation study based on Human Mortality Database shows superior performance of the proposed model compared to other conventional approaches.