Data from: Fast robust SUR with economical and actuarial applications

The seemingly unrelated regression (SUR) model is a generalization of a linear regression model consisting of more than one equation, where the error terms of these equations are contemporaneously correlated. The standard Feasible Generalized Linear Squares (FGLS) estimator is efficient as it takes into account the covariance structure of the errors, but it is also very sensitive to outliers. The robust SUR estimator of Bilodeau and Duchesne (Canadian Journal of Statistics, 28:277-288, 2000) can accommodate outliers, but it is hard to compute. First we propose a fast algorithm, FastSUR, for its computation and show its good performance in a simulation study. We then provide diagnostics for outlier detection and illustrate them on a real data set from economics. Next we apply our FastSUR algorithm in the framework of stochastic loss reserving for general insurance. We focus on the General Multivariate Chain Ladder (GMCL) model that employs SUR to estimate its parameters. Consequently, this multivariate stochastic reserving method takes into account the contemporaneous correlations among run-off triangles and allows structural connections between these triangles. We plug in our FastSUR algorithm into the GMCL model to obtain a robust version.看似不相关回归（seemingly unrelated regression, SUR）模型是由多个方程构成的线性回归模型的推广形式，此类模型中各方程的误差项存在同期相关性。标准可行广义最小二乘（Feasible Generalized Linear Squares, FGLS）估计量因考虑了误差项的协方差结构而具备估计有效性，但对异常值极具敏感性。Bilodeau与Duchesne于2000年发表于《加拿大统计学期刊》（28卷：277-288页）的稳健SUR估计量可适配异常值场景，但其计算复杂度较高。本文首先提出一种用于求解该估计量的快速算法FastSUR，并通过模拟仿真实验验证了其优异性能；随后给出异常值检测诊断方法，并结合一则经济学领域的真实数据集展开示例演示。接下来，我们将FastSUR算法应用于一般保险的随机损失准备金框架中，重点关注采用SUR进行参数估计的广义多元链梯（General Multivariate Chain Ladder, GMCL）模型。故而，该多元随机准备金方法能够考量赔付流量三角形间的同期相关性，同时允许各流量三角形间存在结构关联。最后，我们将FastSUR算法嵌入GMCL模型，得到了其稳健改进版本。