Raw mortality data often exhibit irregular patterns due to randomness. Graduation refers to the act of smoothing crude mortality rates. In this paper, we propose a flexible and robust methodology for graduating mortality rates using adaptive P-splines. Since the observed data at high ages are often sparse and unreliable, we use an exponentially increasing penalty. We use mortality data of England and Wales and model male and female mortality rates jointly by means of penalties, achieving borrowing of information between the two sexes.

中文翻译：

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使用自适应 P 样条对男性和女性死亡率进行联合建模

由于随机性，原始死亡率数据通常表现出不规则的模式。毕业是指平滑粗死亡率的行为。在本文中，我们提出了一种灵活且稳健的方法，用于使用自适应 P 样条计算毕业死亡率。由于在高年龄时观察到的数据通常稀疏且不可靠，因此我们使用指数增加的惩罚。我们使用英格兰和威尔士的死亡率数据，通过惩罚的方式对男性和女性的死亡率进行联合建模，实现两性之间的信息借用。